Na

Fa

b

a

ARR2AA

KCN

1

ptrtstlac

detbmmm

vI[

d

m

0h

Journal of Process Control 24 (2014) 73–80

Contents lists available at ScienceDirect

Journal of Process Control

j ourna l ho me pa ge: www.elsev ier .com/ locate / jprocont

onlinear observer of the thermal loads applied to the helium bath of cryogenic Joule–Thompson cycle

. Bonnea,b,∗, M. Alamirb, P. Bonnaya

UMR-E 9004 CEA/UJF-Grenoble 1, INAC, SBT, 17 rue des Martyrs, 38054 Grenoble, FranceCNRS-University of Grenoble, Control Systems, Department of Gipsa-Lab, 11 rue des mathmatiques, 38402 Saint-Martin d’Hères, France

r t i c l e i n f o

rticle history:eceived 19 July 2013eceived in revised form

a b s t r a c t

In this paper, a nonlinear observer of the thermal loads applied to the helium bath of a cryogenic refrig-erator is proposed. The thermal loads represent a time-varying thermal disturbance expected to takeplace in future tokamaks refrigerators such as those used in the cooling systems for the International

5 November 2013ccepted 23 December 2013vailable online 25 January 2014

eywords:ryogenic helium refrigerator

Thermonuclear Experimental Reactor (ITER) or the Japan Torus-60 Super Advanced (JT-60SA). The pro-posed observer can serve as a monitoring tool for cryogenic operators and/or in observer-based advancedcontrol strategies. The observer is based on a part of the nonlinear model of the refrigerator. The paperdetails how the physical model of the Joule–Thompson cycle is obtained and the structure of the observerand validates its performance using experimental data.

onlinear load observer

. Introduction

Large superconducting tokamak devices produce significantulsed heat loads on magnets, due to huge eddy currents encoun-ered in the magnetic system, to AC losses and to neutron fluxadiations coming from the plasma. Such high pulsed loads disturbhe cryogenic plant that are cooling magnets, and make it neces-ary to use appropriate control strategies. The aim is to maintainhe stability of the overall process subject to the variable thermaload and to satisfy operational and safety constraints (turbine oper-tional temperature range, maximum capacity of the helium tank,ompressor suction and discharge pressures, etc.).

Currently, technological solutions (such as thermal buffers, asescribed in [1], and by-pass valves) are studied to smooth theffect of the thermal disturbance on the cryoplant and to avoidhe over-dimensioning of the process. These solutions have toe combined with specific control algorithms, resulting in opti-ally designed closed-loop systems that can operate near theiraximum capacity without the need for too conservative securityargins.The recent interest in advanced control methodologies has moti-

ated many studies on modelling and control of cryogenic plants.n particular, several dynamic simulators have been proposed by2–6] for operator training, dimensioning and/or control design.

∗ Corresponding author at: UMR-E 9004 CEA/UJF-Grenoble 1, INAC, SBT, 17 ruees Martyrs, 38054 Grenoble, France. Tel.: +33 438784885.

E-mail addresses: francois.bonne@cea.fr, bonne.francois@free.fr (F. Bonne),azen.alamir@grenoble-inp.fr (M. Alamir), patrick.bonnay@cea.fr (P. Bonnay).

959-1524/$ – see front matter © 2014 Elsevier Ltd. All rights reserved.ttp://dx.doi.org/10.1016/j.jprocont.2013.12.015

© 2014 Elsevier Ltd. All rights reserved.

Based on a better dynamic modeling of the underlying process,advanced control schemes have been proposed which were oftendedicated to a particular key variable. For instance, scalar modelpredictive control (MPC) of the helium bath temperature at 1.8 Kusing a Joule–Thomson expansion valve has been proposed in [7].In [8], the problem of control of the bath pressure is addressed,while in [9,10], the high pressure level is monitored in order tocontrol the bath level. In [11], the optimal multivariable control ofa refrigerator is proposed, considering pulsed heat loads. One ofthe key quantities that affect the behaviour of the cryogenic plantis the thermal loads that are applied to the circuits to be cooled.These loads are unpredictable from the cryogenic plant side andhave to be considered as an unmeasured disturbance that has to bereconstructed using appropriate observers.

In the present contribution, a nonlinear observer is devel-oped for an experimental helium refrigerator facility (availableat CEA1-INAC2-SBT3, Grenoble, France) to estimate the thermalloads applied to the helium bath. This estimation can be used forsupervision task and/or to be incorporated into advanced controlstrategies. The paper is organized as follows: the cryogenic plantis first presented in detail in Section 2. Section 3 introduces thedynamic model of the Joule–Thompson cycle, the subsystem where

helium is liquefied in a bath and vaporized by a thermal load. Sec-tion 4 shows the validity of the model by comparing simulationresults with experimental data. Section 5 details the design of the

1 CEA: Commissariat à l’Energie Atomique et aux Energies Alternatives2 Institut NAnosciences et Cryogénie3 Service des Basses Températures

74 F. Bonne et al. / Journal of Process Control 24 (2014) 73–80

F pressob

ndtt

ie

2

S4dt

m

--

-

--

-

eN(towcTwpLhfi

3

tviw

ters (e.g. density or specific heat). Particular names for input andoutput variables will be given at the end of each section. Inputswill referred to with the letter u if they are manipulable and w if

S1

S2

S3

S4

NS1

CV155

NEF1NEF1

NEF2 Stt20 7

NEF34 CV156

NEF5

NEF6

CV167

LN2

GN2

CV952 CV953

CV956

NC1

NC2

NCR22

h

ig. 1. Views of the cryogenic plant of CEA-INAC-SBT, Grenoble. (a) The screw comox.

onlinear observer. Finally, experimental validation of the observeresign is proposed in Section 6 using the cryogenic facility men-ioned above. The paper ends with a conclusion that summarizeshe contribution and gives ideas for future work.

In the following, all physical quantities are expressed with thenternational system of units (SI) except for pressure which isxpressed in bars.

. Overview of the cryogenic plant

Fig. 1 shows an overview of the cryogenic plant of the CEA-INAC-BT, Grenoble. This plant provides a nominal cooling capacity of00 W at 4.4 K in the configuration in which this study have beenone. It is dedicated to physical experiments (cryogenic componentesting, turbulence and pulsed heat load studies, etc.).

The process flow diagram of the cryoplant is shown in Fig. 2. Oneay notice the following main elements:

Two volumetric screw compressors, a set of control valves. Several counterflow heat exchangers, a liquid nitrogen pre-cooler.

A cold turbine expander which extracts work from the circulatinggas (Stt207).

A so-called turbine valve (CV156). A Joule–Thomson expansion valve for helium liquefaction(CV155).

A phase separator (NS1), connected to the loads (simulated hereby the heating device referred as NCR22).

Note that the plant can be viewed as the interconnection of fourlementary subsystems: the Warm Compression Station (S4), theitrogen Pre-Cooler (S3), the Brayton Cycle (S2) and the JT cycleS1), delimited by dotted lines in Fig. 2. The system is currently con-rolled by three independent controllers. The output temperaturef the turbine expander is controlled with a PI controller workingith the turbine valve CV156. The valve CV155 is rather used at a

onstant opening set by the user, depending on the application.he level of liquid helium in the tank is controller by a PI controller,orking with the heating device NCR22. Finally, the high and lowressures (in red and blue pipes, respectively) is controlled by anQ controller, like the one described in [12]. In order to observe theeat deposited in the helium bath, attention will be focused on therst subsystem, namely the JT cycle.

. Model of the Joule–Thompson cycle

In order to model the Joule–Thompson cycle, three elemen-

ary objects have to be modelled: the Joule–Thompson expansionalve, the heat exchanger and the phase separator. A particular-ty of our elementary objects to model is that they are all dealing

ith gas or liquid. They consequently may have the same set of

r of the warm compression station. (b) The cold box. (c) Internal detail of the cold

inputs and outputs (i.e. inlet and outlet temperature) or parame-

Fig. 2. Functional overview of the 400 W at 4.4 K helium refrigerator available atCEA-INAC-SBT, Grenoble. The components named CV are controlled valves, used tocontrol the system. The label Stt stands for the cryogenic turbine while NS is usedfor the phase separator. NC’s are helium compressors while NEF’s stand for heatexchangers.

F. Bonne et al. / Journal of Process Control 24 (2014) 73–80 75

T Hin, P H

in P Cin

MHout MG

out, MLout

CV155

tuama(g

3

htoh

vit

M

C

X

wbbflttctflic

H

wte

�

F

M

f

y

MCout, T C

out, P Cin MG

in, MLin, P C

out

Mvap

h

NCR22

Fig. 3. Synoptic view of a Joule–Thompson valve.

hey depict uncontrollable boundary conditions. The letter y will besed for outputs. Note that all helium properties (such as densitynd specific heat) used by models are given by HEPAK©, an Excelacro which computes them using fundamental state equation

ccording to the literature referenced in [13]. Value of parameterssuch as volumes and surfaces) and variables (such as densities) areathered in Table 3.

.1. Joule–Thompson valve

Since the gas is cooled enough by the cryogenic turbine and theeat exchangers, the liquefaction of the gas consists of an isen-halpic expansion through the Joule–Thompson valve. Dependingn the boundary conditions TH

in, PH

inand PC

indepicted in Fig. 3, liquid

elium is produced.The model of the JT valve expresses the flowrate MH

out into thealve and the mixture quality �. The manipulable input of the valves called CV155. Like any other control valve, the flow MH

out throughhe JT valve can be expressed as [14]:

Hout = 2.4 × 10−5 · Cv ·

(1 − X

3 · Xc

)√� · PH

in· X (1a)

v = Cvmax

Rv

(exp(

CV155

100log Rv

)−(

1 − CV155

100

))(1b)

= min

(PH

in− PC

in

PHin

, Xc

), Xc = �

1.4Xt (1c)

here � is the heat capacity ratio and � the density of the fluid,oth under pressure PH

inand temperature TH

in. Xt is a constant given

y the manufacturer. The min(.) function in Eq. (1c) comes from theow behaviour: if the flow is subsonic, the X parameter is a func-ion of the pressure difference. If the pressure ratio is high enough,he flow is sonic and the X parameter is bounded by the so-calledritical ratio, Xc. Rv and Cvmax are valve sizing constants, respec-ively, the rangeability and the flow coefficient. A part of the gasow expressed in Eq. (3.1) is liquefied during the expansion. Assum-

ng isenthalpic expansion into equilibrate liquid-gas mixture, onean write:

in = Hout = � · HGout + (1 − �) · HL

out (2)

here Hin and Hout denote enthalpies of the fluid before and afterhe expansion; HG

out and HLout denote the saturated gas and liquid

nthalpies. Therefore, the mixture quality can be written as:

= Hin − HLout (3)

HGout − HL

out

inally, the output gas and liquid flows can be written as:

Gout = (1 − �) · MH

out, MLout = � · MH

out (4)

In the following, the model of the JT valve is written in theollowing compact form:

vjt = f vjt(uvjt, wvjt) (5)

Fig. 4. Synoptic view of a phase separator.

where outputs, control effort and boundary conditions are:

yvjt =

⎛⎜⎝

MHout

MGout

MLout

⎞⎟⎠ , uvjt = CV155, wvjt =

⎛⎜⎝

THin

PHin

PCin

⎞⎟⎠ (6)

3.2. Phase separator

After helium liquefaction, the behaviour of the liquid accu-mulation into the phase separator needs to be expressed. Thiscomponent will be modelled assuming liquid–gas thermodynamicequilibrium.

Using the notations of Fig. 4 regarding the boundary conditionsML

inMG

inand PC

in, the phase separator model expresses the behaviour

of the liquid level h, the outflow rate MCout and temperature TC

out .Assuming cylindrical tank, the mass of liquid in the bath can bewritten as:

ML = �L · S · h (7)

where �L, S and h, respectively, denote the density of the liquid,the cross-section of the tank and the height of liquid helium in thetank. By differentiation of Eq. (7) w.r.t. time, one can obtain:

h = ML − �L · S · h

�L · S(8)

and according to the mass conservation law:

ML = MLin − Mvap (9)

The quantity of vaporized helium by heating is equal to:

Mvap = NCR22

Lv(10)

where NCR22 and Lv denote the thermal loads applied to the bath(simulated here by a heating device) and the liquid helium latentheat. Considering that density variation could be omitted as neg-ligible in the case of a small pressure deviation, combining Eqs.(8)–(10) leads to:

h = MLin− (NCR22/Lv)

�L · S(11)

With the similar approach, the quantity of gas in the tank can beexpressed as:

MG = �G · S · (hmax − h) (12)

in which hmax denotes the height tank. By differentiation of(12) w.r.t. time:

MG = �G · S · (hmax − h) − �G · S · h (13)

and according to the mass conservation law:

MG = MGin + Mvap − MC

out (14)

76 F. Bonne et al. / Journal of Process Control 24 (2014) 73–80

T Cout, P C

in, MCout P H

in, T Hin, MH

out

T C , P C , MC P H , T H , MH

Fv

M

Tb

T

T

P

Ibf

x

y

w

x

3

td

P

Tm

-

---

beew[

�

�

T Hin

P Hin

MHout

T Cout

P Cin

MCout

T Hout

P Hout

MHin

T Cin

P Cout

MCin

T H(t, z)

Qex(t, x)

T C(t, z)

0 Lz

Fig. 6. Continuous heat exchanger. TH(t, z) and TC(t, z) denote the evolution of thetemperature with the time and the z coordinate. Qex(t, z) represents the heat fluxfrom hot flow to cold flow. TH(t, 0) is equal to TH

inand TC(t, L) is equal to TC

in.

Qex1 · · · Qex

i · · · QexN−1 Qex

N

T H0 T H

1 · · · T Hi · · · T H

N−1 T HN

T C0T C

1· · ·T Cj· · ·T C

N−1T CN

Fig. 7. Spatially discretized heat exchanger. TH and TC denote the temperatures at

in out in out out in

Fig. 5. Synoptic view of a counter flow heat exchanger.

inally, combining Eqs. (12)–(14) and considering again the densityariation as negligible leads to:

Cout = MG

in +�G

�L

(ML

in −NCR22

Lv

)(15)

he temperature of the outflow, assuming saturated vapour, is alge-raically given by Hepack©:

Cout = f (PC

in) (16)

he output pressure PCin

is propagated through PCout algebraically:

Cout = PC

in (17)

n what follows, the dynamic model of the phase separator wille recalled according to Eqs. (15), (11), (16) and (17), with theollowing condensed form:

˙ ps = f ps(xps, wps) (18a)

ps = gps(xps, wps) (18b)

here states, outputs and inputs boundary conditions are:

ps = h, yps =

⎛⎝ h

MCout

TCout

⎞⎠ , wps =

⎛⎜⎜⎜⎜⎝

NCR22

MLin

MGin

Pin

⎞⎟⎟⎟⎟⎠ (19)

.3. Heat exchanger

The counter flow heat exchanger is the subsystem which allowso recover the enthalpy of the evaporated cold gas, in order to coolown a warmer gas.

Depending on the boundary conditions THin

, TCin

, MHin

, MCin

, PHin

andCin

presented in Fig. 5 the model expresses the output temperaturesHout and TC

out , pressures PHout and PC

out , flowrates MHout and MC

out . Theodel is derived using the following assumptions:

Pressures are assumed to be linearly decreasing along the chan-nels.

Flowrates are assumed homogeneous all along the channels. Aluminium longitudinal conductivity is assumed to be negligible. Aluminium transverse conductivity is included in the overall heattransfer coefficient.

One’s attention will be focused first on the temperatureehaviour. Since the system is transferring heat along each heatxchanger channel, partial differential equation (PDE) based mod-lling is the appropriate choice. The temperature is to differentiateith respect to the time and the spacial coordinate, as described in

15], according to the notation of Fig. 6.

H H H ∂THH H ∂TH

ex

Cp S∂t= M Cp

∂z− Q (20a)

CCpCSC ∂TC

∂t= MCCpC ∂TC

∂z+ Q ex (20b)

i j

the edges of zone i and N − i + 1. Qi represents the heat flux from hot flow to coldflow. TH

0 is equal to THin

and TC0 is equal to TC

in.

Q ex = kdS

dz(TH − TC ) (20c)

where �H and CpH are helium properties under pressure PH andtemperature TH, �C and CpC being under PC and TC. SH and SC denotethe cross-section the tube containing the in the heat exchanger, ineach pipe. S is the exchange surface and k the overall heat transfercoefficient. In order to express the derivative of the temperatureonly w.r.t. time, a spacial discretization has to be done, using a finitenumber N of elementary zones, as it is done in [16], [17] or [18] (seeFig. 7). The dynamic behaviour of zone (or cell) i can be expressedby the following couple of differential equations:

�Hi

CpHi VH

NTH

i = MHCpHi (TH

i−1 − THi ) − Q ex

i (21a)

�Cj

CpCj VC

NTC

j = MCCpCj (TC

j−1 − TCj ) + Q ex

i (21b)

Q exi =

kS�TM

N, j = N − i + 1 (21c)

�TM denotes the mean difference in temperature between the hotfluid and the cold fluid. This methodology leads to a model with 2NODEs.

According to the stated modelling assumptions, outflows areequal to inflows:

MHN = MH

0 = MH, MCN = MC

0 = MC (22)

Pressures output can be modelled by:

2 2

PH

N = PH0 − KH · MH , PC

N = PC0 + KC · MC (23)

where KH and KC denote the pressure drop coefficient due to fric-tions.

F. Bonne et al. / Journal of Process Control 24 (2014) 73–80 77

y2→ 11,3 , y2← 1

3 y2→ 12 , y2← 1

1,2

wex16 , yex1

2,4 wex11,5 , yex1

3

wex12,4 , yex1

6 wex13 , yex1

1,5

wvjt1,2 , yvjt

1

uvjt

wvjt3 , yvjt

2,3

wps14 , yps1

2,3 wps12,3 , yps1

4

yps11

wps11

NEF1

CV155

NS1

Fig. 8. Detailed Joule–Thompson Cycle. Variables having the letter y depicts theoutput of an object, corresponding to the input w of another one, subscripts cor-r ex1 ps1 vjt

i

e

x

y

I

x

3

etn

io

x

y

wd

1 2 3 4 5 6 7

1.121.141.161.18

P (

Bar

s)

(a)

1 2 3 4 5 6 7

55.5

66.5

T (

K)

(b)

1 2 3 4 5 6 7

8

10

T (

K)

(c)

1 2 3 4 5 6 7

−0.06−0.04−0.02

00.02

dh/d

t (m

/s)

(d)

time (hours)

Fig. 9. Model validation result. Red lines represent computed values when blacklines represent the measured outputs. (a) Pressure inside the phase separator. (b)Inlet temperature of the JT valve. (c) Output temperature of the JT cycle. (d) Filtered

esponding of the elements of the vector (e.g. w2,4 = y2,3). u is the manipulatednput.

In the following, the dynamic model of the heat exchanger isxpressed according to Eqs. (21)– (23) using the compact form:

˙ ex = f ex(xex, wex) (24a)

ex = gex(xex, wex) (24b)

n which states, outputs and boundary conditions are:

ex =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

TH1

TCN

...

THN

TC1

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

, yex =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

THN

TCN

MHN

MCN

PHN

PCN

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

, wex =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

TH0

TC0

MH0

MC0

PH0

PC0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(25)

.4. Complete Joule–Thompson cycle model

Now that the three components of the JT cycle have been mod-lled, the only thing which remains to be done is to assemble themo obtain the complete model. To do so, one must achieve the con-ections depicted in Fig. 8, following arrows depicting pipes.

When rewriting all the objects submodel and when identify-

ng variables which are boundary variables for one submodel bututput of another, it ends up with the following global model:

˙ 1 = f 1(x1, u1, w1, y1←2) (26a)

1 = g1(x1, u1, w1, y1←2) (26b)

here x1 gathers the state vectors, u1 gathers the actuator, w1

enotes the thermal loads (simulated by the heating device NCR22,

bath level derivative. (For interpretation of the references to color in this figurelegend, the reader is referred to the web version of the article.)

to be estimated) while y1←2 represents the signals coming from theprevious subsystem (the Brayton cycle). Precisely:

x1 =(

xex

xps

), u1 = uvjt, w1 = wps

1 (27)

y1 gathers the measured outputs. Eq. (28) points out which outputis measured or not:

y1 = Cm

⎛⎜⎝

yps

yex

yvjt

⎞⎟⎠

︸ ︷︷ ︸y1

all

(28)

in which Cm is a matrix that select outputs available for measure-ment. In our case, the available outputs are the elements 1, 3, 4, 5,8 and 9 of the vector y1

all, corresponding to every expressed tem-

peratures and pressures for every objects model. Unfortunately, noflowrate measurement is available on that part of the process.

4. Model validation

Before the model derived in the preceding section is usedfor observer design, it needs to be validated by comparing themodel-based prediction to experimentally measured signals. Thevalidation is done focusing on the following key variables (seeFig. 9):

- The helium bath pressure yex6 .

- The inlet temperature of the JT valve wvjt1 .

- The output temperature of the JT cycle yex2 .

- The bath level derivative xps (filtered).

7 rocess Control 24 (2014) 73–80

fwitcia

wtcacenp

5

nwdbcTd

x

y

ws

w

Ie

x

y

wiro

x

y

wt

y2→ 11

y2→ 12,3

u1

w1

y1

x1

w1um

System

Observer

The experimental validation of such an observer have beendone for several scenarios including discontinuous heats loads. The

8 F. Bonne et al. / Journal of P

A limited number of elementary zones N = 4 (see Eq. (21)) is usedor the model of the heat exchanger, since it is an heat exchangerith a low temperature difference between the hot and the cold

nputs, while the temperature mean �TM (see Eq. (21c)) is choseno be the algebraic mean. The overall heat transfer coefficient k isalculated for the nominal flow with correlations that can be foundn [19]. For the model validation, it will be considered as a constant,nd estimated by the observer afterwards.

A careful look at Fig. 9 shows that the nonlinear model (Eq. (26))ell represents the dynamics of the process, except from time 5:15

o 5:30. It can be explained by lack of knowledge on the heat transferoefficient (chosen to be constant, to be estimated by the observer),nd also by the lack of information concerning the Joule–Thompsonycle input temperature, which is not measured (and will also bestimated). Note also that even filtered, the bath derivative is veryoisy, due to the measurement method. This leads to a specificroblem to be examined later on.

. Observer design

In this section, the nonlinear observer is derived based on theonlinear state space dynamic model (Eq. (26)). The heating power1 is assumed to be unknown. Since some of the boundary con-itions are not available from existing instrumentation, they wille considered as unknown as well. The overall heat transfer coeffi-ient k of the heat exchanger will also be considered to be estimated.ransforming these signals into disturbances in Eq. (26) leads to aynamic model of the form:

˙ 1 = f 1′ (x1, u1, w1m, w1

um) (29a)

1 = g1′ (x1, u1, w1m, w1

um) (29b)

here w1m and w1

um, respectively, denote measured and unmea-ured disturbances, precisely:

1um =

⎛⎜⎝

w1

y1←21

k

⎞⎟⎠ , w1

m = y1←22,3 (30)

n order to synthesize the observer gain, the model has been lin-arized around the steady state operating point of interest, namely:

˙ 1′ = f 1′(x10, u1

0, w1m0, w1

um0) = 0 (31a)

10 = g1′(x1

0, u10, w1

m0, w1um0) (31b)

hich is chosen to be the operating point obtained with the max-mal admissible continuous heat load (i.e. 450 W in the case of ourefrigerator). According to the Taylor’s series, limited to the firstrder:

˙ 1 = A1x1 + [ B1 B1m B1

um ]

⎛⎜⎝

u1

w1m

w1um

⎞⎟⎠ (32a)

˜1 = C1x1 + [ D1 D1m D1

um ]

⎛⎜⎝

u1

w1m

⎞⎟⎠ (32b)

w1um

here x, u, wm, wum, y, respectively, denotes the deviations fromhe linearization point. Considering the linearized state space

Fig. 10. Synoptic view of the observer. The observer output is the estimation ofthe extended state �, gathering the estimation of the state and unknown inputs(i.e. the thermal heat load, the hot input temperature and the overall heat transfertcoefficient).

model, the extended system is defined including the unmeasureddisturbances as components of the extended state vector [20]:(

x1

w1um

)=[

A1 B1um

0 0

]︸ ︷︷ ︸

A1aug

(x1

w1um

)︸ ︷︷ ︸

�

+[

B1 B1m

0 0

] (u1

w1m

)︸ ︷︷ ︸

�

(33a)

y1 =[

C1 D1um

]︸ ︷︷ ︸C1

aug

(x1

w1um

)+[

D1 D1m

]( u1

w1m

)(33b)

which is a 10 states Linear Time Invariant dynamic system for whicha standard optimal Kalman state estimator can be derived by solv-ing the standard infinite time Riccati equations.4 The system isconsidered to be observable since local observability can be shownby noticing that the rank of the observability matrix of the pair(A1

aug, C1aug) is column full-rank.

The linear correction gain matrix L will be used on an extendednon linear model based on the function f 1′ involved in Eq. (29).This extended non-linear model has been written in the same man-ner than Eq. (33), by adding unmeasured disturbances to the state

vector, leading to a function f1

that can be written like:

f1 =(

x1

w1um

)=(

f 1′(x1, u1, w1m, w1

um)

0

)(34)

where y1 and � denote vectors coming from the experiment,according to the previous definitions of and �. Using this nonlinearmodel, the observer can finally be written under the form:

ˆ� = f1(�, �) − L(y1

exp − g1′(�, �)) (35)

Fig. 10 represents the system and the observer, and show howthey are connected.

6. Experimental validation

4 In Matlab formalism, one can use the following command L =lqr(AT

aug, CTaug, Q, R)T where Q and R are weighting matrices.

F. Bonne et al. / Journal of Process Control 24 (2014) 73–80 79

Table 1Assigned weights of measured variables y1 used by the observer, corresponding tothe diagonal of the Q matrix used in the Matlab command lqr(.). It can be noticed thatpressure sensors have been ruled out by a high weigh because of a calibration prob-lem leading to unsatisfactory results. T and P stand for temperature and pressure,respectively.

Out. Variable Physical meaning Weight

1 yps11 Level of liquid helium 105

2 yps13 Outflow T of the tank 1014

3 yex11 Hot outflow exchanger T 10−1

4 yex15 Hot outflow exchanger P 1014

5 yex16 Cold outflow exchanger P 1014

6 yex62 Cold outflow exchanger T 10−1

Table 2Assigned weights to the extended state �, corresponding to the diagonal of the Rmatrix used in the Matlab command lqr(.).

St. State Physical meaning Weight

1 xps11 Level of liquid helium 1

2 xex11 Heat exchanger temperature 1 1

.

.

....

.

.

....

7 xex16 Heat exchanger temperature 6 1

8 w1um1 Heat load 105

wca

mtaJ

0 2 4 6 8 100

100

200

300

400

500

600

time (hours)

NC

R22

(W

atts

)

DepositedEstimated

0 1 2 3 4 5 6 7 80

100

200

300

400

500

600

time (hours)

NC

R22

(W

atts

)

DepositedEstimated

Fig. 11. Observation results. Black lines represent the heating power NCR22 (w1um2)

the observation is rather unaffected by the large deviation of the

TP

9 w1um2 HX hot input temperature 101

10 w1um3 HX heat transfer coefficient 103

eighting matrices Q and R of the MatLab command lqr(.) has beenhosen to be diagonal matrices. The weights assigned to sensorsnd state are detailed in Tables 1 and 2.

Fig. 11 shows two scenarios where the evolution of the esti-ated heat w1 is compared to the measurement. Note while

um2

hese measurements are available on our test-bed, the heat pulsesre unmeasured on the targeted applications such as the ITER or theT60SA installations. It is also worth mentioning that the quality of

able 3hysical variables of the model, their associate typical value and unit.

Component Variable

Type Name Typic

Valve Variable � 1.46

� 150

Xc 0.92

Hin 2.03 ×Hout 2.03 ×HG

out 3.06 ×HL

out 1.09 � 0.478

Constant Cvmax 0.5

Rv 11

Xt 0.72

Tank Variable h 0.60

Mvap 15.7 ×ML 12.0

�L 121

Lv 1.96 ×MG 654 ×�G 19.7

Constant S 0.166hmax 0.8

Heat exchanger Variable �H 144

CpH 3.94 �C 10.1

CpC 6.12 ×Qex 2.60 ×

Constant S 5.00

k 105

VH 2.30 ×VC 2.30 ×N 4

KH 0.260KC 0.260

applied to the bath when the red lines represent the nonlinear observation ˆNCR22

(w1um2). (For interpretation of the references to color in this figure legend, the reader

is referred to the web version of the article.)

thermal load, moving the system far from the linearization point.Some spikes that could be seen at times 1 h, 1,25 h, 1h, 45 and

2,5 h in Fig. 11b are due to discontinuities on the control of the JT

Meaning

al

Heat capacity ratio J/KGas density kg/m3

Pressure ratio 1 104 Input gas enthalpy J/kg 104 Output gas enthalpy J/kg 104 Saturated gas enthalpy J/kg 104 Saturated liquid enthalpy J/kg

Mixture quality 1Flow coefficient –Rangeability 1xt 1Liquid height m

10−3 Vaporized liquid kg/sLiquid mass kgLiquid density kg/m3

104 Liquid latent heat J/kg 10−3 Gas mass m

Gas density kg/m3

Cross-section m2

Tank height mHot gas density kg/m3

103 Hot gas specific heat J/kg/KCold gas density kg/m3

103 Cold gas specific heat J/kg/K 103 Exchanged power J/s

Exchange surface m2

Exchange coefficient J/s/m2/K 10−3 High pressure tube volume m3

10−3 Low pressure tube volume m3

Number of discrete zones 1 High pressure drop coefficient bar/kg2/s2

Low pressure drop coefficient bar/kg2/s2

8 roces

val

7

bbvTePTeoflsSEm

A

Ctc

R

[

[

[

[[

[

[

[

[

0 F. Bonne et al. / Journal of P

alve at these moments. Since the high frequencies of the systemre not captured by the model, discontinuities on control variableeads to large but time-limited estimation error.

. Conclusion and perspectives

A nonlinear observer of the thermal load applied to a heliumath of a cryogenic refrigerator has been proposed. The approach isased on physical modelling of a Joule–Thompson cycle. This pro-ides a tool that could be used by operators to monitor the process.he observer can also be used in order to improve the control strat-gy of the refrigerator, as advanced control schemes (such as Modelredictive Control) need the knowledge of the state to be used.he quality of the observer could be improved by using 2D mod-ls to describe the phase separator and heat exchangers behaviourr by having more variables available for measurement (especiallyowrates and boundary conditions). Furthermore, it have beeneen in Fig. 9d that our level measurement method is very noisy.ince the bath level represents the integral of the thermal loads (seeq. (11)), it would have been useful to have a better measurementethod to improve convergence time.

cknowledgements

The authors would like to thank Michel Bon-Mardion, Jean-Noelheynel et Lionel Monteiro for their participation to experimen-al campaign, and every co-workers from the SBT for their time toorrect and discuss this paper.

eferences

[1] C. Hoa, M. Bon-Mardion, P. Bonnay, P. Charvin, J. Cheynel, B. Lagier, F. Michel, L.Monteiro, J. Poncet, P. Roussel, B. Rousset, R. Vallcorba-Carbonell, Investigations

of pulsed heat loads on a forced flow supercritical helium loop, Cryogenics 52(7-9) (2012) 340–348.

[2] B. Bradu, P. Gayet, S. Niculescu, Modeling, simulation and control of large scalecryogenic systems, in: Proceedinsg of the IFAC world congress, Séoul, 2008, pp.13265–13270.

[

[

s Control 24 (2014) 73–80

[3] R. Maekawa, K. Ooba, M. Nobutoki, T. Mito, Dynamic simulation of the heliumrefrigerator/liquefier for LHD, Cryogenics (45) (2005) 199–211.

[4] B. Bradu, P. Gayet, S. Niculescu, A process and control simulator for large scalecryogenic plants, Control Engineering Practice (2009) 1388–1397.

[5] C. Deschildre, A. Barraud, P. Bonnay, P. Briend, A. Girard, J. Poncet, P. Roussel, S.Sequeira, Dynamic simulation of an helium refrigerator, in: Proceedings of theCryogenic Engineering Conference, 2008.

[6] C. Regier, J. Pieper, E. Matias, Dynamic modeling of liquid helium cryostat atthe canadian light source, Cryogenics 50 (2010) 118–125.

[7] E.B. Vinuela, J.C. Cubillos, C. de Prada Moraga, Linear model based predictivecontrol of the LHC 1.8 K cryogenic loop, in: Proceedings of the 1999 CryogenicEngineering Conference, 1999.

[8] R. Maekawa, S. Takami, M. Nobutoki, Adaptation of advanced control to thehelium liquefier with c-prest, in: Proceedings of the International CryogenicEngineering Conference, 2008.

[9] B. Bradu, P. Gayet, S.-I. Niculescu, Control optimization of a lhc 18 kw cry-oplant warm compression station using dynamic simulations, in: Proceedingsof Cryogenic Engineering Conference CEC/ICMC, 2009.

10] V. Ganni, P. Knudsen, Optimal design and operation of the helium refrigeratorsystem using the ganni cycle, in: Proceedings of the Cryogenic EngineeringConference, 2009.

11] F. Clavel, M. Alamir, P. Bonnay, A. Barraud, G. Bornard, C. Deschildre, Multi-variable control architecture for a cryogenic test facility under high pulsedloads: model derivation, control design and experimental validation, Journalof Process Control 21 (7) (2011) 1030–1039.

12] F. Bonne, M. Alamir, P. Bonnay, B. Bradu, Model based multivariable controllerfor large scale compression stations. Design and experimental validation onthe LHC 18 kw cryorefrigerator, in: Proceedings of the Cryogenic EngineeringConference, 2013.

13] CRYODATA, USER’s GUIDE to HEPAK, 1999.14] P. Thomas, Simulation of Industrial Processes for Control Engineers,

Butterworth-Heinemann, 1999.15] X. Li, Observation et Commande de quelques sytèmes paramètres distribués,

Université Claude Bernard Lyon 1, 2009 (PhD thesis).16] S. Bracco, I. Faccioli, M. Troilo, A numerical discretization method for the

dynamic simulation of a double-pipe heat exchanger, International Journal ofEnergy 1 (3) (2007) 47–58.

17] M. Ansari, V. Mortazavi, Simulation of dynamical response of a countercurrentheat exchanger to inlet temperature or mass flow rate change, Applied ThermalEngineering 26 (17–18) (2006) 2401–2408.

18] A. Zavala-Rio, R. Santiesteban-Cos, Reliable compartmental models for double-pipe heat exchangers: an analytical study, Applied Mathematical Modelling 31

(9) (2007) 1739–1752.

19] F. Clavel, Modélisation et contrôle d’un réfrigérateur cryogénique. Applicationà la station 800 W à 4.5 K du CEA Grenoble, EEATS, 2011 (PhD thesis).

20] H. Kwakernaak, R. Sivan, Linear Optimal Control Systems, Wiley-Interscience,1972.