Interdisciplinary Center for Applied Mathematics
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1 Feedback Control of Boussinesq Equations with Applications to Energy Efficient Buildings Optimal Sensor Location V I. Akhtar, J. Borggaard, J. Burns, E. Cliff, W. Hu, L. Zietsman URC S. Ahuja, S. Narayanan, A. Surana Room with Disturbance Interdisciplinary Center for Applied Mathematics CDPS 211 Wuppertal, Germany July 211
2 Energy Efficiency & Buildings WHY BUILDINGS
3 Energy Efficiency & Buildings WHY BUILDINGS AND NO HIS
4 Building Energy Demand Buildings consume 39% of total U.S. energy 71% of U.S. electricity 54% of U.S. natural gas Building produce 48% of U.S. Carbon emissions Commercial building annual energy bill: $12 billion he only energy end-use sector showing growth in energy intensity 17% growth % growth projected through 225 Energy Breakdown by Sector Energy Intensity by Year Constructed Sources: Ryan and Nicholls 24, USGBC, USDOE 24
5 Impact of Energy Efficient Buildings HUGE A 5 percent reduction in buildings energy usage would be equivalent to taking every passenger vehicle and small truck in the United States off the road. A 7 percent reduction in buildings energy usage is equivalent to eliminating the entire energy consumption of the U.S. transportation sector.
6 Impact of Energy Efficient Buildings HUGE A 5 percent reduction in buildings energy usage would be equivalent to taking every passenger vehicle and small truck in the United States off the road. A 7 percent reduction in buildings energy usage is equivalent to eliminating the entire energy consumption of the U.S. transportation sector. DESIGN, CONROL AND OPIMIZAION OF WHOLE BUILDING SYSEMS IS HE ONLY WAY O GE HERE? WHY? 84% of energy consumed in buildings is during the use of the building REQUIRES COMBINING - MODELING, COMPLEX MULI-SCALE DYNAMICS, CONROL, OPIMIZAION, SENSIIVIY ANALYSIS, HIGH PERFORMANCE COMPUING ALL HE HINGS HA APPLIED MAHEMAICIANS DO
7 raditional HVAC System
8 raditional HVAC System
9 raditional HVAC System
10 raditional HVAC System
11 Conference Room Control Problem u (t) = inflow temperature sensors / region ( q) x: q x controlled region is around conference table ( t, x ) b ( x ) u ( t) 2 (, x) v(, x) (, x) (, x) ( x) ( ) t t t t g w t t c y( t) C( q) ( t, ) c( x) ( t, x) dx n( t) q ( ) s
12 Conference Room Control Problem u (t) = inflow temperature sensors / region ( q) x: q x controlled region is around conference table ( t, x ) b ( x ) u ( t) 2 (, x) v(, x) (, x) (, x) ( x) ( ) t t t t g w t t c y( t) C( q) ( t, ) c( x) ( t, x) dx n( t) q ( ) s WHA IS A GOOD FORMULAION OF HE CONROL PROBLEM
13 Conference Room Control Problem u (t) = inflow temperature sensors / region ( q) x: q x controlled region is around conference table ( t, x ) b ( x ) u ( t) 2 (, x) v(, x) (, x) (, x) ( x) ( ) t t t t g w t t c y( t) C( q) ( t, ) c( x) ( t, x) dx n( t) q ( ) s ( t) D ( t, ) CONROLLED OUPU??
14 Conference Room Control Problem u (t) = inflow temperature sensors / region ( q) x: q x controlled region is around conference table ( t) D ( t, ) d( x) ( t, x) d x D ( t, x ) b ( x ) u ( t) AVERAGE ROOM EMPERAURE (WEIGHED)? c p
15 Conference Room Control Problem u (t) = inflow temperature sensors / region ( q) x: q x controlled region is around conference table ( t) D ( t, ) d( x) ( t, x) d x D ( t, x ) b ( x ) u ( t) AVERAGE ROOM EMPERAURE (WEIGHED)? [ ( t)]( x) D( t, x) d( x) ( t, x) L ( ) LOCAL IN SPACE EMPERAURE (WEIGHED)? c 2 p
16 Conference Room Control Problem u (t) = inflow temperature sensors / region ( q) x: q x controlled region is around conference table ( t) D ( t, ) d( x) ( t, x) d x D ( t, x ) b ( x ) u ( t) AVERAGE ROOM EMPERAURE (WEIGHED)? c p? WHERE SHOULD ONE PLACE SENSORS?
17 Other Controls: Floor Heating Strips sensors / region ( q) x: q x controlled region is around conference table u (t) = inflow temperature ( t, x ) b ( x ) u ( t) 2 (, x) v(, x) (, x) (, x) ( x) ( ) t t t t g w t t c t (, x) b ( ) ( ) c F x uf t
18 Other Controls: Chilled Beams sensors / region ( q) x: q x controlled region is around conference table u (t) = inflow temperature ( t, x ) b ( x ) u ( t) 2 (, x) v(, x) (, x) (, x) ( x) ( ) t t t t g w t t c ( t, x) b ( ) ( ) cb CB x ucb t t (, x) b ( ) ( ) c F x uf t
19 Other Controls: Chilled Beams sensors / region ( q) x: q x controlled region is around conference table u (t) = inflow temperature ( t, x ) b ( x ) u ( t) 2 (, x) v(, x) (, x) (, x) ( x) ( ) t t t t g w t t c ( t, x) b ( ) ( ) cb CB x ucb t t (, x) b ( ) ( ) c F x uf t? MORE COMPLEX PHYSICS?
20 u (t) = inflow temperature Boussinesq Equations sensors / region ( q) x: q x controlled region is around conference table ( t, x ) b ( x ) u ( t) 2 (, x) v(, x) (, x) (, x) ( x) ( ) t t t t g w t t c
21 u (t) = inflow temperature Boussinesq Equations sensors / region ( q) x: q x controlled region is around conference table ( t, x ) b ( x ) u ( t) 2 (, x) v(, x) (, x) (, x) ( x) ( ) t t t t g w t t 2 t v( t, x) v( t, x) v( t, x) v( t, x) p( t, x) c ged ( t, x) g v ( x) w v ( t) div v( t, x)
22 Boussinesq Equations sensors / region ( q) x: q x controlled region is around conference table u (t) = inflow temperature ( t, x ) b ( x ) u ( t) 2 (, x) v(, x) (, x) (, x) ( x) ( ) t t t t g w t t 2 t v( t, x) v( t, x) v( t, x) v( t, x) p( t, x) v( t, x) b ( x) u ( t) v c v c ged ( t, x) g v ( x) w v ( t) div v( t, x)
23 Linearized Model 2 t v v g w t 2 t v v v v v v p () ged g v w v ( t)
24 Linearized Model 2 t v v g w t 2 t v v v v v v () ged g v w v ( t) IN HE PROPER SPACES FOR VARIOUS CONROL INPUS ( ) (t) ( t) v( t) u ( t) w ( t) ( ) v(t) v( t) ( t) u ( t) w ( t) v V V V V p V
25 Specific Structure d ( ) ( ) ( ) ( ) t t u t dt ( t) ( t) u ( t) v v v v v w( t) w ( t) v v
26 Specific Structure: d ( ) ( ) ( ) ( ) t t u t dt ( t) ( t) u ( t) v v v v v w( t) w ( t) v v d ( t) ( t) u ( t) ( ) dt ( t) ( t) u ( t) v v v v v w( t) w ( t) v v
27 heory Slide ( ) (t) ( t) v( t) u ( t) w ( t) ( ) v(t) v( t) ( t) u ( t) w ( t) v V V V V V
28 heory Slide ( ) (t) ( t) v( t) u ( t) w ( t) ( ) v(t) v( t) ( t) u ( t) w ( t) v V V V V ( ) x(t) x( t) u( t) w( t),, v v v V
29 heory Slide ( ) (t) ( t) v( t) u ( t) w ( t) ( ) v(t) v( t) ( t) u ( t) w ( t) v V V V V ( ) x(t) x( t) u( t) w( t),, v v v heorem (J. Burns & W. Hu) In a suitable space X, the system () is well posed. Moreover, a LQR feedback law u(t)=-b* that exponentially stabilizes () also locally stabilizes the full non-linear Boussinesq equations. V
30 raditional HVAC System
31 raditional HVAC System ADD ACUAOR MODEL / DYNAMICS
32 Actuator Dynamics d ( t) ( t) u ( t) ( ) dt ( t) ( t) u ( t) v v v v v u ( t) H z ( t) u v ( t) H v z v ( t)
33 Actuator Dynamics d ( t) ( t) u ( t) ( ) dt ( t) ( t) u ( t) v v v v v u ( t) H z ( t) u v ( t) H v z v ( t) ( ) z (t) z ( t) v ( t) a a a ( ) z (t) z ( t) v ( t) av v av v av v
34 Actuator Dynamics d ( t) ( t) u ( t) ( ) dt ( t) ( t) u ( t) v v v v v u ( t) H z ( t) u v ( t) H v z v ( t) ( ) z (t) z ( t) v ( t) a a a ( ) z (t) z ( t) v ( t) av v av v av v COMPOSIE SYSEM x( t) ( t) v( t) z ( t) z ( t) v
35 Composite System ( ) x(t) x( t) v( t) w( t) full A H a H v v v av
36 Composite System ( ) x(t) x( t) v( t) w( t) full A H a H v v v av a av v
37 Composite System ( ) x(t) x( t) v( t) w( t) full A H a H v v v av a av v
38 Composite System ( ) x(t) Ax( t) Bu( t), x() x X ADD ACUAOR MODEL / DYNAMICS ( ) x (t) A x ( t) B v( t), x () x X a a a a a a a a u( t) Hx ( t) a
39 Composite System ( ) x(t) Ax( t) Bu( t), x() x X ADD ACUAOR MODEL / DYNAMICS ( ) x (t) A x ( t) B v( t), x () x X a a a a a a a a u( t) Hx ( t) a COMPOSIE SYSEM x(t) Ax( t) BHx ( t), x() x X a x (t) A x ( t) B v( t), x () x X a a a a a a a
40 Composite System z( t) x( t) x ( t) Z X X a ( ) z(t) Az( t) Bv( t), z() z Z c a A A BH B A B a a
41 Composite System z( t) x( t) x ( t) Z X X a ( ) z(t) Az( t) Bv( t), z() z Z c a A A BH B A B a a ISSUES WIH COMPOSIE SYSEMS
42 Composite System Example ( ) x(t) x( t) u( t) d x ( t) 1 x ( t) 1 1 ( a) x a(t) v( t) dt x2( t) x2( t) 1 u( t) ( ) Hxa t x2() t x () t BH
43 Composite System Example ( ) x(t) x( t) u( t) d x ( t) 1 x ( t) 1 1 ( a) x a(t) v( t) dt x2( t) x2( t) 1 u( t) ( ) Hxa t x2() t x () t BH () and ( a ) are controllable but
44 Composite System Example ( ) x(t) x( t) u( t) d x ( t) 1 x ( t) 1 1 ( a) x a(t) v( t) dt x2( t) x2( t) 1 u( t) ( ) Hxa t x2() t x () t BH () and ( a ) are controllable but ( ) ( ) ( ) d xt xt x c ( t ) 1 x ( t ) uc( t ) dt 1 1 x2( t) x2( t) 1 ( c ) is not stabilizable
45 Boussinesq Equations sensors / region ( q) x: q x controlled region is around conference table u (t) = inflow temperature ( t, x ) b ( x ) u ( t) 2 (, x) v(, x) (, x) (, x) ( x) ( ) t t t t g w t t 2 t v( t, x) v( t, x) v( t, x) v( t, x) p( t, x) v( t, x) b ( x) u ( t) v c v c ged ( t, x) g v ( x) w v ( t) div v( t, x)
46 Building DPS Control ProblemS MODEL REDUCION MODEL IDENIFICAION PDE OPIMIZAION PROBLEM SELECION SENSOR LOCAION ACUAOR LOCAION ACUAOR DYNAMICS CONROLLER DESIGN ROBUSNESS UNCERAINY MULI-SCALE
47 Building DPS Control ProblemS COMPUAIONAL ISSUES MODEL REDUCION CONROLLER REDUCION SIMULAION OPEN LOOP CLOSED LOOP MODEL REDUCION MODEL IDENIFICAION PDE OPIMIZAION PROBLEM SELECION SENSOR LOCAION ACUAOR LOCAION ACUAOR DYNAMICS CONROLLER DESIGN ROBUSNESS UNCERAINY MULI-SCALE
48 What About REAL Buildings?
49 What About REAL Buildings? PRACICAL SIMULAION, DESIGN & CONROL OOLS
50 What About REAL Buildings? PRACICAL SIMULAION, DESIGN & CONROL OOLS MESH ZOOM MESH
51 HP 2 C Enabled Model Reduction Cluster, Grid, HPC High Performance High Productivity Computing High Fidelity Physics- Based Simulation ools ADVANCED MODEL REDUCION MEHODS HP 2 C ENABLED MODELING & COMPUER DESIGN OOLS BUILDING PHYSICS DESIGN PARAMEERS Hierarchal Reduced Order Design & Control Models INERACIVE INEROPERABLE Building Description Architecture, Materials HVAC, Loads, Lights Occupancy Sensors, Actuators design model COMPUER DESIGN OOL
52 HP 2 C Enabled Controller Reduction High Fidelity Physics- Based Control Design ools DISRIBUED PARAMEER CONROL HP 2 C BUILDING PHYSICS DESIGN PARAMEER S Building Description Architecture, Materials HVAC, Loads, Lights Occupancy Sensors, Actuators NEW INFORMAION ABOU WHA MUS BE SENSED OR WHA MUS BE ESIMAED WHERE O PLACE SENSORS VENS ROBUSNESS SENSIIVIY x x x u( t) k ( ) ( t, ) d ADVANCED CONROLLER REDUCION MEHODS Holistic Reduced Order Controller h h h h h h z (t) A z ( t) F ( q)[ y( t) C ( q) z ( t)] e e e e e e x x x h u( t) k ( ) ( t, ) d e
53 Disturbance Rejection: ICAM CUBE Small room (4m x 4m x 3m) with displacement ventilation and chilled ceiling Inputs Control: chilled ceiling heat flux Disturbance: floor heat flux Outputs emperature measurements, on two walls parallel to the XY planes Occupied zone averaged temperature (used to define LQR cost) Boussinesq equations; k-ε model Grid: 6.6 x 1 4 nodes Re = 2.2x1 4 (w.r.t. room h=3m, diffuser vel.=.1m/s) Gr = 4.7x1 9 (w.r.t. room h=8m) Re 2 /Gr =.1 < 1 => buoyancy dominated ime-step = 2 sec
54 Disturbance Rejection: ICAM CUBE z(t) A z( t) B u( t) G v( t) r r r (t) Dz () t r y(t) C( q) z( t) Ew( t)
55 Results for a Real Building Re = 1.2x1 5 (w.r.t. room h=8m, diff. v=.2m/s) Gr = 9x1 1 (w.r.t. room h=8m) Re2/Gr =.16 < 1
56 Results for a Real Building
57 More Complex Physics 3 CONROLLERS; 1 EMP SENSOR; 2 CONROLLED OUPUS 12 th ORDER H 2 CONROLLER H 2 O Distribution
58 here is no Free Lunch (Energy)
59 here is no Free Lunch (Energy)
60 HANK YOU CONAC INFORMAION John A. Burns Interdisciplinary Center for Applied Mathematics West Campus Drive Virginia ech Blacksburg, VA
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