JOURNAL OF AUTOMATIC CONTROL, UNIVERSITY OF BELGRADE, VOL. 19:19-26, 2009©
Fuzzy Modeling and Control for Detection and Isolation of Surge in Industrial Centrifugal Compressors Ahmed Hafaifa, Ferhat Laaouad and Kouider Laroussi Abstract— This paper provides the possible application of the fuzzy approaches in fault detection and isolation area for a class of complex industrial processes with uncertain interval parameters. The main idea of fuzzy fault detection and isolation is to build a model of a diagnosis procedures, using rules-based Fuzzy Expert System, capable to minimize false alarms enhance detectability and isolability and minimize detection time by hardware implementation to improve reliability, safety and global efficiency. This paper illustrates an alternative implementation to the compression systems supervision task using the basic principles of model-based fault detection and isolation associated with the self-tuning of surge measurements with subsequent appropriate corrective actions. Using a combination of fuzzy modeling approach makes it possible to devise a fault-isolation scheme based on the given incidence matrix. Simulation results of a fault detection and isolation for a compression system are provided, which illustrate the relevance of the proposed FDI method. Index Terms — Compression system, Surge control, Fuzzy control, Fuzzy fault detection and isolation
I. INTRODUCTION
A
COMPRESOR
transfers kinetic energy from an aero-
mechanically-driven rotor to a steady flow of gas. The pressure of the gas is raised by converting the acceleration imparted by the rotating parts of the compressor via diffusion, in normal operation of a compressor; the flow is nominally steady and axisymmetric. The pressure rise is dependent on the speed of rotation, but the efficient range is limited. As the flow through the compressor is throttled from the design point to the stall limits, the steady axisymmetric flow pattern becomes unstable. This instability can take on one of two forms, either surge or rotating stalls depending upon the compressor speed. The performance of a compressor is plotted as pressure ratio
This manuscript was received originally on November 10 2009 and accepted in the final form in Decembaer 2009. This work was supported by the Algerian Ministry of Higher Education and Scientific Research with the Industrial Automation and Diagnosis Systems Laboratory, Faculty of Science and Technology, University of Djelfa 17000 DZ, Algeria. Ahmed Hafaifa is with the Industrial Automation and Diagnosis Systems Laboratory, Science and Technology Faculty, University of Djelfa 17000 DZ, Algeria. Fax: +213 2787 5055 Phone: +213 5 5523 3674, e-mail:
[email protected] Ferhat Laaouad is at the Applied Automation Laboratory of the department of Industrial Process Automation, Faculty of Hydrocarbons and Chemistry, University of Boumerdes 35000 DZ, Algeria, e-mail:
[email protected] Kouider Laroussi is at the Industrial Automation and Diagnosis Systems Laboratory, Faculty of Science and Technology, University of Djelfa 17000 DZ, Algeria, e-mail: kouider-laro@hotmail. com
DOI: 10.2298/JAC0901019H
versus mass flow for different rotational speeds. The plot is divided into two regions by the stall (or surge) line. This line defines the operation limits of the compressor. To the left of the stall line the flow is no longer steady. In this region, large oscillations of the mass flow rate may occur (surge) or severe self-induced circumferential flow distortions may rotate around the annulus (rotating stall), or a combination of both may appear [1]. The general topic of this work is the detection and the isolation of surge in centrifugal compression systems. In particular it deals with fuzzy modeling and control of the surge phenomenon. Surge is an unstable operating mode of a compression system that occurs at low mass flows. Surge not only limits compressor performance and efficiency but it can also damage the compressor and auxiliaries due to the large thermal and mechanical loads involved. Surge in centrifugal compressors cannot, in general, be avoided when a unit trip or a major upset occurs, but the energy of surge should be minimized. Surge is a dramatic collapse of flow within a centrifugal compressor witch results in reverse flows within the machine and attached piping and can cause damage to bearings and other components. During normal and slowly changing operations, surge can be avoided by recycling gas through the surge control valve to maintain a minimum flow [2]. However, when a trip or major upset occurs, flow rate drops and the primary means by which surge energy can be reduced is to lower the head (suction to discharge pressure difference) at which the compressor reaches the surge (minimum stable) flow condition. The head across a compressor during a trip or upset is dependent on the response of the entire system, including changing performance of the compressor, transient flows within the piping, control system responses, and capacity in opening rate of surge and other automatic valves, such as vent or blow down valves, and check valves. This paper describes tools and techniques that can and have been used to model transient flows and performance, mechanical and control responses, and time dependent head in compressor systems. The tools used by fuzzy logic include a method of finite time step programs that simulate control systems, valve actuators, and the opening (or closing) rate of valves with the resulting flows. The volumes and lengths of station piping, scrubbers, and coolers including temperature effects of are accounted. Fuzzy logic models also track the performance of centrifugal compressors at different speeds, account for the rotation inertia of compressor trains, and evaluate the thermo physical properties of gas streams. The presented approach is based on the use of the fuzzy model. As was introduced in [3], by applying a Takagi-
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HAFAIFA A ET AL. FUZZY MODELING AND CONTROL FOR DETECTION AND ISOLATION OF SURGE IN….
Sugeno-type fuzzy model with interval parameters, one is able to approximate the upper and lower boundaries of the domain of functions that result from an uncertain system. The fuzzy model is therefore intended for robust modeling purposes; on the other hand, studies show it can be used in fault detection as well in [4 - 7]. The novelty lies in defining of confidence bands over finite sets of input and output measurements in which the effects of unknown process inputs are already included. Moreover, it will be shown that by the data pre-processing, fuzzy model parameter-optimization problem will be significantly reduced. By calculating the normalized distance of the system output from the boundary model outputs, a numerical fault measure is obtained. In many applications the FDI (Fault Detection and Isolation) system is a crucial issue that has been theoretically and experimentally investigated with different types of approaches, as can be seen from the survey work; [4, 7 - 9]. It has been widely acknowledged that the FDI system can be split into two steps: generation of residuals, which are ideally close to zero under no fault conditions, minimally sensitive to noises and disturbances, and maximally sensitive to faults, and residual evaluation, namely design of decision rules based on these residuals. The main idea of the proposed approach is to use the fuzzy model in a fault detection and isolation system as a residual generator, and then combine the fuzzy model outputs for the purpose of fault isolation. Due to data preprocessing, the decision stage is able to handle robustly a wide range of the effects of system disturbances. The paper presents an application of the fuzzy model in fault detection and isolation for the compression system with uncertain interval parameters. II. COMPRESSION SYSTEM MODELLING A. Modeling. Compressions systems are used, for example, as part of a gas turbine for jet and marine propulsion or power generation, in superchargers and turbochargers for internal combustion engines, and in a wide variety of industrial processes [1, 10 – 14]. In this paper we focus on centrifugal compressors that are used in natural gas pipeline transportation. This type of centrifugal compressor can exhibit a variety of instabilities under different operating conditions. Moore and Greitzer [15, 16] developed a phenomenological model for rotating stall and surge. This pioneering work modeled the compression system with just three components: 1. The first component is the inlet duct that allows infinitesimally small disturbances at the duct entrance to grow until they reach an appreciable magnitude at the compressor face. 2. The second component is the compressor itself, modeled as an actuator disk, which raises the pressure ratio by doing work on the fluid. 3. The third component is the plenum chamber (or diffuser) downstream, which acts as a large reservoir and responds to fluctuations in mass flow with fluctuations in pressure behind the actuator disk. In this paper, we are considering a compression system consisting of a centrifugal compressor, valve, compressor
duct, plenum volume and a throttle (Fig. 1). The throttle can be regarded as a simplified model of a turbine [17]. The system is showed in Figs 1a and 1b.
Figure 1a: Compression system: 1. Suction Bearing and Seal Assembly / 2. Inlet Guide Vane / 3. Inlet Housing / 4. Stators / 5. Discharge Volute / 6. Discharge Bearing and Seal Assembly / 7. Impellers
Figure 1b: Compression system
JOURNAL OF AUTOMATIC CONTROL, UNIVERSITY OF BELGRADE, VOL 19, 2009. The model to be used for controller design is in the form: dp p dt
= p p =
(
k P01 m - kt ρ01 V p
Pp - P01
)
4 ( k -1 ) ∆ hideal k A1 dm P01 1 + ηi (m, N) - Pp = m = dt Lc C p T01 dN 1 ηt mtur C p , t ∆ Ttur - 2 r22 σ π N m =N = dt 2J π 2πN
(1) Where pp is the plenum pressure, K is a numerical constant, p01 is the ambient pressure, ρ01 is the inlet stagnation density, Vp is the plenum volume, m is the compressor mass flow, kt is a parameter proportional to throttle opening , A1 is the area of the impeller eye (used as reference area), Lc is the length of compressor and duct, ηi is the isentropic efficiency, B is the spool moment of inertia, ∆hideal is the total specific enthalpy delivered to fluid, cp is the specific heat capacity at constant pressure, cv is the specific heat capacity at constant volume, T01 is the inlet stagnation temperature and k is the ratio of specific heats k=cp/cv. The present work has analytically integrated the right hand side of equation (1), presented by three ordinary differential equations of Moore and Greitzer model gives rise to modeling the compression system, the first for the non-dimensional total-to-static pressure rise ∆p across the compression system, the second for the amplitude of mass flow rate fluctuations m and the third for the nondimensional, spool moment of inertia [15, 16]. The two first equations of (1) are equivalent to the model of [18]. Note that the Moore-Greitzer model [15, 16] does not attempt to explain what physical mechanism triggers these instabilities. Rather, it attempts to determine the favorable conditions under which the disturbances will grow, and what can be done to suppress the instabilities. Its simplicity, mathematical elegance, and generality have led to wide acceptance and use of this model by researchers in industry, government and academia [19]. It is also used in surge control research with the belief that rotating stall is a precursor to surge, and with the expectation that elimination of rotating stall will also eliminate the development of surge. The instabilities within compression systems can be studied using energy considerations. As shown by Gysling in [20], the rate of energy input by the compressor to the fluid (over and above the steady state input) may be written as: δE = ∫ δ (∆P)δφdA Annulus (2) If this integral is positive, energy is added to the fluid by the compressor, and the disturbances will grow in amplitude. In the performance map, the slope of the curve is negative to the right side of the peak. In this region, a small increase in mass flow rate δφ will decrease pressure, so that δ(∆p) is negative. The above integral is thus negative for the performance map that is to the right of the pressure peak. The side of the performance map to the left of the peak pressure point is, by similar arguments, unstable. Even though compression systems are designed to operate to the right of
21
the pressure peak making them inherently stable, small disturbances such as change in the compressor RPM rises in back pressure or inlet stall, can push the operating point momentarily to the left of the pressure peak, triggering instabilities. Control systems in current generation of compressor open bleed ports in the plenum chamber downstream. This has the dual effect of increasing the mass flow rate, and/or dropping the pressure rise, thereby shifting the operating point to the right (stable) side of the performance map [19]. B. Compression system surge Given the potential risk of damage due to violent surge oscillations, compressions systems are usually operated at a safe distance from the surge line. The imposed safety margin limits the achievable pressure rise and ability to operate at off-design conditions [21]. Surge initiation measurements are performed to determine the stability of the compression system. In Fig. 2.a, 2.b, and 2.c the initiation of surge is shown. By closing the throttle valve YT, the mass flow is decreased, and surge is initiated as can be seen in the pressure ratio. The initiation of surge is defined as the point at which the amplitude of the pressure trace starts to grow and a distinct oscillation frequency appears which is pointed out in the lower right plot. Upon surge initiation a surge development zone B is recognized in which the surge develops to fully developed surge that occurs in zone C. Due to a constant shaft power, the reduction of mass flow results in a slight increase of the mean rotational speed, N. The Fig. 2 also shows that while the throttle valve is kept at a certain value, the rotational speed still increases due to inertia of the rotating parts.
Figure 2: a) Throttle valve; b) Rotational Speed; d) Detailed pressure ratio at surge initiation
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HAFAIFA A ET AL. FUZZY MODELING AND CONTROL FOR DETECTION AND ISOLATION OF SURGE IN….
The plots in Fig. 2 presents the system signals is surge incept as A: normal operation, B: surge development, D: Fully developed surge. The development of the surge is sketched in a compressor map in Fig. 3. As the throttle is closed the compressor performance follows the steady-state characteristic, a constant rotational speed scenario is followed (for simplicity). At a certain pressure ratio the surge limit line is reached and a surge cycle is initiated. Before reaching the limit cycle some smaller cycles occur, as is seen in the pressure ratio in the B-zone of Fig. 2.c.
line that is specific for the compressor, and a surge limit line that is specific for the whole compression system. For N > 19000 [rpm] sketched in Fig. 3, reduction of mass flow leads to the surge limit. Due to a quick change of mass flow and pressure a fully- developed surge is obtained without any sign of rotating stall. For N < 19000 [rpm], a reduction of mass flow first results in an abrupt rotating stall. Since the rotating stall limit and the surge limit are very close, the system does not operate stable on the rotating stall characteristic. The unstable flow eventually develops to a fully-developed surge.
6 5.5 5
Pressure ratio [1]
4.5 4 3.5 3 2.5
Figure 4.a: Sudden drop in pressure ratio
2 1.5 1 -2
-1
0
1
2 3 4 Mass Flow [kg/s]
5
6
7
8
Figure 3: Sketch of the surge cycle
The stability line of the compressor is determined also for transient measurements into surge. Since the instrumented orifice cannot be used for reliable mass flow measurements, the relation between the throttle valve position and the mass flow, Equation (2), is used to determine the surge initiation mass flow. First, for different rotational speeds the surge initiation points, i.e. pressure ratio, rotational speed and throttle valve position at initiation, are determined. Then the mass flow is calculated. As can be seen, the determination of mass flow via the throttle valve position is reasonable, since the surge initiation points are close to the lowest measured steady mass flows. Only the initiation of the curve with rotational speed N = 18000 [rpm] shows an overestimation of the mass flow determined via the transient measurements. This is due to the fact that the fit that is used to determine the transient mass flow is not applicable for flow coefficient ΦC < 0.15. For this compression system, the development of surge after the surge initiation point as described above only applies for rotational speeds larger than ≅ 19000 [rpm]. For rotational speeds smaller than ≅ 19000 [rpm] fully developed prior to surge, a drop in compressor pressure ratio is measured, as shown in Fig. 4. This is a characteristic of abrupt rotating stall. After the drop in pressure ratio, in zone B, which a distinct frequency is found in indicates that the system is not at a stable operating point but rather experiences a surge around the assumed rotating stall characteristic. Furthermore, it develops to fully-developed surge, as shown in zone C. The stability of the present compression system is explained by assuming that there exists a rotating stall limit
Figure 4.b: Detail of the drop in pressure ratio
Figs 4.a and 4.b presents the system signals is incepted, rotating stall and surge as A: normal operation, B: pressure drop and surge development. III. SURGE CONTROL ALGORITHM. Fuzzy fault detection and isolation method defining the surge point over a wide range of changing conditions makes it possible to set the control line for optimum surge protection without unnecessary re-cycling function. This method automatically compensates for changes in pressure rise, mass flow, temperature, and compressor rotor speed. To manipulate and measure system behavior, the control of compression system should include sensors that provide information about the momentary flow conditions and actuators to influence those conditions, and a controller that determines appropriate control action after comparing measured data with the desired flow conditions. A schematic representation of a general surge control system is given in Fig. 5. This algorithm allows for use of the surge control system in this paper, resulting in minimized recycle or blow-off flow. This method reduces the initial cost and simplifies engineering, testing, operation, and maintenance associated with the system when compared to alternative
JOURNAL OF AUTOMATIC CONTROL, UNIVERSITY OF BELGRADE, VOL 19, 2009. methods. The input signals required to facilitate use of the surge control algorithm on centrifugal compressors are the suction flow differential pressure, suction pressure and discharge pressure as indicated in Fig. 2. Using the fuzzy logic model it was possible to analyze the deficiencies of the original surge control algorithm by observing the “real” surge margin calculated from the compressor performance, the objective of an anti-surge controller should not be limited to basic independent machine protection. The antisurge control performance as an integral part of the machine performance control must be considered. Compressor
Engine
User / output gas
Input gas Recirculate gas Control valve action Controller
Sensor
Actuator
-
actual conditions
+
23
before and after surge initiation. A surge controller typically measures a function of pressure rise versus flow. The controller operates a surge valve to maintain sufficient forward flow to prevent surge [23]. IV. FAULT DIAGNOSIS USING FUZZY SYSTEMS. In the course of developing fault diagnosis schemes the use of analytical redundancy implies that a mathematical model of the system is used to describe the inherent relationship (or redundancy) contained among the system inputs and outputs which may be used to generate the residuals for fault diagnosis. The resulting approaches are usually referred to as ‘analytical redundancy’ based fault diagnosis or ‘model based’ methods. This is the approach we take here. The evaluation of the residual signals generated by the models is performed using an expert supervisory scheme. The heuristic knowledge of faults and processing experience can be incorporated into the expert system in the form of rules easily, and thus its advantages are the transparency of operation and simple integration of a priori knowledge. Basically, the rule based expert supervisory system performs two functions. U
Compression System
Y
Figure 5: Schema of general surge control system
Storing real surge points, applying fuzzy logic control of the recycle valve (variable gain depending on operating region), and compensating for interaction between surges, overload and process control can significantly expand the operating window. This allows operation very close to the actual surge lines (4-8%) under all process conditions. Straight line surge control, even with variable slope, must make allowance for the poor fit to actual surge points by using a wider margin of surge line (15-20%) to limits the operating range of the compressor in surge. Interim remedial actions to improve the surge control constants were carried out until an advanced complex control system was installed. An identical steady-state model that was built separately helped to design and test the revised compressor surge control algorithm prior to commissioning on the compressor [22]. Centrifugal compressors will surge when forward flow through the compressor can no longer be maintained, due to an increase in pressure across the compressor, and a momentary flow reversal occurs. Once surge occurs, the reversal of flow reduces the discharge pressure or increases the suction pressure, thus allowing forward flow to resume again until the pressure rise again reaches the surge point. This essentially one dimensional instability affects the compression system as a whole and results in a limit cycle oscillation in the compressor map. This surge cycle will continue until some change is made in the process or compressor conditions. Fig. 1 shows a pressure trace for a compressor system, which was initially operated in a steady operating point. By throttling the compressor mass flow, the machine is run into surge. This figure illustrates the difference between pressure variations
Fuzzy Model
Residual
Desired operation mode
r
Fault Isolation Logic
Fault Detection Figure 6: FDI approach in compression system using fuzzy model
The first function is to determine the bank of models to generate the residuals: Due to our understanding of the plant, we already have some knowledge of faults, the possible types of faults and the possible combinations of different faults. According to this, N + 1 models have been established to describe the characteristics of the se possible situations in which the plant may operate. Besides, we may know that there are some relationships among these fault situations. For instance, suppose fault i and fault j are two different types of faults, if fault i and only fault i is happening at present, then the next possible situation may be the existence of fault j together with fault i, or no fault, but not the situation where only fault j exists under the assumption that no two faults can happen or extinguish simultaneously. Therefore, at each instant only a subset of N + 1 situations may happen and thus only a subset of models may be required to generate residuals which can be summarized as the rule: IF (fault situation i is currently in existence) THEN (the subset Si of models {Mj}jN = 0 will be used as the model bank to generate residuals afterwards. In particular, the rules may be also written: IF (there is no fault), THEN (the model bank consists of no fault model and all possible single fault models) or, IF (there is a single
F
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HAFAIFA A ET AL. FUZZY MODELING AND CONTROL FOR DETECTION AND ISOLATION OF SURGE IN….
fault), THEN (all single fault models except the one that has been isolated will be removed from the model bank, and the multiple fault models including the isolated fault and another possible fault will be added). The initial situation is usually assumed to be no fault. Using this type of knowledge the computation complexity can be reduced and the possibility of false alarms can be decreased as well. The second function is the Fault diagnosis by decision logic: Once the residuals are generated, the fault decision logic is used to diagnose the faults. A sequence of minimum residual indices is first generated by: I ∗ (k ) = arg min(ri (k )) (3)
Where I*(k) denotes the index of the model whose residual is minimum, that is the most appropriate model to represent the plant at instant k, note that when the plant situation changes, the input/output characteristics of the system will change as well. Therefore, the change of index, the change of most suitable model may be used to indicate the change of plant situations, by the occurrence of new faults. However, the new index after the change does not mean that the corresponding fault situation has occurred. This is because during this transient phase, the residuals may change drastically and some of them may happen to be very small for a short time and become large later since they are not the proper fault situations. In order to isolate the fault situations correctly, some logic rules are used to guarantee that a fault will be isolated only if it lasts at least for T0 seconds. The results of isolation can be represented by a fault index F* as shown in Fig. 5. This strategy may affect fault sensitivity to some extent, but it is effective to isolate faults accurately which is more important. A. Fuzzy model of compression system The linearization of the two first equations of the model (1) of the compression system around an operating point M(Ppc0, mc0, ut0, ub0) gives:
− B PpC 0 P B x = = m + ub m 1 / B 1 / Bm mC − V / B
(4)
where: ŧ = twH with wH is the frequency of Helmholtz [18] defined by the following equation:
WH = a
AC ( a = γRTa ,Bm = B / G ). VP LC
The parameters B and G are defined as follows:
Yt 2 wH LC LA G= t C LC At B=
(5) (6)
where B is the parameter of stability of Greitzer [24, 25]. It follows that, the linear model (4) conducting state following:
δx( t ) = Aδx( t ) + Bδu( t ) (7) δy( t ) = Cδx( t ) + Dδu( t ) With: δx = x − x0 ,δu = u − u0 ,δy = y − y0 and x0 = [5.74 1.25 12.3]
T
u0 = [0.2
0.13 -1.15] T 0] T
y0 = [2.23 6.6
The matrices A, B, C and D evaluated in this equilibrium point is given by:
0.2076 A = 3.7181 - 9.6780
0.1323
0.2076 B = 0 0
1.2365
1 C = 0 0
- 0.4892 - 1.539
- 3.6891 - 3.8230
0 1 0
0 0 0 0
- 2.312 0
0 0 1
0 0 0 D= 0 0 0 12.1074 0.3582 0 The fuzzy model proposed for the compression system is expressed symbolically by IF-THEN rules of the form: IF Z1(t) and Fi1 and … and Zn and Fin, i = 1, 2, …..r THEN
x ( t ) = Ai x( t ) + Bi u( t ) + K i ( y( t ) − y( t )) (8) y ( t ) = C x ( t ), i = , ,.., r 1 2 i i
B. The Takagi_Sugeno Fuzzy System Developing mathematical models for nonlinear systems can be quite challenging. However, Takagi-Sugeno fuzzy systems are capable of serving as the analytical model for nonlinear systems due to its universal approximation property, that is, any desired approximation accuracy can be achieved by increasing the size of the approximation structure and properly defining the parameters of the approximator [3, 26]. A Takagi-Sugeno fuzzy system can be defined by: r = x ( t ) hi ( z( t ))[ Ai x( t ) + Biu( t )] ∑ i =1 r y( t ) = h ( z( t )C x( t ) ∑ i i i =1
(9)
The fuzzy model developed for the compression system is then expressed symbolically in the form or rules as shown bellow: Rule 1: IF: δx1(t) is N and δx3(t) is N
JOURNAL OF AUTOMATIC CONTROL, UNIVERSITY OF BELGRADE, VOL 19, 2009.
THEN:
= δ x (t )
A1 δ x (t ) + B1 δ u (t )
= δ y (t )
C1 δ x (t ) + D1 δ u (t )
Rule 2: IF: δx1(t) is NM and δx3(t) is NM
δ x (t ) A2 δ x (t ) + B2 δ u (t ) THEN: = = δ y (t ) C2 δ x (t ) + D2 δ u (t )
line and thereby avoiding the unstable demonstrated in the simulations above. 200
1.7
50
1.6
0 0
5
THEN:
= δ x (t )
A5 δ x (t ) + B5 δ u (t )
= δ y (t )
C5 δ x (t ) + D5 δ u (t )
Rule 6: IF: δx1(t) is P and δx3(t) is Z THEN:
= δ x (t )
A6 δ x (t ) + B6 δ u (t )
= δ y (t )
C6 δ x (t ) + D6 δ u (t )
Rule 7: IF: δx1(t) is N and δx3(t) is Z
δ x (t ) A7 δ x (t ) + B7 δ u (t ) THEN: = = δ y (t ) C7 δ x (t ) + D7 δ u (t ) Rule 8: IF: δx1(t) is Z and δx3(t) is PM
δ x (t ) A8 δ x (t ) + B8 δ u (t ) THEN: = = δ y (t ) C8 δ x (t ) + D8 δ u (t ) Rule 9: IF: δx1(t) is NM and δx3(t) is Z = δ x (t ) A9 δ x (t ) + B9 δ u (t ) THEN: = δ y (t ) C9 δ x (t ) + D9 δ u (t ) The matrices (Aj, Bj, Cj, Dj), j = 1-9 are determined from the combination of operating points considered in the rule j. V. SIMULATION RESULTS Simulations of the proposed fuzzy fault detection and isolation approach to the surge control will now be presented. The idea is to control the compressor speed with feedback from the mass flow so that the compressor can operate in a stable mode even to the left of the surge
x 10
1.15
10 Time [s]
15
20
1.5
0
5
10 Time [s]
15
20
10000
Speed [rpm]
1.1
8000 Drive torque [Nm]
6000
1.05
4000
1 0.95
Pressure rise
1.8
100
-50
operation
1.9 Mass flow [ks/s]
150
4
Rule 3: IF: δx1(t) is Z and δx3(t) is Z = δ x (t ) A3 δ x (t ) + B3 δ u (t ) THEN: = δ y (t ) C3 δ x (t ) + D3 δ u (t ) Rule 4: IF: δx1(t) is PM and δx3(t) is PM = δ x (t ) A4 δ x (t ) + B4 δ u (t ) THEN: = δ y (t ) C4 δ x (t ) + D4 δ u (t ) Rule 5: IF: δx1(t) is P and δx3(t) is P
25
2000 0
5
10 Time [s]
15
0
20
5
10 Time [s]
15
20
Figure 7: Surge initiation in compression system 1.6
110
Pressure rise
Mass flow [ks/s]
100 90
1.58
80 70 60 1.02
1.56 0
5
x 104
10 15 Time [s]
5
10 15 Time [s]
20
10000 8000
1.01
6000
1
4000
Speed [rpm]
0.99 0.98
0
20
2000 0
5
10 15 Time [s]
20
0 0
Drive torque [Nm] 5
10 15 Time [s]
20
Figure 8: The compression system without control of surge
The simulations results are presented in this section, the first is the results of simulations of the compression system in surge situation at 20 KRPM of compression speed and the simulations of the compression system without control of surge, and the second simulation is the compression system with control of surge using fuzzy fault detection and isolation method. The response of the compression system in surge situation at 20 KRPM of compressor speed is shown in Fig. 7 and the response of the system without control of surge is shown in Fig. 8.The response of the compression system with control of surge using fuzzy fault detection and isolation is shown in Fig. 9. A fuzzy fault detection and isolation of the compression system was constructed using data from the simulation model. Fuzzy fault and detection of different complexities were studied. A model FDI controller with a longer prediction horizon and a small; ; the large the computational time but also the better the results control weighting factor provides good performance in terms of surge detection and isolation and reduced error. However, the observation on the variation of the controller output provided an interesting result. Implementing such a controller on a real-time system would probably be
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HAFAIFA A ET AL. FUZZY MODELING AND CONTROL FOR DETECTION AND ISOLATION OF SURGE IN….
prohibitive due to the fact that there are limitations on the incremental variation of the compression system. 110
[8]
1.6
Pressure rise
Mass flow [ks/s]
100
1.58
90
[9]
80 70 60
1.02
1.56 0 x 10
5 4
10 Time [s]
15
20
0
5
10 Time [s]
15
20
[11]
Drive torque [Nm]
8000 6000
Speed [rpm]
1
[12]
4000
0.99 0.98
[10]
10000
1.01
2000 0
5
10 Time [s]
15
[7]
20
0
0
[13] 10 20 Time [s]
30
[14]
Figure 9: The use of the fuzzy FDI to the surge control
VI. CONCLUSION A novel approach of the fault detection and isolation for a class of nonlinear systems was presented. The fuzzy Takagi Sugeno model, formerly used in robust identification of nonlinear functions, was applied in the residual generation stage of the FDI model a family of system responses from the confidence band, already including the effects of uncertainties, that is based only on the input-output data. Applying low-pass filtering when obtaining the input-output data set makes it possible to use relatively simple fuzzy structure of the fault detection and isolation, without any significant loss in Fault Detection stage efficiency. The detection and the isolation of surge in the compression system illustrate that by using the Fuzzy FDI based approach, surge can be successfully isolated, even though only a limited amount of the input output data is available. Investigating the performance changes resulting from different choices of filter structure, applying the proposed method on a broader range of systems, and investigating possible extensions to frequency-based methods and fault tolerant control deserves further attention. In addition, the results of the simulated example demonstrate the quality of performance coupled with simplicity of application, which is very important from the application point of view. REFERENCES [1] [2] [3] [4] [5] [6]
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