Monitorability Analysis for a Gas Turbine Using Structural Analysis

9 IFAC Fault Detection, Supervision and Safety of Technical Processes, Beijing 2006

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MONITORABILITY ANALYSIS FOR A GAS TURBINE USING STRUCTURAL ANALYSIS Cristina Verde** and Marino Sgmchez-Parra*

*~Instituto de Ingenieria, UNAM, 0~510, DF Mdxico verde @servidor. unam. mx *Instituto de Investigaciones Eldctricas PB 1-475 Cuernavaca, Morelos, 062001 Mdxico msanchez@iie, org. mx

Abstract: This paper studies the monitorability of a gas turbine in a combined cycle power plant with respect to sensors fault. The analysis is carried on using the structural analysis where the structurally relations between known and unknown variables are obtained from a nonlinear complex dynamic model described by 37 equations with twelve sensors. The structure decomposition in just-constrained and over-constrained subsets allows a separation of the monitorability analysis in two parts and a reduction in the dimension of the structure from which the causal incidence matched matrices and the redundancy relations are obtained. The analysis concludes that all the turbine variables can be evaluated from measurements and control actions in normal conditions and this capability is only robust with respect to fault in two sensors. Moreover, the monitorability of the structure can be recovered for a set of three sensors fault by redundancy. However, the capability to compute all variables of the structure is destroyed if one of a set of seven sensors is not reliable even some of them are in the over-constrained subsystem. From the justconstrained subsystem one identifies that the plant components without redundancy are associated to the variables of the proper gas turbine. Copyright 9 2006 IFA C. Keywords: Structural Analysis, Gas Turbine of a Combined Cycle Power Plant, System Monitorability.

1. INTRODUCTION Combined cycle power plants (CCPP) are becoming increasingly prevalent in the electric utilities market place. There are different configurations for this plants but the one considered here is integrated by two gas turbine-generator (GT), each one with a heat recovery-steam generator, and a unique steam turbine-generator. The essential and more critical component is the GT because it comprises complex dynamical subsystems which can fail due to faults in sensors, actuators and components and relies heavily on the control sys1 Supported by Instituto de Investigaciones El~ctricas and IN- 102306-2-DGAPA-UNAM

tem affecting the reliability, availability and maintainability of the plant. Therefore, the design of a robust monitoring and supervision system for a GT is a challenge for the safe process community. It is known that the principle to solve a fault diagnosis problem is the redundancy and consistent of data in a system (Patton et al., 2000). However, a general theory to solve the fault detection and isolation (FDI) issues for non linear systems as well as the design of robust residual generators is still missing. Moreover, in the case of a complex and large scale systems, as the gas turbine, it is necessary previous to the design of the diagnosis system to study the process limitations with respect to fault. Consequently, one must develop

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efficient analysis methods to capture the possibilities to solve a diagnostic issue. This analysis must include systematic monitorability analysis when some measurements are not available.

2. STRUCTURAL

Structural analysis is based on graph theory (Blanke et al., 2003) and deals with the structural information between variables in a system and allows to explore system fundamental properties in a very simple way.

On the other hand, to analyze the diagnosis possibility for large scale complex systems, the use of generic structural models is more appropriate than the analytical models. Diverse structural methods have been proposed for FDI issues (Dion et al., 2003). In particular, the Staroswiecki's school suggests the structural analysis (SA) (Blanke et al., 2003) to study the monitorability and the observability of large scale systems. This framework has two advantages: allows to cope with complex system, and requires few parameters information. To know if there is a way to evaluate all the variables of a process, only the structure of the system without explicit mathematical model plays an important role with this approach. The SA is based on causal relationships between variables given in the form of a structure graph or equivalently as an boolean incidence matrix, and it can be easily used in the early design phase of a supervision system. Some methods have been proposed to systematize the generation of residual structure (Dustegor et al., 2004) and a toolbox is now available (Lorentzen and Blanke, 2004). The above mentioned advantages of the SA motivates this work in which one shows the powerful of a structural model to study the monitorability of a GT of a CCPP in normal conditions and its vulnerability with respect to smart sensors with their self-diagnosis. Starting from a set of 37 equations which describes the GT in normal conditions, the structure of the system is decomposed in the parts with redundancy and without it. From these subsystems, the monitorability properties of the GT are discussed. Thus, the gas turbine is identified as the undetectable component with respect to any fault. Moreover, selecting a causal matching matrix one determines the set of sensors which must be redundant by hardware to hold the monitorability of the whole GT. The outline of the paper is as follows. In Section 2 the background of SA and the important definitions related with the monitorability and the ranking of the unknown variables evaluation with a matching are presented. Section 3 introduces a compact description of the gas turbine classifying the known variables in 3 sets. Section 4 presents the monitorability study based on a matched matrix of the gas turbine with its redundancy relations; some aspects of the structure vulnerability are mentioned. Finally, Section 5 gives some conclusions.

ANALYSIS

Definition 1. The structure graph of a system with constraints set g, variables set Z = ~' U K: with ~' unknown, K: known and a parameter set O c Z is a bipartite graph G = {g,Z,s

(1)

with 2 sets of vertices whose set of edges g c C • Z is defined by

(ci, zj) E $ r zj appears in ci

(2)

The corresponding matrix of a structure graph is given by its incidence matrix ( I M ) where the rows of the matrix are the set of constraints, the columns the set of variables and the edge (ci, zj) is associated with a 1 in the intersection of row i and column j, i.e.

I M ( i , j) =

1 if(ci, zj) E g 0 otherwise

A basic concept in the SA is the matching process, which associates variables with constraints from which the unknown variables can be evaluated. The key to match Z with g is the decomposition of I M in subsystems considering explicitly the system dynamic and the following property of a graph.

Theorem 2. (Blanke et al., 2003) Any incidence matrix of a graph with finite external dimension can be uniquely decomposed in three parts: (1) Over-constrained: S + = (C+,Z +) such that Q(C +) = Z +, and a complete matching exists on Z + but not on g implying the existence of redundant information. (2) Just-constrained: S ~ = ( g ~ ~ such that Q(C ~ = Z ~ u Z +, and a complete matching exists on both Z ~ and on g ~ (3) Under-constrained: S - = (g-, Z - ) such that Q(C-) = Z - U Z ~ U Z + and a complete matching exists on g - but not on 2 - . To consider dynamic systems in the framework of SA, each state variable xi includes a constraint

dxi (di) ici= dt which is denoted with symbol di. Moreover, the feasible matchings have to be causal according Definition 3. Definition 3. A Causal matching is a matching that induces an oriented graph without differen-

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tim cycles assuming known a behavior model integrated with non-linear algebraic and differential equations. The mark O in the incidence matrix indicates the edge of the matching (ci, xj).

kl 3

xk 3

I LJva I

t

3

9 Bli~IWI~

kl ~

f

L Stack

SG

Definition ~. The structurally monitorable sub-

k15 X14

system is a subset of SOu $+, associated to a set of constraints Cm, such that a subset of unknown variables Xm can be expressed as a function of known variables K: using Cm, for any input, assuming the derivative causality in the matching. This definition is more general that the introduced in (Blanke et al., 2003) where the fault monitorability is considered. To identify the path size required to evaluate a variable from set K: in the matched matrix, a column R with a pair of indexes is added to the matrix of S+ and S O according to Definition 5.

Definition 5. The pair of indexes (ip, im)j for constraint cj added to the incidence matrix denotes the path size to evaluate a variable and if the path has been used in the matching. The index ip determines the ranking for a variable evaluation starting from K: and the flag im denotes if cj is used in the matching; i.e. im-~ 1 means matched constraint and im = 0 redundant relation. (Ss and Verde, 2006)

2.1 Structural Analysis for Fault Diagnosis The start point of the fault diagnosis analysis is the exhaustive search of all the complete matchings on S + satisfying the causal matching and the respective redundancy equations, notated set 7~n.

Definition 6. The detectable part of a structurally system with respect to a set of fault $" is the subsystem associated to a set of constraints, Cd, such that there exists a set of redundancy constraints which are structurally sensitive to $-. Fact 1. The subsystem S ~ is undetectable with respect to any fault, since there is not any redundant constraint in it. This means any fault which affect the constraints and the known variables of S ~ can not be detected using parity relations. In this case the signal-based methods are an alternative for the FDI issue.

3. GAS TURBINE MODEL DESCRIPTION The GT behavior model (BM) considered at this work is based on the Westinghouse W510b gas turbine of a combined cycle electrical power plant (CCPP) which is integrated by two gas turbines,

x23

Inlet Guide Vanes I-'-I Actualor (IGV's) k5

15

~176

k18 ~-kg

k9

6 %~Yg' , Bo..

klgk9 ~16 x~6

Fig. 1. Gas Turbine System two heat recovery steam generators and a steam turbine. The model may go from cold startup to base load generation, it was validated against real dynamic trends, according to the experience in this kind of units. The main components of the GT model are: compressor, combustion chamber, gas turbine, electric generator and heat recovery, which are shown in Fig. 1. Additionally, two gas fuel control valves (one for the GT and the other for the heat recovery afterburners), compressor bleed valves, and the actuator for the compressor inlet guide vanes are included (Delgadillo and Fuentes, 1996). Related with the BM validation, it was used the first time for testing the GT control algorithms in two control system modernization projects. Subsequently the GT BM was upgraded and nowadays it is a subsystem of a CCPP dynamic simulator. The BM was designed applying the lumped parameter criteria and a simplified control strategy in order to get complete simulation tests. Some liraitations considered in developing the GT model were ideal gas behavior, isentropic compression and expansion of ideal gas, and linear flow characteristic for all control valves. The obtained GT behavioral model consists of 28 constraints: 19 static-algebraic constraints and 9 dynamic-differential constraints. In accordance with (Blanke et al., 2003) for the SA framework an explicit differential equation for each dynamic constraint is added in order to get a complete dynamic structured model with 37 constraints. Then, the system model has 27 unknown variables xi and xi, 19 known variables ki and 29 physical parameters Oi. The equations and the incidence matrix of 37 x 27 associated to the GT can be found in (S~nchez-Parra and Verde, 2006). The set E of known variables is integrated with 3 sets E = E~ U K;~ U K;~

where set K~s = {kl, k2, k3, k4, k6, k9, klO, kll, k12, k13, k14, k15 }

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corresponds to the twelve process sensors; 4 pressure transmitters, 1 magnetic pick-up, 1 thermometer, 5 thermocouples and 1 wattmeter; the set K:~ = {ks,

S +, the analysis is carried on using the following observations. From the redundancy part of 8+ identify the known variables in column K:, which are as well in the subset K~s and define these as set K:R. In the case of the GT, the set is given by

kT, ks, k~6}

is associated to the position transducers from actuators; and the variables of set

K:R = {kl,

](:c - {k17, k18, k19}

For each interest variable kj E K:R which is assumed unreliable, the following observations allows a systematic analysis:

are the control signals (GT inputs).

(1) Detect the first row of the matched low triangular matrix (ira = 1) in which appears the first time the variable kj in column K: during the matching and identify its ranking index ip in column R. Simultaneously detect the row in the redundancy part in which the variable kj is used and identify its ranking index i R in column R. (2) Compare the indexes ip and iR and If ip >_ iR the monitorability of the corresponding matched variable is not affected, since the value of kj can be calculated using ~r. If ip < iR, the monitorability of the matched variable is lost, because the value of kj is required before it could be computed from the set T~R in a computed sequence.

4. MONITORABILITY ANALYSIS FOR THE

GT Following the Dulmage-Mendelsohn decomposition of the incidence matrix of the system (Pothen and Fan, 1990), the sets S o and S + and S are obtained. In the particular case of the GT, subsystem S - - 0, then the just constrained and over constrained subsystems are the start point of the analysis. Since, the set 2 '+ can affect in general the subsystem S o , one starts the analysis considering S +. For the analysis of $+ and S ~ it is assumed that each sensor of set K:, is smart and has a self detection system, as it is suggested by (Clarke and Fraher, 1996).

If a sensor kj ~ ER and it is used in the matched sequence, the monitorability is as well lost, since there is not option to calculate its value.

Based on the over-constrained system S +, different matching can be obtained. However, there are some matchings which are usefulness for the SA, since they include differential cycles. Using the software SaTool by (Lorentzen and Blanke, 2004), one got the causal low triangular matched matrix given in Table 1 in which the set of redundancy relation T~R is included and separated using a double line and the symbol r denotes a edge of a matching.

Considering these simple remarks for the members of K:~ involved in Table 1, one gets: 9 The set X + is monitorable in normal conditions and there is a redundancy degree without fault since exist 10 constraints, from which is possible to evaluate some variables to hold the monitorability of the system when the self detector of a sensor generates a fault alarm. 9 Assume kl 6 K:R is unreliable or failed. Then, the matched set {x3,x6, x9} with ranking execution 0 requires an alternative path to estimate kl with ranking 0. This could be achieved using c4 because it has a ranking 0. Moreover, because x12 and x15 are matched with ranking 1, the evaluation of kl with the redundant constraint at ranking 0 can be used for the evaluation of both x12 and x15. Consequently the monitorability can be preserved if kl is unreliable. Similarly results are obtained, if one assumes that k3 or k6 are unreliable or failed, since the set {x3, x9} has ranking 0 and k3 or k6 could be matched using ca with ranking 0. In both cases the monitorability of S + is preserved assuming only one unreliable sensor.

If one is interested in the monitorability of the GT when a sensor is unreliable, there are two ways to carry on the study. One way is to consider the variable associated with the unreliable sensor as an unknown element and to make a new matching. The other way is based on the analysis of the original monitorability state for reliable sensors and managing the ranking order in the matching sequence with all the constraints of S+.

4.1

k3, k4, k6, kll, k12, k14} C K:s

Test of Monitorability for S +

A full GT monitorability analysis requires to determine if there exits at least a set of causal paths which preserves the complete evaluation of the unknown variables in the structure assuming unreliable sensors. Based on a causal matching sequence (low triangular match) of subsystem

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Table 1. M a t c h e d S u b s y s t e m S + R

Known Set

Unknown Subset k '+

C

~5 X27 X3 X13 X6 X9 X22 X21 X23 X25 ~26 X15 X7 ~1 Xl0 X12 X20 X24 X14 X2

0,1 c7 0 0,1 d9 0, I c2 0,1 d4

0

0

kl, k2, k3, k5 ks

0 0

O, 1 (:18

0,1 0,1 O, 1 O, 1 O, 1 0,1

kl, kl0, kll kl, k3, k6, k7 k14 k2

0

c~ ds C22

klO, k14 k3,klo, k15 k9, klO,k14, k16

C23

C24 c25

<> 1 1 1

I , I C10

1,1 ds 1,1 cl i,I

c~

1,1 1, 1 1,1 2,1 2,1 0,0 1,0 3, 0 3, 0 1,0 1, 0 2,0 2, 0 2, 0 1, 0

c8 c~.i dv co d2

]C

k5, k17 k16

r 0 1

1

kl, kl0, kl 1 0 kl 0 kl, k8, k9 k13 0 0 0 kl, k3, k4, k6 k5

<> O

1

0 1

r 1

O 1

1

0

1

0

c4

dl 1 cll c12 c13 c17 d6 c26 c27 c28 1

1 1

1

1

1

1

1 1

1

1

0 k6

k8,k18 kll,kl2 0 O k11, k14 k16, k19

1

1 1

1 1

1 1

1 1 1 1

9 A different s i t u a t i o n occurs if kll E ](:R is unreliable or failed. Since, x 6 is m a t c h e d with a r a n k i n g execution 0, and the evaluation of kll by constraint c17, has ranking 1, the variable x6 cannot be m o n i t o r e d even the r e d u n d a n c y of t h e structure. Similar result is o b t a i n e d if k14 is unreliable, because x22 is m a t c h e d with ranking 0 and t h e r e d u n d a n c y constraint c27 which allows to evaluate k14 has ranking 2. 9 If the assumed failed sensor is k4 or k12, m o n i t o r a b i l i t y will not be affected because b o t h known variables are not included in the m a t c h e d sequence. 9 Since t h e set of sensors {kz,k9, klo, k13, k15} does not a p p e a r in the r e d u n d a n t p a r t of S +, if one of their fails, the m o n i t o r a b i l i t y of 5 '+ is destroyed. This result justifies why in practice h a r d w a r e r e d u n d a n c y is used for these sensors.

1

Table 2. IM of S u b s y s t e m $0 R

C

Subset X ~ X4 X19 X17 X16 X18 X8 Xll

umi

|

Hnl WDI Wml nnnl Wnl mm,~ll

1

1 1 1

Subsets ]CA'+ k2 k2

kl,klo,xl 1

1

1

1

1 1

k6, XlO,//:12 kl, Xl kl, k3, X3 1 k2, k13, x15

m e m b e r of k '~ is included in the r e d u n d a n c y relations. 9 T h e s u b s y s t e m S O corresponds to constraints associated to the gas t u r b i n e c o m p o n e n t of G T . In p a r t i c u l a r the involved variables are: t h e rotor acceleration, the compressor energy, e x h a u s t gas enthalpy, combustion chamber gas enthalpy, exhaust gas density, energy friction losses and speed rate. 9 If kl is failed, a correct calculation via the r e d u n d a n c y relation c4 from $ + can be subs t i t u t e d in c15 E C ~ Similar result applies for k6 and k3. This means the m o n i t o r a b i l i t y of X ~ is not destroyed if only one of the variable {kl, k3, k6} is calculated in 8 + and used in S ~ This r e m a r k cannot be concluded if the SA is performed considering only S + .

4.2 Monitorability Analysis for S ~ F r o m the s u b s y s t e m S o given in Table 2 one can identify t h a t 9 T h e variables are m o n i t o r a b l e only in normal conditions and none fault which affect t h e constraints is undetectable, because any

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Annuel IAR Jahrestagung, Karlsruhe, Germany. pp. 187-192. Patton, Ron, P. Frank and R. Clark (2000). Issues of Fault Diagnosis for Dynamic Systems.

It is to note that some of the variables of X ~ as the gas enthalpies, gas densities and energy are necessary for the calculation of the gas turbine efficiency. Therefore, to avoid a poor estimation or misinterpretation of the turbine state, set {ki, k2, k3, k5, klo, ki3, kis} has to be estimated with a high reliability.

Chap. Introduction, pp. 1-6. Springer. London. Pothen, Alex and Chin-Fu Fan (1990). Computing the block triangular form of a sparse matrix. A CM Trans. on Mathematical Software 303-324. SAnchez-Parra, Marino and Cristina Verde (2006). Analytical redundancy for a gas turbine of a combined cycle power plant. In: American

5. CONCLUSIONS It has been used the structural analysis and the matching sequence procedure to study the variables monitorability in a large scale system. The study concerns with the conditions which preserve the variables monitorability of a Gas Turbine which belongs to a Combined Cycle Power Plant. The analysis shown that the monitorability of the system is destroyed if one sensor of the set {k2, k9, kl0, kll, k13, k14, k15} fails; the structure is only robust with respect to sensors {k4, ki2}. However, the monitorability could be preserved if one of the sensors {ki, k3, k6 } fails and a software sensor is designed using the redundancy of the structure. These properties are obtained taking into account the ranking of a matched structure. It is shown that the structural analysis can be used simultaneously to study the possibilities to get a parity equations for fault diagnosis design and to select a robust structure for a fault tolerant control of a gas turbine before one designs the control algorithms. 6. REFERENCES Blanke, Mogens, M. Kinnaert, J. Lunze and M. Staroswiecki (2003). Diagnosis and fault Tolerant Control. Springer. Berlin. Clarke, D. W. and P. M. A. Fraher (1996). Modelbased validation of a dox sensor. Control Eng. Practice 4(9), 1313-1320. Delgadillo, M. A. and J. E. Fuentes (1996). Dynamic modeling of a gas turbine in a combined cycle power plant. Document 5117, in spanish. Instituto de Investigaciones E14ctricas, M ~xico. Dion, J.M., C. Commault and J. van der Woude (2003). Generic propertie and control of linear structured systems: a survey. A utomatica 39, 1125-1144. Dustegor, D., V. Cocquempot and M. Staroswiescki (2004). Structural analysis for fault detection and isolation: an algorithm study. In:

2nd IFAC Symposium on System Structure and Control. pp. 134-139. Lorentzen, T. and M. Blanke (2004). Industrial use of structural analysis - a rapid prototyping tool in the public domain. In: Colloque

Control Conference. 7. APPENDIX A 7.1 UNKNOWN VARIABLES X Combustion chamber gas density, x i; Combustion chamber gas rate density, x2; Compressor inlet air flow, x3; Turbogenerator speed rate, x4; Compressor IGVs position rate, x5; Combustion chamber gas temperature, x6; Combustion chamber gas rate temperature, x7; Compressor energy, xs; Compressor bleed air flow, x9; Compressor outlet air flow, xio; Starting motor energy, xii; Combustion chamber gas fuel flow, x i2; Gas turbine fuel gas valve position rate, x i3; Combustion chamber inlet gas flow, xi4; Combustion chamber outlet gas flow, xi5; Combustion chamber gas enthalpy, xi6; Gas turbine exhaust gas density, xiT; Gas turbine exhaust gas enthalpy, xis; Gas turbine energy friction losses, xig; Electrical generator power angle , x20; Electrical generator power rate angle, x2i; Heat recovery gas rate temperature, x22; Heat recovery gas density, x2a; Heat recovery gas rate density, x24; Heat recovery outlet gas flow, x25; Afterburners gas fuel flow, x26; Afterburner fuel gas valve position rate, x27.

7.2 KNOWN VARIABLES IC Compressor discharge pressure, ki; Turbogenerator speed, k2; Atmospheric pressure, k3; Outlet temperature, k4; Compressor inlet guide vanes position, k5; Compressor air discharge temperature, k6; Compressor air bleed valve position, kT; Gas turbine fuel gas valve position, ks; Inlet fuel gas valves pressure, k9; Heat recovery pressure, kio; Exhaust gas temperature (EGT), kii; Blade path temperature(BPT), ki2; Electrical generator power output, ki3; Heat recovery gas temperature, ki4; Heat recovery gas outlet temperature, kis; Afterburner fuel gas valve position, ki6; Inlet guide vanes control signal, kiT; Gas turbine fuel gas valve control signal, kis; Afterburner fuel gas valve control signal, k19.

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