Model Predictive Control for a Thermostatic Controlled Systemkom.aau.dk/project/smartcool/restricted_files/2012.11.06-AAU/WP1_pres.pdfNov 06, 2012 · Model Predictive Control for

Model Predictive Control for a ThermostaticControlled System

Ehsan Shafiei

November 6, 2012

S. E. Shafiei, H. Rasmussen and J. Stoustrup MPC for thermostatic controlled systems

Outline

� Supermarket refrigeration system

Booster configurationModel validation

� Set-point control

Algorithmic pressure controlModel predictive control

I Reduced order observerI COP predictionI Convex programming

� Results


Booster configuration

RefrigerationSystem

DistributedControllers

SupervisoryControl

GridInterface

local feedbacks

Supervisory required feedbacks

Outer Control Loop

Price Signal

Set-point commands

Control signals

Closed loop model including local controlsSystem (grid node) data

OutdoorTemperature

CondenserCP_HP

COMP_HI

COMP_LO

BPV

REC

EV_MT

EV_LTEVAP_LT

EVAP_MT

1 2

2´

3

3´

4´

4

1b 2b 5

6

7

8

Cold room dynamics:

MCpfoodsdTfoods

dt= −Qfoods/cr

MCpcrdTcr

dt= Qload + Qfoods/cr − Qe

Qfoods/cr = UAfoods/cr (Tfoods − Tcr )

Qload = UAload(Tindoor − Tcr )

Qe = mr (hoe − hie)

mr = OD KvA√ρsuc(Prec − Psuc)105


Power consumption

Electrical power consumption:

Wc =1

ηmemref (ho,c − hi ,c)

ho,c = hi ,c +1

ηis(his − hi ,c)

ηis = c0 + c1(fc/100) + c2(Pc,o/Psuc)


Coefficient of performance

Coefficient of performance (COP):

COP =Qe,tot

Wc,tot

COP =xMT (hoe,MT − hie,MT ) + xLT (hoe,LT − hie,LT )

1ηMT

(his,MT − hi ,c,MT ) + xLTηLT

(his,LT − hi ,c,LT )

ηMT = ηme,MTηis,MT

ηLT = ηme,LTηis,LT


Model validation

0 50 100 150 200 250 3000

1

2

3

4

5

6

7

8

T dc,3

[o C]

measurementestimation

0 50 100 150 200 250 3000

2

4

6

8

10

12

time(min)

Wco

mp[kW

]

measurementestimation


Set-point control structureRefrigeration

SystemDistributedControllers

SupervisoryControl

GridInterface

local feedbacks

Supervisory required feedbacks

Outer Control Loop

Price Signal

Set-point commands

Control signals

Closed loop model including local controls

power reference

PI

Gn

G1

G2

∆T1

∆T2

∆Tn

RefrigerationSystem

Supervisory controller

Power consumption feedback

T1

T2

Tn

System (grid node) data

OutdoorTemperature

Compressors

Thermostaticcontrollers

Set-pointcontrol

Cold rooms

Refrigeration System

PIcontrollers

Distributed controllers


Algorithmic pressure control scheme

Algorithm 1 Calculate the set-point value for each suction pressure

if Psuc > Psuc,min and max(ODavr ) > γOD thenDecrease the pressure set-point

else if Psuc < Psuc,max thenIncrease the pressure set-point

elseDo not change the pressure set-point

end


Linear system formulation

Linear model formulation:

x = Ax + B1u + B2d

A =

[−UAfoods/cr

MCpfoods

UAfoods/cr

MCpfoodsUAfoods/cr

MCpcr−UAfoods/cr+UAload

MCpcr

]

B1 =

[0−1

MCpcr

], B2 =

[0

UAloadMCpcr

]

Constraints:

Tfoods,min ≤ Tfoods ≤ Tfoods,max

0 ≤ Qe ≤ Qe,max

x =[Tfoods Tcr

]T, u = Qe , d = Tindoor


Reduced order observer

Rewriting linear dynamics:[x1

x2

]=

[a11 a12

a21 a22

] [x1

x2

]+

[b11 b12

b21 b22

] [ud

]

Estimator equation:

˙x1 = Ao x1 + Bo,1uo + Bo,2d + L(yo − Co x1)

Ao = a11, Bo,1 =[a12 b11

], uo =

[x2 u

]T, Bo,2 = b12, Co = a21

yo = x2 − a22x2 − b21u − b22d


MPC synthesis

Discrete-time multivariable linear system:

x [k + 1] = Adx [k] + Bd,1u[k] + Bd,2d [k]

x =[xT1 xT2

]TSoft constraints:

Tmin − ε∆Tfoods ≤ x1 ≤ Tmax + ε∆Tfoods

ε ≥ 0

where

Tmin = Tfoods,min + Tsafe , Tmax = Tfoods,max − Tsafe


MPC synthesis

Cost of energy:

Jec =

∫ TN

T0

epWc,totdt

Discrete-time formulation:

Jec =N−1∑k=0

∥∥∥∥∥ep Qe,tot

COP

∥∥∥∥∥2

2

where Qe,tot =m∑i=1

Q ie with m indicating the number of cold rooms

Penalizing the rate of change:

J∆u =N−1∑k=1

∥∥∥R∆u

(Qe [k] − Qe [k − 1]

)∥∥∥2

2


MPC algorithm

Algorithm 2 MPC implementationPrediction

Load

COP and Toutdoor from previous horizonep and Toutdoor predictions

Compute

COP prediction based on its previous horizon values and Toutdoor

Solve

minimizeu,ε

Jec + J∆u + ρεε (over the horizon)

subject to x[k + 1] = Ad x[k] + Bd,1u[k] + Bd,2d [k]x1 ≥ Tmin − ε∆Tfoodsx1 ≤ Tmax + ε∆Tfoodsε ≥ 0

0 ≤ u ≤ Qe,max

Update

u[k] = first move in obtained ux[k + 1] = Ad x[k] + Bd,1u[k] + Bd,2d [k]

Tref ,cr = x2[k + 1] where x =[x1 x2

]T


Simulation set-up

0 2 4 6 8 10 12 14 16 18 20 22 2410

11

12

13

14

15

16

17

18

19

20

time(hour)

T outd

oor [o C]

0 2 4 6 8 10 12 14 16 18 20 22 2425

30

35

40

45

50

55

60

65

70

time(hour)

e p [EUR

/MW

h]

Outdoor temperature (top) and electricity price (bottom)


Simulation set-up

MPC sampling time = 15 min

prediction horizon = 24 h

ρε = 5

R∆u = 0.1 ·diag(1, 1, 1, 1, 1, 0.5, 1, 0.025, 0.1, 0.025, 0.1)

Tsafe,dc = 0.5 ◦C, Tsafe,fr = 1 ◦C

A 5 min moving average as well as γOD = 0.9 are used for theimplementation of Algorithm 1


Results

0 2 4 6 8 10 12 14 16 18 20 22 240

2

4

6

8

10

12

Wc,tot[kW

]

time(hour)

Etot

= 64 [kWh] and Ec = 32.5

Power consumption in case of traditional fixed set-point control. The total

energy consumption and corresponding electricity cost are Etot = 64 [kWh]

and ec = 32.5.


Results

0 1 2 3 4 5 60.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

Tem

pera

ture

[o C]

time(hour)

T

dc,1

Tdc,3

Tfood,1

Tfood,3

0 1 2 3 4 5 6−25

−24

−23

−22

−21

−20

−19

−18

−17

Tem

pera

ture

[o C]

time(hour)

T

fr,1

Tfr,2

Tfood,1

Tfood,2

Cold room temperatures. Dashed red lines indicate the temperature limits. Left: Air temperatures of the first and

third display cases, Tdc , and corresponding food temperatures. Right: Air temperatures of the first and second

freezing rooms, Tfr , and corresponding food temperatures.


Results

0 2 4 6 8 10 12 14 16 18 20 22 240.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

Wc,tot[kW

]

time(hour)

Etot

= 49.8 [kWh] and Ec = 21.4

Power consumption after applying MPC (Algorithm 2) together with al-

gorithmic suction pressure control (Algorithm 1). The total energy con-

sumption and electricity cost are Etot = 50 [kWh] and ec = 21.4 (34%

reduction).


Results

0 2 4 6 8 10 12 14 16 18 20 22 240.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

T food

s in d

ispl

ay c

ases

[o C]

time(hour)0 2 4 6 8 10 12 14 16 18 20 22 24

−24

−23

−22

−21

−20

−19

−18

T food

s in fr

eezi

ng ro

oms

[o C]

time(hour)

Actual food temperatures in display cases (left) and freezing rooms (right).

The temperature limits for display cases are [1, 5] except the lower one

which is [1, 3], and for freezing rooms are [-24, -18].


Results

12 13 14 15 16 17 18 19 202.5

3

3.5

4

4.5

5

5.5

6

6.5

7

CO

P

Toutdoor

[oC]

COP Estimation

0 2 4 6 8 10 12 14 16 18 20 22 242.5

3

3.5

4

4.5

5

5.5

6

6.5

7

CO

P

time(hour)

COP Prediction

measurementprediction

COP estimation and prediction. Left: Estimation of the system COP as

a linear function depending on outdoor temperature. Right: Prediction of

the system COP based on the linear estimation.


Results

0 2 4 6 8 10 12 14 16 18 20 22 2420

22

24

26

28

30

32

34

36

38

Suct

ion

pres

sure

[bar

]

time(hour)

Suction pressures of two LT and MT sections resulted from applying Al-

gorithm 1.


Results

0 2 4 6 8 10 12 14 16 18 20 22 240.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

T food

s est

imat

ions

in d

ispl

ay c

ases

[o C]

time(hour)

T_{actual}: ______ T_{estimation}: .........

T

food, dc1

Tfood, dc3

Tfood, dc7

0 2 4 6 8 10 12 14 16 18 20 22 24

−24

−23

−22

−21

−20

−19

−18

T food

s in fr

eezi

ng ro

oms

[o C]

time(hour)

T_{actual}: ______ T_{estimation}: .........

T

food, fr1

Tfood, fr2

Tfood, fr4

Estimation of the food temperatures by reduced order observer. The im-

posed safety margin prevent the violation of temperature constraints due

to the estimation error.


Results

0 1 2 3 4 5 63.4

3.6

3.8

4

4.2

4.4

4.6

4.8

5

5.2

Air

tem

per

atu

re in

dis

pla

y ca

ses

[oC

]

time(hour)

T

dc,1

Tdc,5

0 1 2 3 4 5 6−32

−30

−28

−26

−24

−22

−20

−18

−16

Air

tem

per

atu

re in

fre

ezin

g r

oo

ms

[oC

]

time(hour)

T

fr,1

Tfr,2

Air temperatures (Tcr ) of the first and fifth display cases as well as the

first and second freezing rooms. Local thermostatic controllers regulate

the temperatures around the set-points provided by MPC.


The end

Questions?


Documents

Model Predictive Control for a Thermostatic Controlled Systemkom.aau.dk/project/smartcool/restricted_files/2012.11.06-AAU/WP1_pres.pdfNov 06, 2012 · Model Predictive Control for