On Active Surge Control of Compression Systems via Characteristic Linearization and Model Nonlinearity Cancellation

Transcription

1 34 J. Eng. Tehnol. Si., Vol. 46, No. 3, 4, On Ative Surge Control of Compreion Systems via Charateristi Linearization and Model Nonlinearity Canellation Yohannes S.M. Simamora,3, Harijono A. Tjokronegoro, & Edi Leksono Instrumentation and Control Graduate Program, Institut Teknologi Bandung Engineering Physis Researh Group, Institut Teknologi Bandung Jalan Ganesha, Bandung 43, Indonesia 3 Mehanial Engineering Study Program, Politeknik Purbaya Jalan Panakarya, Talang, Kabupaten Tegal 593, Indonesia simamora@me.purbaya.a.id Abstrat. A simple approah of ative surge ontrol to ompreion systems is presented. Speifially, nonlinear omponents of the preure ratio and rotating speed states of the Moore-Greitzer model are transferred into the input vetors. Subsequently, the ompreor harateristi is linearized into two modes, whih desribe the stable region and the unstable region respetively. As a result, the system s state and input matries both appear linear, to whih linear realization and analysis are appliable. A linear quadrati regulator plus integrator is then hosen as losed-loop ontroller. By simulation it was shown that the modified model and harateristis an desribe surge behavior, while the losed-loop ontroller an stabilize the system in the unstable operating region. The lastmentioned was ahieved when maflow was 5.38 per ent le than the surge point. Keywords: ative surge ontrol; ompreion systems; linear quadrati regulator; nonlinear model; nonlinearity anellation. Introdution The harateristi and operating area of a ompreor an be desribed by a ompreor map as shown in Figure. The harateristi is a funtion of nondimensional preure ratio ( ) against nondimensional ma flow ( ), as shown in Figure. A surge line divides the ompreor operating area into a stable region and a surge/unstable region. When ompreion takes plae, maflow is redued so that it moves toward the unstable region. If ma flow is left to pa through the surge line, the ompreor will enter its unstable operating region and undergoes surge. Surge is a ompreor instability phenomenon when preure ratio and maflow flutuate with a large amplitude. The flutuations result in a limit yle in the ompreor map. Sine it redues effiieny and may even damage Reeived September 9 th,, Revised February 7 th, 4, Aepted for publiation May th, 4. Copyright 4 Published by ITB Journal Publisher, ISSN: , DOI:.564/j.eng.tehnol.si

2 On Ative Surge Control of Compreion Systems 343 the ompreor, the presene of surge is highly undesired in any ompreor operation. The traditional and most-used approah in dealing with surge is surge avoidane. While its implementation strategies may vary, the basi idea of surge avoidane is simply to prevent the ompreor from entering the unstable operating region. This an be ahieved by defining a surge margin lose to the surge line in the stable region, as shown in Figure. However, this approah has an inherent drawbak, i.e. the limitation of the ompreor operating point. Moreover, the ompreor annot reah its maximum effiieny sine it lies at the surge line. Ative surge ontrol takes a very different approah from surge avoidane. This approah is intended to deal with surge instead of avoiding it. Therefore, the ompreor an perform stably even in its unstable operating region. This means that the stable operating area of the ompreor is extended and it an now reah its maximum effiieny. Figure Compreor Map. Sine being introdued by Epstein, et al. [], various aspets of ative surge ontrol has been developed. An overview about the development in model, sensor and atuator an be found in [,3]. Several approahes to approximate

3 344 Yohannes S.M. Simamora, et al. ompreor harateristis are tested and disued in [4]. In the area of ontrol strategies, several methods have been applied, namely proportional-integral ontrol [5], globally stabilizing swithing ontrollers [6], bakstepping [7,8], and linear quadrati regulation (LQR) with integral ation [9]. It is worth mentioning that the ontrol strategies in [5-] are applied to a entrifugal ompreor model with spool dynamis. In this study, an equivalent linear form of state and input matries was obtained by applying nonlinearity anellation [] to the model and linearizing the ompreor harateristi. As a framework, the Moore-Greitzer ompreor model with spool dynamis, developed in [5,6], was used. Throttle and torque were hosen as atuators as in [6]. Any lo ourring in the atuators, suh as frition lo, was not taken into onsideration. Finally, an LQR plus integrator was hosen as the ontroller. Model. Model and Compreor Charateristi The ompreion system under study onsists of entrifugal ompreor, dut, plenum and throttle, as shown in Figure. This sheme is related to the Moore- Greitzer model with spool dynamis, whih was developed by Gravdahl [5] and nondimensionalized by Leonea, et al. [6] as: b a, b a th where ψ is nondimensional preure ratio, non-dimensional maflow, ω ψ,ω is ompreor harateristi, nondimensional rotational speed, γ th ψ and τ are throttle and torque inputs, respetively, while (. ) refers to derivation of a state against nondimensional time ξ. The parameters a, b and represent ompreor physial quantities, whih are defined as [6]: () 3 L a = V p ()

4 On Ative Surge Control of Compreion Systems 345 and: b = L, (3) A 3 Lr = a p J, (4) where L is dut length, V p plenum volume, A induer (impeller eye) area, r radius of impeller tip, p preure at inlet, a soni veloity at inlet, and J moment of inertia. Figure Shemati diagram of ompreion system []. The ompreor harateristi is the ratio between the preures at the ompreor s downstream and upstream flow (inlet), denoted in Figure as p and p, respetively. Several approahes to approximate ompreor harateristis are disued in [4]. In this paper, a physial-based ompreor harateristi is used. This harateristi was derived by Gravdahl [5], experimentally validated by Gravdahl, et al. [], and has been used in [6]. In nondimensional form, it is defined as [6]: d, whereη,ω,, (5) is ompreor isentropi effiieny, and σ a slip fator between induer and impeller tip. Parameters κ and d are given by [5-6]: p κ =, (6) v

5 346 Yohannes S.M. Simamora, et al. a d = T p, (7) where p and v are speifi heat at onstant preure and volume respetively, while T is inlet temperature. Isentropi effiieny is defined as [5-6]: Δh,id ea l η,ω =, (8) Δh +Δh,id ea l lo where Δh is speifi enthalpy delivered to the fluid without taking, ideal aount of the loes [5]. In nondimensional form, it is defined as [6]: Δh ideal =σω On the other hand, defined as [5]: where,. (9) lo Δh is the sum of loes in the ompreion system that is lo Δh =Δh +Δh +Δh + Δh, () Δh ii and respetively, while ii di if df Δh di are inidene loes at the induer and diffuser Δh fi and Δh fd are frition loes at the induer and diffuser respetively. In nondimensional form, these loes are defined as [6]: and: Δh ii = fω f, () Δh di = σω f3, () Δh if = f 4, (3) Δh df = f 5, (4) where f i,i =...5 are nondimensional parameters. To find the details of the dimensional parameters represented by f i, the reader may onsult [5]. In ompreion system operation, the ompreor harateristi as defined in (5) applies for positive maflow. For negative maflow, the ompreor harateristi is defined as [6]:

6 On Ative Surge Control of Compreion Systems 347 ψ,ω = μ +,, (5) ψ where and [6]: and [6] : κ κ ψ,ω = + η σdω, (6) η = η,ω =. Surge Line σ =. (7) σ + σ + f The surge line is defined as the line that paes through the maxima of the ompreor harateristi [5]. Hene, from ψ /,ω =, it is found that the surge line for any speed line as desribed in Figure will pa through the extreme point: ψ = k ω ψ, ω sl sl f = sl where index sl refers to the surge line and [6]: f f +σf. (8) k f is a onstant that is defined as 3 k f =. (9) f + f3 + f4 + f5 As implied by (8), the ompreor s maximum preure ratio exists right at the surge point. The ompreor s maximum isentropi effiieny is also ahieved at this point [6]. However, a small redution in maflow from this point will lead the ompreor into surge..3 Model and Compreor Charateristi Modifiations Both the ompreor model () and itsharateristi (5) are highly nonlinear. In order to apply linear ontrol theory to (), the following approahes are used: (i) Canellation of the nonlinear omponents [], whih exist at statesψ and ω in (), by transferring them into the input vetors. (ii) Linearization of the ompreor harateristi (5) around the surge point in the stable region and in the unstable region as well. By approah (i), states ψ and ω in () appear linear, that is:

7 348 Yohannes S.M. Simamora, et al. a a, v b b, () v where v and v are newly defined ontrol inputs given by: and: v = γth ψ, () v =τσω. () Approah (ii) is applied based on the fat that surge point (8) is the point where the system is in steady-state ondition. In this ondition, the following relations apply [6]: ω = ψ v v ψ, (3) where index refers to the steady-state ondition. Inspired by Leonea, et al. [6], linear ompreor harateristis are derived by defining a small operating range of eah region relative to the steady-state point. For the unstable region, suh range is defined by: and: x =ψ ψ, (4) x =, (5) x = ω ω 3, (6) vˆ = v v, (7) vˆ = v v. (8) Auming the operating range defined by (4)-(8) is linear, the ompreor harateristi an be defined as: ψ x x =ψ,ω ψ,ω. (9), 3 Furthermore, linearization (5) near and ω using Taylor expansion results in:

8 On Ative Surge Control of Compreion Systems 349 ψ,ω ψ,ω + ω ω The value of ψ, ω ψ ψ,ω ω,ω is obtained by substituting. (3) = and ω = ω ω = ω into (5). On the other hand, substituting = and into: ψ,ω κ = + η κ and: σω σ ω + 4 f + f f f ω f f + f σ f ω f + σω f ψ,ω κ = + η ω κ σω +,ω σdω 3,ω σdω σ dω f ω f + σω f 3 4 κ k ω k f ω + k results in ψ /, ω and ψ / ω, ω f 5 + f + f 5, (3), (3), respetively. Substituting (3) into (9) results in the ompreor harateristi at the unstable (surge) region, that is: where: and:,ω=β +γω ψ, (33) ψ β =,ω γ =,ω, (34) ψ ω. (35)

9 35 Yohannes S.M. Simamora, et al. Remark. Aording to [5], surge ours at points with positive slope in the ompreor harateristi urve. However, alulation of (3) uses <, thus x.therefore ψ, ω / in (3) must be negative, so that (33) has a positive slope. Substituting (4)-(8), differentiation of (4)-(6) and (33) into () results in a linear ompreor model for the surge region: a a ˆ v b αβb αγb ˆ v. (36) Parameter α > in (36) is a oeffiient that guarantees ψ = ψ in (3) will be ahieved, where: ψ α = β + γω. (37) For the stable region, a small operating range relative to the steady-state point is defined by: and: x r = ψ ψ, (38) x x r =, (39) 3r = ω ω, (4) ˆ, (4) vr = v v ˆ. (4) vr = v v The index r in (38)-(4) refers to the stable region. Auming the range defined by (38)-(4) is linear, the ompreor harateristi for the unstable region an be defined as [6]: ψ x,x =ψ +x,ω +x ψ, ω. (43) r 3r r 3r Substituting (38)-(4) into (3) and subsequently substituting its result into (43) results in: ψ,ω =β +γ. (44) ω

10 On Ative Surge Control of Compreion Systems 35 Remark. The differene between (33) and (44) lies in their respetive operating regions. In (44), >, thus x r >. Sine ψ, ω/ is negative, the slope in (44) will be also negative. The linear ompreor model for the stable region is obtained by substituting (38)-(4), differentiation of (38)-(4) and (44) into (), that is: a a b αr βb αrγb vˆ vˆ r r. (45) Parameter α r > in (45) is a oeffiient that guarantees thatψ = ψ in (3) is ahieved, where: ψ α r = β+ γω. (46).4 Defining Swithing Funtion for Compreor Models By observation on (36) and (45), it was shown that the general form of these two models is similar. Hene, it is of interest to find a swithing funtion that determines when the system is in stable mode or in unstable mode. The harateristi equation of the state matrix in (36) or (45): is given by: a A b αβb αγb, (47) +λ +αβbλ+ab = λ. (48) By inspetion of (47) it is found that open-loop stability depends solely on the sign of A,, both in (36) and (45). This is beause α (or α r ), a, b and are all onstant and positive, while β is negative. If A, is positive, A will be stable, sine its eigen values are all negative. Conversely, if A, is negative, A is unstable, sine at least one of its eigen values is positive. These fats are in aordane with the definition of negative and positive slope in (33) and (44).

11 35 Yohannes S.M. Simamora, et al. Hene, a swithing funtion is adequate to define stable and unstable region mode. A swithing funtion is then inserted to A,, suh that: where: A f αβ b, (49) f s, = s if < =. (5) else Using (5), (37) and (46) an now be stated into a single funtion, whih is: α =. (5) f s β ψ + γω 3 Controller Design Based on (3), (5) and (5), the ontrol strategy is formulated as follows: (i) open-loop ontrol is employed from initial ondition to surge line (ii) losed-loop ontrol is employed from surge line to unstable operating point 3. Filtering the Control Ation In losed-loop ontrol, the nonlinear omponent that originally exists in the third-state of () will re-appear. However, as indiated in (), its presene will produe a large additional load for the ontroller to handle. Suh a situation will be disadvantageous for system performane, espeially when the ontrol load hanges abruptly. To antiipate suh problems, a linear low-pa filter with the following impulse response: f lp f = K t / T lp fe, (5) is introdued to filter v (). The filter will keep hanges inv within a small range. Gain Klpf is set suh that the maximum value produed by nonlinearity of v an be moderated, while onstant T is set to expedite diipation of the nonlinearity as time t goes to infinity. Hene, the losed-loop ontrol inputs in the unstable region are obtained by inorporating () and (7):

12 On Ative Surge Control of Compreion Systems 353 γ th v = v ˆ ψ, (53) and by inorporating (), (8) and (5): τ =v vˆ + flpf σω. (54) 3. LQR plus Integrators Design From the general form of state spae equations: x = Ax+ Bv ˆ+ Gw, (55) y = Cx + Dvˆ, (56) with all the outputs aumed to be measurable. In order to inorporate error, an augmented state is introdued [3]: where e and x Ax Bv =, (57) e y r ef Cx y ref are the output error and set point vetors, respetively. In the form of (55), (57) an be written as: x Ax Bv, (58) where: y ref A A =, (59) T T ref = y ref B = B, (6) y. (6) Furthermore, the standard quadrati performane index form of (58) is written as [4]: T T x Qx ˆ v Rv ˆ J, (6) where Q and R are positive semi-definite and positive definite matries, respetively. Control signal v ˆ is given by:

13 354 Yohannes S.M. Simamora, et al. and: ˆ v Kx, (63) K= R B T P, (64) where P is a solution of the following Riati equation [3]: T T P A+ A P PBR B P+ Q=, (65) with: T C Q yc Q =. (66) Qe Qe in (66) is the weighting matrix of e. Substituting the result of (65) in (64), the ontrol gains are obtained: K = K sfb K, (67) e where K sfb and K int are state feedbak ontrol gain and integrator gain, respetively. Reall that A and B are 6 6 and 6 matries, respetively, then K is a 6 matrix. In this ase, whereas K e exists the 4 th therefore defined as: sfb xkee+ Ksfb exist at the st -3 rd olumn of K, -6 th olumn. The losed-loop ontrol signal is vˆ = K K x, (68) where K is referene gain. ref ref ref 4 Simulation The numerial values used in this simulation were taken from [6]. The Riati equation in (65) was solved using the built-in Riati funtion of Silab It was hosen that R =.5 and Q = diag(5,.5,,.5,.5 ). It was obtained that: and: K sfb =, (69)

14 On Ative Surge Control of Compreion Systems K e =. (7) For the linear low-pa filter defined in (5), the numerial values K lpf. 5 andt 5 were hosen. The simulation was arried out using Silab 5.3. and the 4 th Order Runge-Kutta was used as the numerial method. From initial ondition ψ =.88, =.4, ω =.397, the ompreor was brought to steady-state ondition ψ =.35, =.86, ω =.493. Using the dimensional values in [5], the steady-state nondimensional values represent ambient preure p =.3 5 Pa, maflow m =.9 kg/s and rotating speed = 5 rpm, respetively. The ompreor harateristi for the stable region in suh point is given by: r,, (7) while the surge region is defined by: ,. (7) At nondimensional time ξ =.75, maflow was redued to =.76, whih is 5.38 perent le than the allowed minimum value for ω =.493. This means the ompreor enters its unstable operating region. Several indiators of the ompreor s behavior are represented in Figures 3-7. System response without the losed-loop ontrol is represented by a dotted line, while its ounterpart is represented by a solid line. As shown in Figures 3-4, the modified model and harateristi an desribe the surge behavior. On the other hand, the losed-loop ontrol managed to keep the ompreor stable when being operated in the unstable region. It should be noted that maflow is atually diffiult to measure [5]. Hene, inorporating a maflow observer as proposed in [5] to the model ould give more realisti results.

15 356 Yohannes S.M. Simamora, et al. Figure 3 Preure ratio against nondimensional time. Figure 4 Maflow against nondimensional time.

16 On Ative Surge Control of Compreion Systems 357 Figure 5 Rotating speed against nondimensional time. Figure 6 Throttle opening against nondimensional time.

17 358 Yohannes S.M. Simamora, et al. Figure 7 Torque against nondimensional time. 5 Conlusion An ative ontrol of entrifugal ompreors via model nonlinear omponent anellation and harateristis linearization has been presented. By simulation it was shown that the modified model and harateristi an desribe the surge behavior. The losed-loop ontroller managed to keep the ompreor stable when operated in the unstable region. Inherently, linear realization of the ative surge of ompreion systems has limitations within the range of the unstable operating region as to what it an ope with. However, the simpler design and analysis used an be seen as a trade-off for its nonlinear ounterpart. On the other hand, its ahievement in dealing with surge is still a departure from the surge avoidane approah. Referenes [] Epstein, A.H., Ffows Williams, J. & Greitzer, E.M., Ative Suppreion of Aerodynami Instabilities in Turbomahines, Journal of Propulsion and Power, 5(), pp. 4-, 989. [] Gravdahl, J.T. & Egeland, O., A Moore-Greitzer Axial Compreor Model with Spool Dynamis, Pro. of the 36th IEEE Conferene on Deision and Control, San Diego, CA, 997.

18 On Ative Surge Control of Compreion Systems 359 [3] Willems, F., Modeling and Bounded Feedbak Stabilization of Centrifugal Compreor Surge, PhD Thesis, Tehnishe Universiteit Eindhoven, Eindhoven, Netherlands,. [4] Grong, T.S., Modeling of Compreor Charateristis and Ative Surge Control, MS Thesis, Norwegian University of Siene and Tehnology, Trondheim, Norway, 9. [5] Gravdahl, J.T., Modeling and Control in Centrifugal Compreor, PhD thesis, Norwegian University of Siene and Tehnology, Trondheim, Norway, 998. [6] Leonea, A., Haddad, W.M. & Li, H., Globally Stabilizing Swithing Controllers for A Centrifugal Compreor Model with Spool Dynamis, IEEE Trans. Control Systems Tehnology, 8(3), pp ,. [7] Bøhagen, B. & Gravdahl, J.T., Ative Surge Control of Compreion System Using Drive Torque: A Bakstepping Approah, In: Pro. of the 44th IEEE Conferene on Deision and Control, Seville, Spain, 5. [8] Bøhagen, B. & Gravdahl, J.T., Ative Surge Control of Compreion System Using Drive Torque, Automatia, 44(4), pp.35-4,8. [9] Uddin, N. & Gravdahl, J.T., Piston-Atuated Ative Surge Control of Centrifugal Compreor Inluding Integral Ation, In: Pro. of the th International Conferene on Control, Automation and Systems, Gyeonggi-do, Korea,. [] Gravdahl, J.T., Egeland, O. & Vatland, S.O., Drive Torque Atuation in Ative Surge Control of Centrifugal Compreors, Automatia, 38 (), pp ,. [] Khalil, H., Nonlinear Systems, Third Edition, Prentie-Hall, Upper- Saddle River, NJ,. [] Gravdahl, J.T., Willems, F., De Jager, B. & Egeland, O., Modeling for Surge Control of Centrifugal Compreors: Comparison with Experiment, In: Pro. of the 39th IEEE Conferene on Deision and Control, Sidney, Australia,. [3] Åstrom, K.J. & Murray, R.M., Feedbak Systems: An Introdution for Sientists and Engineers, Prineton University Pre, Prineton, NJ, 8. [4] Naidu, D.S., Optimal Control Systems, CRC Pre, Boa Raton, FL, 3. [5] Bøhagen, B. & Gravdahl, J.T., On Ative Surge Control of Compreors Using A Ma Flow Observer, In: Pro. of the 4st IEEE Conferene on Deision and Control, Las Vegas, NV,.