Faults isolation and identification of Heat-exchanger/ Reactor with parameter uncertainties

Abstract

This paper deals with sensor and process fault detection, isolation (FDI) and identification of an intensified heat-exchanger/reactor. Extended high gain observers are adopted for identifying sensor faults and guaranteeing accurate dynamics since they can simultaneously estimate both states and uncertain parameters. Uncertain parameters involve overall heat transfer coefficient in this paper. Meanwhile, in the proposed algorithm, an extended high gain observer is fed by only one measurement. In this way, observers are allowed to act as soft sensors to yield healthy virtual measures for faulty physical sensors. Then, healthy measurements, together with a bank of parameter interval filters are processed, aimed at isolating process faults and identifying faulty values. Effectiveness of the proposed approach is demonstrated on an intensified heat-exchanger/ reactor developed by the Laboratoire de Génie Chimique, Toulouse, France.

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Nowadays, safety is a priority in the design and development of chemical processes. Large research efforts contributed to the improvement of new safety tools and methodology. Process intensification can be considered as an inherently safer design such as intensified heat exchangers (HEX) reactors in [ 1 ], the prospects are a drastic reduction of unit size and solvent consumption while safety is increased due to their remarkable heat transfer capabilities.

However, risk assessment presented in [2] shows that po

tential risk of thermal runaway exists in such intensified process. Further, several kinds of failures may compromise safety and productivity: actuator failures (e.g., pump failures, valves failures), process failures (e.g., abrupt variations of some process parameters) and sensor failures.

Therefore, supervision like FDI is required prior to the implementation of an intensified process.

For complex systems (e.g. heat-exchanger/reactors), fault

detection and isolation are more complicated for the reason that some sensors cannot be placed in a desirable place, and for some variables (concentrations), no sensor exists. In addition, complete state and parameters measurements (i.e. overall heat transfer coefficient) are usually not available. Supervision studies in chemical reactors have been reported in the literature concerning process monitoring, fouling detection, fault detection and isolation. Existing approaches can be roughly divided into data based method as in [ 3 ], neural networks as in [ 4 ] and model based method as in [ 5,6,7,8,9 ]. Among the model based approach, observer based methods are said to be the most capable [ 10,11,12,13,14 ] if analytical models are available. Most of previous approaches focus on a particular class of failures. This paper deals with integrated fault diagnosis for both sensor and process failures. Using temperature measurements, together with state observers, an integrated diagnosis scheme is proposed to detect, isolate and identify faults. As for sensor faults, a FDI framework is proposed based on the extended observer developed in [ 15 ]. Extended high gain observers are adopted in this paper due to its capability of simultaneous estimation of both states and parameters, resulting in more accurate system dynamics. The estimates information provided by the observers and the sensors measurements are processed so as to recognize the faulty physical sensors, thus achieving sensor FDI. Moreover, the extended high gain observers will work as soft sensors to output healthy virtual measurements once there are sensor faults occurred. Then, the healthy measures are utilized to feed a bank of parameter intervals filters developed in [ 11 ] to generate a bank of residuals. These residuals are processed for isolating and identifying process faults which involves jumps in overall heat transfer coefficient in this work.

It should be pointed out that the contribution of this work does not lie with the soft sensor design or the parameter interval filter design as either part has individually already been addressed in the existing literature. However, the authors are not aware of any studies where both tasks are combined for integrated FDI, besides, there is no report whereby parameter estimation capacity of the extended high gain observer is used to adapt the coefficient, rather than parameter

FDI, thus together with sensor FDI framework forms the contribution of this work.

2 2.1

System modelling

Process description

The key feature of the studied intensified continuous heatexchanger/reactor is an integrated plate heat-exchanger

technology which allows for the thermal integration of several functions in a single device. Indeed, by combining a reactor and a heat exchanger in only one unit, the heat generated (or absorbed) by the reaction is removed (or supplied) much more rapidly than in a classical batch reactor. As a consequence, heat exchanger/reactors may offer better safety (by a better thermal control of the reaction), better selectivity (by a more controlled operating temperature). 2.2

Dynamic model

Supervision like FDI study can be much more efficient if a

dynamic model of the system under consideration is available to evaluate the consequences of variables deviations and the efficiency of the proposed FDI scheme.

Generally speaking, intensified continuous heat-exchanger/ reactor is treated as similar to a continuous reactor [ 16,17 ], then flow modelling is therefore based on the same hypothesis as the one used for the modelling of real continuous reactors, represented by a series of N perfectly stirred tank reactors (cells). According to [ 18 ] , the number of cells N should be greater than the number of heat transfer units, and the heat transfer units is related with heat capacity flowrate. The modelling of a cell is based on the expression of balances (mass and energy) which describes the evolution of the characteristic values: temperature, mass, composition, pressure, etc. Given the specific geometry of the heat-exchanger/reactor, two main parts are distinguished. The first part is associated with the reaction and the second part encompasses heat transfer aspect. Without reaction, the basic mass balance expression for a cell is written as: {Rate of mass flow in – Rate of mass flow out = Rate of change of mass within system} The state and evolutions of the homogeneous medium circulating inside cell are described by the following balance:

2.2.1 Heat balance of the process fluid (J. s−1)

ρkpVpkCpk dTpk = hkpAk(Tpk − Tuk) + ρkpFpkCpk(Tpk−1 − Tpk) (1) p dt p where ρkp is density of the process fluid in cell k (in kg. m−3), Vpk is volume of the process fluid in cell k (in m3), Cpkspecific heat of the process fluid in cell k (in J. kpg−1. K−1) , hkp is the overall heat transfer coefficient (in J. m−2. K−1. s−1).

2.2.2 Heat balance of the utility fluid (J. s−1)

ρkuVukCpku ddTtku = hkuAk(Tuk − Thk) + ρkuFukCpku(Tuk−1 − Tuk) (2) whereρku is density of the utility fluid in cell k (in kg. m−3), Vuk is volume of the utility fluid in cell k (in m3), Cpkspecific u heat of the utility fluid in cell k (in J. kg−1. K−1) , hku is overall heat transfer coefficient (in J. m−2. K−1. s−1).

The eq. (1) (2) represent the dynamic reactor comportment.

The two equations represent the evolution of two states (Tp: reactor temperature and T : utility fluid temperature).The u heat transfer coefficient (h) is considered as a variable which undergoes either an abrupt jumps (by an expected fault in the process) or a gradual variation (essentially due to degradation). The degradation can be attributed to fouling. Fouling in intensified process is tiny due to the micro channel volume and cannot be a failure leads to fatal accident normally, but it may influence the dynamic of the process and it is rather difficult to calculate the changes online. In this paper, we treat the parameter uncertainty as an unmeasured state, and employ an observer as soft sensor to estimate it, unlike other literature, the estimation here is not for fouling detection but for more accurate model dynamics, and to ensure the value of the variable is within acceptable parameter, (e.g., upper and lower bounds of the process variable value).

To rewrite the whole model in the form of state equations,

due to the assumption that every element behaves like a perfectly stirred tank, we suppose that one cell can keep the main feature of the qualitative behavior of the reactor. For the sake of simplicity, only one cell has been considered.

Let us delete the subscript k for a given cell.

Define the state vector as x1T = [x11, x12]T = [Tp, Tu]T, unmeasured state x2T = [x21, x22]T = [hu, hp]T , ddhtp = dhu = ε(t) , ε(t) is an unknown but bounded function refers dt to variation of h, the control input u = Tui, the output vector T of measurable variables yT = [y1, y2]T = [Tp, Tu] , then the equation (1) and (2) can be rewritten in the following state-space form: Where, F1(x1) = ( ρpCppVp

ẋ 1 = F1(x1)x2 + g1(x1, u) { ẋ 2 = ε(t) y = x1

(3) A (Tp − Tu) 0

A ρuCpuVu 0 (Tu − Tp) ) , (Tpi−Tp)Fp and g1(x) = ( Vp ) , Tpi, Tui is the output of previ(Tui−Tu)Fu

Vu ous cell, for the first cell, it is the inlet temperature of process fluid and utility fluid.

In this case, the full state of the studied system is given as:

ẋ = F(x1)x + G(x1, u) + ̅ε(t) { y = Cx (4) Where x = [xx1] , F(x1) = (0 F1(x1)) , G(x1, u) = 2 0 0 (g1(0x, u)) , C = (I 0), ̅ε(t) = (ε(0t))

3 Fault detection and diagnose scheme

3.1 Observer design for sensor FDI

The extended high gain observer proposed by [ 15 ] can be used like an adaptive observer for estimation both states and parameters simultaneously, in this paper, the latter capability is utilized to estimate incipient degradation of overall heat transfer coefficient (due to fouling), thus guaranteeing a more accurate approximation of the temperature. It is quite useful in chemical processes since parameters are usually with uncertainties and unable to be measured. { y = Cx ẋ = F(x1)x + G(x1, u) (5) where x = (x1, x2)T ∈ ℛ2n, x1 ∈ ℛn is the state, x2 ∈ ℛn is the unmeasured state, x2 = ϵ(t), u ∈ ℛm, y ∈ ℛp are input and output, ϵ(t) is an unknown bounded function which may depend on u(t), y(t), noise, etc., and F(x1) = ( 0 0

0 F1(x1)) , G(x1, u) = (g1(x, u) ), C(I 0), 0

F1(x1) is a nonlinear vector function, g1(x, u) is a matrix

function whose elements are nonlinear functions. Supposed that assumptions related boundedness of the states, signals, functions etc. in [ 15 ] are satisfied, the extended high gain observer for the system can be given by: x̂̇ = F(x̂1)x + G(x̂1, u) − Λ−1(x̂1)Sθ−1CT(ŷ − y) (6) { ŷ = Cx̂ Where: Λ( ̂1) = [ 0 1( ̂1) 0 is the unique symmetric positive definite matrix satisfying the following algebraic Lyapunov equation: θSθ + ATS

θ + Sθ A − CTC = 0 0 0

I 0 Where A = [

] , θ > 0 is a parameter define by [ 15 ] and the solution of eq. (7) is:

Sθ = [ θ 1

I − θ12

I − θ12 I] 2 θ3

I

Then, the gain of estimator can be given by:

H = Λ−1(x̂1)Sθ−1CT = Λ(x̂1) [θ2F1−1(x̂1) ] 2θI

Notice that larger ensures small estimation error. However, very large values of are to be avoided in practice due to noise sensitiveness. Thus, the choice of is a compromise between fast convergence and sensitivity to noise.

3.2 Sensor fault detection and isolation scheme

The above observer could guarantee the heat-exchanger/re

actor dynamics ideally. Then, a bank of the proposed observers, together with sensor measurements, are used to generate robust residuals for recognizing faulty sensor.

Thus, we propose a FDI scheme to detect, meanwhile, isolate and recovery the sensor fault.

3.2.1 Sensor faulty model

pected output when it is healthy, that is: A sensor fault can be modeled as an unknown additive term in the output equation. Supposed θsj is the actual measured output from jthsensor, if jthsensor is healthy, θsj=yj , while if jthsensor is faulty, θsj = y jf = yj + fsj, ( for t ≥ tf and lim|yj − θsj| ≠ 0.That means yjf is the actual output of the jtt→h∞sensor when it is faulty, while yj is the exis the fault), y ;

i θsi = { yif = yi + fsi; jthsensorwhen itis faulty (10) jthsensorwhen itis faulty

Consider a nonlinear system as the form: With this formulation, the faulty model becomes:

{ y = Cx + Fsfs ẋ = F(x1)x + G(x1, u) + ̅ε(t) (11) is the fault distribution matrix and we consider that fault vector ∈ ℛ

( is the ℎ element of the vector) is also a bounded signal. Notice that, a faulty sensor may lead to incorrect estimation of parameter. That is why we emphasized healthy measurement for parameter fault isolation as mentioned above.

3.2.2 Fault detection and isolation scheme

The proposed sensor FDI framework is based on a bank of observers, the number of observers is equal to the number of sensors. Each observer use only one sensor output to estimate all the states and parameters. First, assumed the sensor used by ith observer is healthy, let yi denotes the ith system output used by the ithobserver. Then we form the observer as: 1 ≤ i ≤ p{ x̂̇ i = F(x̂1i)x + G(x̂1i, u) + Hi(yi − ŷii) ŷi = Cx̂i (12) tion :

Then we get:

Theorem 1: Define eix = x̂i − x, eiy = Ceix, eiyj = ŷji − yj, rji(t) = ‖ŷji − yj‖, μi = ‖rji(t)‖ ≔ sup‖rji(t)‖, for t ≥ 0.

Where i denotes the ithobserver, ŷii, ŷjidenotes the ith, jth estimated system output generated by the ithobserver, Hi is the gain of ith observer determined by the following equaHi = Λ−1(x̂1)Sθ−1CT = Λ(x̂1) [θi2F1−1(x̂1) i

] 2θiI

If the lth sensor is faulty, then for system of form (4), the

observer (12) has the following properties: For i ≠ l , ŷi = y asymptotically For i = l, ŷi ≠ y

Proof: If the lth sensor is faulty, then:

For i ≠ l, means that fsi = 0, yi = θsi , we have: tl→im∞eix = lim(x̂i − x) = 0 (13)

t→∞ we have:

Then the vector of the estimated output ŷi generated by ith observer guarantee ŷi = y after a finite time.

For i = l, means that θsl = ylf = yl + fsl, fsl ≠ 0 , the observer is designed on the assumption that there is no fault occurs, because there is fault fsl exit, so the estimation error e lx = 0 asymptotically cannot be satisfied, then : t→∞ lim(x̂i − x) = lim(x̂l − x) ≠ 0 t→∞ (14) ė lx = F(x̂1i, u) elx − HiG(x̂1i, u, fsl) elx (15)

Then the vector of the estimated output ŷi generated by the

ithobserver is different from y, that is ŷi ≠ y.⊡ As mentioned above, the observers are deigned under the assumption that no fault occurs, furthermore, each observer just subject to one sensor output. Residual rii is the difference between the ithoutput estimation ŷii determined by the ithobserver and the ithsystem output yi, then Theorem

2 formulates the fault detection and isolation scheme. (7) (8)

(9) Theorem 2:

If the lth sensor is faulty, then:

For i ≠ l, we have:

fsi = 0, yi = θsi (16) thus ŷii converges to yi asymptotically, we get:

rii = ‖ŷii − yi‖ ≤ μi (17) For i = l, we have: fsl ≠ 0, θsl = ylf = yl + fsl ≠ yl, then ŷll could not track yl correctly: rll = ‖ŷll − yl‖ ≥ μl (18) Therefore, in practice, we can check all the residuals rii, for 1 ≤ i ≤ p, if rii ≥ μi denotes that ithsensor is faulty, then the sensor fault detection and isolation is achieved.

The residuals are designed to be sensitive to a fault that

comes from a specific sensor and as insensitive as possible to all the others sensor faults. This residual will permit us to treat not only with single faults but also with multiple and simultaneous faults.

Let rsi denotes the fault signature of the ithsensor, define: rsi(t) = { 1 ifrii ≥ μi; ithsensoris faulty 0 if rii ≤ μi; ithsensorishealth (19)

3.2.3 Fault identification and handling mechanism

1) Fault identification

Supposed there are m healthy sensors and p − m faulty ones, then to identify the faulty size of ith sensor, use m estimated output ŷim generated by m observers which use healthy measures, 1 ≤ m ≤ p − 1, m ≠ i , define ̂fsi as the estimated faulty value of the ithsensor, then: ∆ ̂fsi = m1 ∑im=1|ŷim − θsi| → fsi (20)

2) Fault recovery

As mentioned above, the extended high gain observer is also worked as a software sensor to provide an adequate estimation of the process output, thus replacing the measurement given by faulty physical sensor. θsi is the actual measured output from ithsensor: yi θsi = {yif = yi + fsi (21) Let m observers use healthy measurements as the soft sensor for ithsensor, define: y̅i = 1 m m ∑ ŷm i

(22) i=1

If ithsensor is healthy, let the sensor actual output as θsi

its output, while if it is faulty, let y̅i to replace θsi , that is: yi = {θsi, ifithsensorhealthy (23)

y̅i, if ithsensorfaulty

3.3 process fault diagnose

In order to achieve process FDD, healthy measurements are

fed to a bank of parameter intervals filters developed in [ 11 ] to generate a bank of residuals. These residuals are processed for identifying parameter changes, which involves variation of overall heat transfer coefficient in this paper.

The main idea of the method is as follows.

The practical domain of the value of each system parameter is divided into a certain number of intervals. After verifying

all the intervals whether or not one of them contains the faulty parameter value of the system, the faulty parameter value is found, the fault is therefore isolated and estimated. The practical domain of each parameter is partitioned into a certain number of intervals. For example, parameter hp is partitioned into q intervals, their bounds are denoted by h(p0), h(p1), … , h(pi), … , h(pq) . The bounds of ithinterval are h(i−1)and h(pi) , are also noted as hbpi and hapi, and the nomip nal value for hp denotes by hp0 .

To verify if an interval contains the faulty parameter value of the post-fault system, a parameter filter is built for this interval. A parameter filter consists of two isolation observers which correspond to two interval bounds, and each isolation observer serves two neighboring intervals. An interval which contains a parameter nominal value is unable to contain the faulty parameter value, so a parameter filter will not be built for it.

Define Eq. (3) into a simple form as:

{ẋ 1 = F1(x1)x2 + g1(x1, u) = {ẋ 1 = f(x1, hp, u) (24) y = x1 y = x1

The parameter filter for ith interval of hp is given below.

The isolation observers are: x̂̇ ai = f(x̂1, hapi0, u) + H(y − ŷai) { ŷ̇ ai = cx̂̇ ai

εai = y − cx̂̇ ai { ŷ̇ bi = cx̂̇ bi εbi = y − hx̂̇ bi x̂̇ bi = f(x̂1, hbpi0, u) + H(y − ŷbi) Where: hapi0(t) = { hp0, t < tf h(pi), t ≥ tf hp0, t < tf , hbpi0(t) = {h(i−1), t ≥ tf ,(27) p

The isolation index of this parameter filter is calculated by:

νi(t) = sgn(εai)sgn(εbi) (28) As soon as νi(t) = 1, the parameter filter sends the ’noncontaining’ signal to indicate that this interval does not contain the faulty parameter value. And if the fault is in the ith interval. Let: ĥA = 1 (haiA + hbiA) (29)

2 to represent the faulty value, fault isolation and identification is then achieved. 4

Numerical simulation

A case study is developed to test the effectiveness of the proposed scheme. The real data is from a laboratory pilot of a continuous intensified heat-exchanger/reactor. The pilot is made of three process plates sandwiched between five utility plates, shown in Fig.1. More Relative information could be found in [ 2 ]. As previously said, the simulation model is considered just for one cell which may lead to moderate inaccuracy of the dynamic behavior of the realistic reactor.

However, this point may not affect the application and demonstration of the proposed FDD algorithm encouraging results are got.

(25) (26) sign; (c) the heat exchanger/reactor after assembly.

The constants and physical data used in the pilot are given

in table1. ture in utility fluid is time-varying between 15.6℃ and 12.6℃, which is a classical disturbance in the studied system, as shown in Fig.2. The inlet temperature in process fluid is 76℃. Initial condition for all observers and models are supposed to be T̂ 0 = T̂ 0 = 30℃, hA = 214.8 W. K−1 .

Fig.2 utility inlet temperature

4.2 High gain observer performance

To prove the convergence of the observers and show their

tracking capabilities, suppose the heat transfer coefficient subjects to a decreasing of ℎ = (1 − 0.01 )ℎ and followes by a sudden jumps of 15 at = 100 . These variations and observer estimation results are reported in Fig.3.

4.3 Sensor FDI and recovery demonstration

In order to show effectiveness of the proposed method on sensor FDI, multi faults and simultaneous faults in the temperature sensors are considered in case 1 and case 2 respectively. Besides, the pilot is suffered to parameter uncertainties caused by heat transfer coefficient decreases with ℎ = (1 − 0.01 )ℎ. Two extended high gain observers are designed to generate a set of residuals achieving fault detection and isolation in individual sensors. Observer 1 is fed by output of sensor to estimate the whole states and parameter while observer 2 uses output of sensor . Advantages of the proposed FDI methodology drop on that if one sensor is faulty, we can use the estimated value generated by the healthy one to replace the faulty physical value, thus providing a healthy virtual measure.

Case 1: abrupt faults occur at output of sensor at t=80s, 100s, with an amplitude of 0.3℃, 0.5℃ respectively.the results are reported in Fig.5-8.

1, red curve demonstrates the estimated value while black one is the measured value.

It is obviously that since t=80s, ̂ (red curve) cannot track (black curve) correctly, while it needs about 0.2s for ̂ to track at t=80s and t=100s. It suggests that faults occur, then the following task is to identify size and location of faulty sensors. Fig.6 and Fig.7 achieves the goal. It takes 0.1s and 0.3s for isolating the faults at 80s, 100s respectively. to generate a health value for faulty sensor . Observer 2 uses only measured to estimate all states and parameters. Therefore, ̂

, ̂ generated by observer 2 are only decided by . In case 1, faults occur only on sensor , sensor is healthy, that is to say ̂

, ̂ generated by observer 2 will be satisfied their expected values. As shown in Fig.8, we can see that since is healthy, estimated value ̂ tracks measured perfectly, while estimated value ̂ (red curve) does not track the faulty measured value (black curve), ̂ (red curve) illustrates the expected value for sensor , we can use estimate ̂ (red curve) to replace measured faulty value

If there are faults occurred only on output of sensor , the

same results can be yield easily. For multi and simultaneous faults on both sensors, we can still isolate the faults correctly. Case 2 will verify this point.

Case 2: simultaneous faults imposed to the outputs of sen

sors as in case 1 and at t=80s with amplitude of 0.6℃.

Results are reported in Fig.9-10. Residuals are beyond their threshold obviously at time 80s, 100s.

It can be seen from Fig.9, Fig .10 that the proposed FDI

scheme can isolate faults correctly, and it takes 0.25s, 0.4s for isolating the faults in sensor at 80s, 100s and 0.2s for isolating that in sensor at t=80s respectively. Compared with Case 1, more times is needed in this Case 2.

4.4 Fast process fault isolation and identification

Process fault is related to variation of overall heat transfer coefficient (h). The heat transfer coefficient is considered as variable which undergoes either an abrupt jumps (by an expected fault in the flow rate) or a gradual variation (essentially due to fouling). For incipient variation, since fouling in intensified heat-exchanger/reactor is tiny and only influence dynamics, we have employed extended high observers to ensure the dynamic influenced by this slowly variation.

Therefore, the abrupt changes in heat transfer coefficient ℎ can only be because of sudden changes in mass flow rate. It implies that the root cause of process fault is due to actuator fault in this system.

Supposed an abrupt jumps in ℎ at t=40 from 214.8 to 167. From Fig.11, at t=40s, unlike sensor fault cases, the residual leaves zero and never goes back, this indicates that process fault occurs. For fast fault isolation and identification, we use the methodology of parameter interval filters developed in [ 11 ]. In [ 2 ], heat transfer coefficient ℎ changes between 130.96 and 214.8, then ℎ is divided into 4 intervals as shown in table 2 and simulation results are shown in Fig.12. It can be seen at t=40s, only index for interval 150-170 goes to zero rapidly, then there is a fault in this interval. The faulty value is estimated by ℎ̂ = (ℎ + ℎ ) = 170) = 160. We can see it is closely to actual faulty value 167, and if more intervals are divided, the estimated value may be closer to the actual faulty value. 2 1 (150 + 1 2

5 Conclusion

An integrated approach for fault diagnose in intensified heat-exchange/reactor has been developed in this paper. The approach is capable of detecting, isolating and identifying failures due to both sensors and parameters. Robustness of the proposed FDI for sensors is ensured by adopting a soft sensor with respect to parameter uncertainties. Ideal isolation speed for process fault is guaranteed due to adoption of parameter interval filter. It should be notice that the proposed method is suitable for a large kind of nonlinear systems with dynamics models as the studied system. Application on the pilot heat-exchange/reactor confirms the effectiveness and robustness of the proposed approach.

Proceedings of the 26th International Workshop on Principles of Diagnosis 260

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