Microprocessor-Based Failure Detection of Heat Pumps

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MICROPROCESSOR-BASED FAILURE DETECTION OF HEAT PUMPS R. Shoureshi* and K. McLaughlin** * Sr il o() / ()j'.\l l'ci/(/I//((/ / E lIgi ll l'l'I'il lg. PII I'I/III' 1.." lIi" I'I.liIL 1\'1',11 L a/ a\'I' III' . /.\" -li 90 i. L'SA **TRII ' Spa('/' T erilllo/()gy (;I'IJ IIP. (JII /, SPaCl' Plllk Rn /olldo Bl'!/ril. CA 902 i 8. L'SA

Ab s tr ac t. The r mo fluid p roce s s e s and sys tem un derg o s patial an d t e mp o r a l v a ri a t ions in b o th fluid an d the rma l p ro pe rti e s. Th e r efo r e , they a r e hi ghly nonlinea r a n d coupled . This pape r pre sen t s a n a ppr oac h t o f a i lur e d e t e c t i on of a h ea t pump. Th e t ec hnique i nv o lv e s model i ng of t he system , design o f a n obse rver and s t a ti s ti ca l a n al y si s f o r f a ilur e i d e nt i f ica tion . Th is p a pe r f ocu ses o n th e f i r s t t wo ma j o r s t e p s . A bo nd gr a ph a ppr o a c h i s u se d to mode l the heat p um p and a n e xt e nd e d l i near obse rve r t ec hn i qu e i s u s ed t o deriv e an o ptimal s t a t e e stima t o r f o r t h e non l in e ar t he r mofl u i d system . A Z-1 5 1 mi c r ocompu ter is u se d t o impl emen t t he o bserv e r an d other required sof twa r e . The r es ul ts o f t he mod e l a nd th e o b se rv e r ar e compa re d wi th experiment a l d a t a a nd v e r y g o o d ag r e eme nt hav e be en o bt a ined. Keyw o rd s. Graph Th eo r y ; Ref r ige rati o n s t ate es t i ma t ion; o b se r v ab i l i t y Kal man F i lte r; s ta te-s p ace me th o d s .

INTRODUCTION Thermofluid processes and system undergo spatial and temporal variations in Doth fluid and thermal properties. The necessary coupling and mutual dependency of these two sets of equiliDrlum and transport properties make analysis of such systems far more complex. As processes have Decome more complex and

1.

A priori knowledge of the normal state of the sys tem.

2.

On-line process observation, identification, and es tima tion .

performance more demanding,

3.

Statistical likelyhood analysis (Baysian approach) for fault detection based on a sequential test on the parameters variances.

there has been an inex-

orable growth in more rational, quantitative approaches to system design and operation. Furthermore , thermofluid processes are mostly found in energy systems. With recent energy crisis more effort has been concentrated on dynamics and control of thermofluid processes and systems . In general, HVAC equipment are good examples of energy systems used in all sectors of society. Previous studies show the effects of the dynamic response of environmental control processes in industrial and residential buildings on energy savings. Recently, interest in using control strategies for failure detection and accommodation has significantly increased. One objective is to appl y microprocessor technology to develop a time-history of failures that a system undergoes when a component fails. S uch time-history records will be of great help to detect and locate the failed component(s) and ease the process of preventing future failures of a similar type. As explained in the next section, the failure detection scheme used in this study composed of four main steps. This paper presents the results of the first two steps namely mode ling and observer design for a hea t pump.

4.

Fault localization using the properties of the identified parameter Variations .

Figure 1 shows a block diagram for implementation of the above scheme on a heat pump. The first step dealing with the normal state of the system depends on a particular system or process. Usually manufacturing data, or a simple experiment on a non-fault system provides the normal state information. A major step in failure detection is on-line process identification. Although the control literature is very rich in the design of an observer for various linear systems; however, there is no analytical method for handling nonlinear systems. This paper presents an approach to design of an observer for a nonlinear heat pump by using temperature-entropy bond graphS. Figure 2 shows a schematic of a heat pump. A heat pump consists of two primary heat eXChangers, evaporator and condenser, an expander, and a compres-

FAILURE DETECTION There are several failure detection techniques presented in the literature . They are mostl y dealing with the state space representation of systems and assume the system can be linearly represented. The failure detection process used in the present stUdy consists of the following steps:

sor. Estimation and on- line identification of the heat pump can De aChieved in real time by derivation of an accurate model and a fast oDserver. The Objective is to apply Dond graph and derive a model for the system of Fig. 2 . Authors in references [2,7, 8 ,11 published in 1984 and 1985] have shown a true variable, temperature-entropy, Dond graph can be applied to thermofluid processes and systems. In

summary, the analysis of every thermofluid system begins with selection of a set of control volumes. In general each control volume has a fluid flow, with incompressible or variable density fluid, it has heat transfer and work transfer with its environment, and it has the inherent dependency between thermal and fluid energy domains. Figure shows a control volume and a bond graph constructed for this control volume. Details about derivation of such bond graph is gi ven by the au thors in ref e rences. The components included in the model are compressor, condenser, evaporator, and capillary tube. Four control volumes are considered for these four components. The bond graph of Fig. 3 is applied and simplified for each of these control volumes. Thereby a complete bond graph of the heat pump is developed and it is shown in Fig. 4. The inner loop of the bond graph represents the fluid energy domain, and the other loop represents the thermal domain. More complex and detailed models can be created and inserted for different components of this model (if necessary) as long as the causalities are preserved. As indicated by the integral causalities of the bond graph, the model is 6th order. Therefore, six nonlinear differential equations that describe dynamics of the hea t pump, can be obtained from the bond graph of Fig. 4 as listed below. e

m

- e

Z

(l)

e

1

R

m - m

ev

2

(L)

1

(l ,

PIECEWISE CLOSED FORM METHOD This method provides an estimate of temperature and pressure of a two phase flow given the specific entropy and specific volume. This method is very useful for real time failure detection applications because of the simplicity of the resulting equations . The complete details of the method is given in reference [14, McLaughlin 1984J. In an entropy-volume plane pOints with constant temperature can be shown to lie in a straight line. The slope of that line has a one to one correspondence to the temperature. It was found that for most refrigerants these constant temperature lines all intersect at almost a pivot point on the s-v plane. Figure 5 shows this plane. Then by derivation of that point (s ,v), for any given point in the two phase region ~s,~) the slope of the line connecting (s,v) pOint and the pivot point (so,v ) o can be ca lcu la ted by

s - s m =

v-v

o

( 7)

o

The temperature is defined uniquely by the slope m, so it is merely a matter of performing a least square fit of the temperature in terms of the slope to arrive at the temperature that corresponds to this slope. In reality, the lines do not all intersect at the same pOint, but rather there is a small neighborhood within Which all of the lines intersect. However, once the temperature region is identified it can be divided to smaller subregions and a pivot point for each subregion can De identified. The accuracy at this technique is so high that even in the extreme cases the error is less than one degree ceicius. MOUEL VERIFICATIUN

T

T

S

I

-T

r

PI

------

Pz

Ri

2

T2

-

-

Z

- T

Pz

(4

r co nct P1 - -------R

O

2

T2

where h is enthalpy, A is fluid momentum, m is mas> f low rate, " is chemical potential, R is the therm", l resistance, and S is entropy. One of the main difficulties in dealing with thermofluid systems is that there are no closed form constitutive laws between properties of the working fluid. In case of a temperature-entropy bond grapn the capacitive field with integral causalities requires temperature and pressure as functions of specific volume and specific entropy. This means, in order to find temperature and pressure of the heat pump from equations (1) through ,6) one requires the following functions:

T

T(v,s)

P

P(v,s)

A new technique for aerivation of such function described in the next section.

1S

Combination of the mode ling results shown by equation (1) through (6) and the piecewise technique for on-line calculations of thermofluid properties of the working fluid in the heat pump forms the basis for a model of the heat pump. This model was implemented on a Z-lS1 microcomputer and the software required for integration of the non1inear equations was developed. The software is in hybrid form, namely combines assembly instructions and BASIC, high level language programming format. To check the accuracy of the model its time response was compared with actual data. Figure 6 shows the time response of the condenser and evaporator during two transient opera tions of the compressor, namely J during off-cycle and on-cycle. As shown, the agreement is very good. The mOdel has a smaller rise time whicn is due to the use of one lump model tor every component. Similar agreements have been obtained Detween model predictions and experilnental results tor pressure variations ana mass flow rate through each component. The comparison with experimental results verifies the accuracy of the model. Figure 7 shows the mOdel predication after a compressor failure. It represents the temperature-entropy diagram of the heat pump. The area underneath of this diagram corresponds to the performance of the heat pump. This figure shows how the performance changes during the transient operation of working fluid caused by the compressor failure.

Failurc [)clcnioll of Hl';11 PUIllPS STATE KECONSTRUCTIOY The task of an observer is to reconstruct the state variables of the system, as accurately as possible, based on a minimum number of measurements. The chosen measurements must satisfy the observability requirements. In case of thermofluid systems those measurements are preferred to be quantities that can be measured without having to Dreak into the system. This is especially important in case of closed loop cycles such as a refrigerator or other HVAC equipment where breaking into the system results in a loss of working fluid and many leakage problems. A schematic diagram for the general operation of an observer is shown in Fig. 8. The observer is located within the dashed lines. The state equations of the plant are given by equations (1) through (6). In genera l they can De wri t ten as:

x;

!.(~,!!.)

\ 8)

Y

~(~)

(9)

where X is the state vector with its components being fluid momentum in the capillary tube, mass flow rate in the condenser, entropies in the evaporator and condenser, and entropies ot wall materi· ais in the evaporator and 1n the condenser.

State equations for the observer can be written as "X

AR

+ RA

K ;

RC

Y

In order to determine the entries of the gain matrix K, which is a 2 Dy 6 matrix, while using linear observer theory, the non linear state equations (8 ) must be linearized. In the linearized form the A matrix is simply the Jacobian matrix of !.(~,!!.).

Y

ex +

+ V

BVB

T

o

(15)

T

11

-1

(16)

It was found that p ; lOUD prOduces a aesired closed loop solution. Furthermore, it was observed that although the optimal feedback gains vary along the trajectory, but the magnitudes of the entries in each of the gain matrices are similar and the signs do not change.

Based on this method the optimum gain matrix was determined and the ODserver was constructed. The results at the observer are compared with the actual measurements in F1gure 9. As shown, the observer performs very well even though the Observer starts with zero initial conditions while the heat pump has non-zero initial state . CONCLUSION A failure detection scheme for thermofluid processes was described from the four main steps in the detection process, the results of the first two steps, namely, modeling and states estimation were described. The results of these two steps will be used to compare the instantaneous state of the heat pump with the no-fault state and calculate the error. Based on the resulting error and statistical analysis such as Baysian likelyhood ratio a decision will be made for detection and localization of a failure. These steps are part of the current study and the results will be reported in future publications. REFERENCES i.

u-

+

The relative magnitudes of the white noise intensities V and W determine the optimum balance between the speed of the state reconstruction and the immunity to measurement noise. Reference [14. McLaughlin 1984J describes a method for selection of V and 11 along the trajectory of operation of the heat pump. There will be a series of gain matrices obtained along the trajectory. Out of those the one which best guarantees the observer requirements auring the whole transient operation should be selectea. The criteria is to have stability from both dynamic sense and computational sense.

To calculate the feedDack gains, the optimal observer problem was solvea [4,5,6,14J. The follow ing is assumed tor the system

U

RC IICR

and

(10 ;

where K is the gain matriX, to be determined, and X and are estimate state and output vectors. The output vector (available measurements) are assumea to be exit temperatures of the condenser and evaporator. These have nonlinear relationships with the s ta te vec tor.

T

T

(I!

D.R. Tree and M.F. McIlride, "The Dynamic Response of Environmental Control Processes in BUildings," Purdue University, March 13-15,

1979.

(L:

11

This means the input vector (!!.) has a deterministic component (U-) and a stochastic component (V). The output noise in rLleasurements 1s shown by vector .!!. Furthermore, it is assumed that y and ~ are stationary white noises with Gaussian distribution and zero mean values . Therefore, the correlation functions can be written as

V6('t)

116('t)

2.

R. Shoureshi, "Dynamic Analysis ano Failure Detection of HVAC Systems Using TemperatureEntropy Bond Graphs," ASHRAE lIorkshop on HVAC ContrOl ~odeling and Simulation, Feb. 2-3, 1984, Atlanta.

3.

D.F. Farris, "Energy Conservation by Adaptive Control for a SoLar Heated Building," Proceedings of the International Conference on Cybernetics and Society, Sept. 1977.

4.

M. Athans, P.L. Falb, "Optimal Control,1! McGraw-Hill, Yew York, 1966.

5.

H. Kwakernaak, R. Sivan, '·Linear Optimal Control Systems," John Wiley, New York 1972.

6.

R. Shoureshi, "Theory and DeSign of Control Systems," Lecture Notes, ME 575, School of Mechanical Engineering, Purdue University, Fall

(14 )

where 6('t) is the airac delta function, and V and 11 are noise powers. The optimal feedback gains are obtained Dy solving the Riccati equation for the dual system, written as

1983.

I j~

R. Shollres hi and K.

7.

R. Shoureshi, K. McLaughlin, '~nalytical and Experimental Investigation of Flow-Reversible Heat Exchangers Using Temperature-Entropy Bond Graphs," Journal of Dynamic Systems, Measurement, and Control, Vol. 106, June 1984.

8.

R. Shoureshi, K. McLaughlin, '~odeling and Dynamics of Two-Phase Flow Heat Exchangers Using Temperature-Entropy Bond Graphs," Proceedings of 1984 American Control Conference, San Diego, California.

9.

:,\kL a ll ~ hlil1

.., '"'"

3

P d

~

~

"1 Eh-nAl.PY

~.C.

Rosenberg, D.C . Karnopp, Introduction to Physical System Dynamics," McGraw-Hill, New York, 1983 .

10.

LR. Hildebrand, "Advanced Calculus for Applications," 2nd Edition, 1976, Prentice-Hall publications.

ll.

R. Shoureshi, K. McLaughlin, "Application of Bond Graph to Thermofluid Processes and Systems," Proceedings of 1985 American Control Conference, Boston, MA, June 1985.

12.

U. Tsach, "Failure Detection and Location Method Applied to a Simulated Fossil Power Plant," Proceedings of 1983 ACC, pp. 946-950.

13.

A. N. Madiwale, B. Freidland, "Comparison of Innovations-Based Failure Detection and Correction Methods," Proceedings of 1983 ACC, pp. 940-945.

14.

K. McLaughlin, '~SME TheSiS, School of Mechanical Engineering, Purdue University, December 1984.

Fig. 2: 11eosu remen t I~ol se

I

Inputs

Schematic and Pressure-Enthalpy Diagrams of a Heat Pump

11eosuremen t

====~lr==--~==~1LH_e_ o t____ "

\i .1

~--- -

~====! NO-Foul t

I

Fig. I:

State

Fault Detection Process

Fig. 3:

Schematic of a General Control Vo lume and Correspond ing Bond Graph.

Failure Detect ioll o f Hea t Pumps

159

Sf

~ c~~~enser

'to .OO

,... _ __

,..--C

I

f :

"

U

,.-___.....;.t:,;h..:,e=r-:mc;.a-=l--'-_, -----L. ... Gy .....- -_ _ domain l ~.'

j

:

,..>

,

doma in L ____J / r

1><-1, ' R........ ~

It)

I

I

evanorator' r-"-----,

T"-'-TF~

I

L..L....,..T'F~.

L _ _ _ _ ....J

~

:J

i ) cO'!P re s sor ,---,MTT--L:,... - - - -

; --- ul fluid

11 I I

I I

......

ClJ

(1~"'F:,

capillary tube

t _ <-!

20 . 00

,

I

J

~

I

'

a.

'

,..>

E

I

-,.&...j'!.fl--,

T

0 . 000

ClJ

--,

r

an

off

ClJ

,

-20 . 00

I

J L_______ !__ ~ I

'~ 1C I i l

I

I

-~o

I

__

00 +-----~------~-----r---0000 . 6000 1 . 200 2 . ~OO

I4TF

t.lme(sec

'-------~~ , /"---'-----~----'

: 1 A1t---- ( : L-l--- J

Fig. 6:

Comparsion of Heat Pump Kodel Response and Experimental Data

I I

•

Cl.

!l Fig. 4:

Bond Graph of a Heat Pump SATURATED VAPOR

SATURATED LIQUID

300

0

'"

\ \

0 ~

\

0

...~

275

250IL____________L -________ 0.2

~'--

__________

Spec i f ie vo lume

Fig- 7:

Fig. 5 :

Piecewise Closed Form Method for Evaluation of Temperature and Pressure

~

___________

0.6

0.4 [KJ/KgOk]

Dynamics of T-S Diagram after Compressor Failure

R. Shollreshi alld K. !\lcLllIKhlill

l(iO

u

"-

-

"-

PLANT

117

/

I

,

i

-

rI

I

I

--- 1;

-- -

[It

" I-

I

0I

~ -

I \,

1

1

I

I

I

,

I"

.0 DEL

1/

I

Fig . 8:

U

u
11)

...
_

_

_

ob~.rv..!!.

1

j

-7 . SO

... :J

...

_

Block Diagram of an Observer for a Nonlinear System

0 . 00

-15 .0

CIbortIf

/

a. E

...
-22 .5

-30 0

+----.------.---..---....

1 . 22

1 . 28

1.3~

1.39

1.~5

timeCsec.J (X10 3 , Fig. 9 :

[\1 I

1

L

I

-

-...

I

n

> y

Comparison of Observer Estimated Evaporator Temperature with Experimental Data.

..J