Modeling and Advanced Control of HVAC Systems Topic: HVAC Modeling - PowerPoint PPT Presentation

Modeling and Advanced Control of HVAC Systems Topic: HVAC Modeling & Control Truong Nghiem ESE, University of Pennsylvania nghiem@seas.upenn.edu January 26, 2011

Outline ◮ Part I: Modeling of HVAC Systems ◮ Part II: Advanced Control of HVAC Systems T. Nghiem HVAC Modeling & Control 2

Part I Modeling of HVAC Systems Fundamentals Zone Model T. Nghiem HVAC Modeling & Control 3

HVAC Modeling: Overview supply air heat gain set-point reheat Thermostat VAV Zone damper zone temperature Sensor ◮ Mathematical model of the plant (Zone block). ◮ HVAC system: exact models are complex (nonlinear, PDE, stochastic, etc.). ◮ Focus: simplified (linearized) first-principles models derived from heat transfer and thermodynamics theories. ◮ Other types of models: regression models, neural networks, look-up tables, etc. T. Nghiem HVAC Modeling & Control 4

HVAC Modeling: Fundamental Equation First Law of Thermodynamics (Conservation of Energy) Heat balance equation: H − W = ∆ E Heat H Energy input to the system. Work W Energy extracted from the system. Internal heat E Energy stored in the system (can only measure/calculate its change). Heat input Zone Heat extracted (Conduction, in- (Supply air, radiation, (Zone air) filtration, etc.) internal heat gain, etc.) T. Nghiem HVAC Modeling & Control 5

Heat Transfer: Concepts ◮ Heat Q : energy transferred across system boundary by temperature difference (J). ◮ Heat flow (rate) ˙ Q : heat transfer rate (W). ◮ Heat flux : heat flow rate through a surface. Heat flux density is heat flux per unit area (W/m 2 ). ◮ Heat capacity C : heat needed to raise temperature of a body mass by 1 ◦ C (J/K). Also called thermal mass , thermal capacitance . ◮ Specific heat (capacity) C p : heat needed to raise temperature of 1 kg of material by 1 ◦ C (J/kg K); C = mC p = ρ VC p . ◮ Energy change by temperature change ∆ E = ρ VC p ∆ T . m (kg/s) and volume flow rate ˙ ◮ Mass flow rate ˙ V (m 2 /s); m = ρ ˙ ˙ V . T. Nghiem HVAC Modeling & Control 6

Heat Transfer: Mass Transfer Heating Supply air temperature T s , return air temperature T r < T s , volume flow rate ˙ V . Heat transfer to the zone is: Q = ˙ ˙ H = ρ ˙ V C p ( T s − T r ) ( W ) Cooling Similarly, with T s < T r , heat extracted from the zone is: Q = ˙ ˙ W = ρ ˙ V C p ( T r − T s ) ( W ) T. Nghiem HVAC Modeling & Control 7

Heat Transfer: Conduction Conduction is the process of heat transfer through a substance such as a wall, from higher to lower temperature. Fourier’s equation (3-dimensional PDE with time): � ∂ 2 T ∂ x 2 + ∂ 2 T ∂ y 2 + ∂ 2 T d T � ρ C p d t = k ∂ z 2 where k : thermal conductivity ( W / mK ). Simplified equation (timeless, one-dimensional): Q = kA ∆ T ∆ x = kAT h − T l ˙ l where A : cross-sectional area ( m 2 ), T h : high temperature, T l : low temperature, l : thickness/length of material. T. Nghiem HVAC Modeling & Control 8

Heat Transfer: Conduction 1 Define R th = kA (thermal resistance) then ˙ QR th = T h − T l Equivalent to an electric circuit: T = potential, ∆ T = voltage, ˙ Q = current, R th = resistance. ˙ R th T h Q T l T. Nghiem HVAC Modeling & Control 9

Heat Transfer: Convection Convection is the heat transfer between a surface and fluid/gas by the movement of the fluid/gas. ◮ Natural convection: heat transfer from a radiator to room air. ◮ Forced convection: from a heat exchanger to fluid being pumped through. Newton’s law of cooling: ˙ Q = hA ∆ T where h : heat transfer coefficient ( W / m 2 K 2 ); A : surface area ( m 2 ), ∆ T : temperature difference between surface and fluid. hA and write ˙ 1 Define R cv = QR cv = ∆ T . ˙ R cv Q T surf T fluid T. Nghiem HVAC Modeling & Control 10

Heat Transfer: Radiation Radiation is the heat transfer through space by electromagnetic waves. Example: radiation between a radiator and a wall that faces it. Fourth-order equation given by the Stefan-Boltzman law (cf. heat transfer textbooks). Approximate linearized equation: ˙ Q = ǫ h r A ( T 1 − T 2 ) where ǫ : emissivity of the surface (0 . 9 for most building materials); h r : radiation heat transfer coefficient ( W / m 2 K 2 ). ǫ h r A and write ˙ 1 Define R r = QR r = ∆ T . ˙ R r T 1 Q T 2 T. Nghiem HVAC Modeling & Control 11

Heat Transfer: Solar Radiation Solar radiation is the radiation heat transfer by sun light. ◮ Direct radiation to the walls, furnitures, etc. in the room. ◮ Then heat transfer from walls, furnitures, etc. to room air. ◮ No direct heat transfer to room air but indirectly through walls, furnitures, etc. ⇒ large time lag. T. Nghiem HVAC Modeling & Control 12

Part I Modeling of HVAC Systems Fundamentals Zone Model T. Nghiem HVAC Modeling & Control 13

Zone Temperature Model Window 1 Door 1 Door 2 Window 2 Room 1 Room 2 Door 3 Door 4 Door 4 Window 4 Room 3 Room 4 Source: [Deng et al. , 2010] ◮ Ignore latent load (humidity), only sensible load (temperature). ◮ No infiltration. ◮ Simplified model with simplified heat transfer equations. ◮ HVAC system of VAV type. T. Nghiem HVAC Modeling & Control 14

Body Mass Temperature At what point is room air temperature (or wall surface temperature) measured? ◮ Exact temperature distribution in a body mass is complex (PDE). ◮ Simplification: mean temperature of all points. ◮ How to measure mean temperature? Sensor placement. ◮ Mean temperature � = temperature that occupants feel. T. Nghiem HVAC Modeling & Control 15

RC Network of a Wall Model heat transfer processes using resistance-capacitance equivalent models (RC network). Zone – wall surface model R r T s T R T s T reduces to R cv where T : zone air temperature, T s : wall surface temperature, R r : radiation resistance, R cv : convection resistance, and 1 1 1 R = R r + R cv . T. Nghiem HVAC Modeling & Control 16

RC Network of a Wall Model heat transfer processes using resistance-capacitance equivalent models (RC network). R c ( n +1) T 1 R 1 T s 1 R c 1 T w 1 R c 2 T w 2 T wn T s 2 R 2 T 2 C 1 C w 1 C w 2 C wn C 2 Zone 1 Wall Zone 2 More accurate model of conduction with large n . Usually use n = 2 or simplify to a single thermal resistance between T s 1 and T s 2 ( n = 0). T. Nghiem HVAC Modeling & Control 17

RC Network for Four Rooms RC network of conduction, convection and radiation between rooms and outside air. T 33 T 29 T 1 T 2 T 30 T 34 T 5 T 7 R window,1 T 21 T 25 T 26 T 22 R window,2 T 13 T 14 T 17 T 18 T 37 T 6 T 8 T 37 T 12 T 9 T 20 T 19 T 15 T 16 R window,4 T 23 T 27 T 28 T 24 T 10 T 11 T 3 T 4 T 35 T 31 T 32 T 36 T 37 Source: [Deng et al. , 2010] T. Nghiem HVAC Modeling & Control 18

Heating/Cooling and Other Gains Add heating/cooling, internal gain and solar radiation to the network. ˙ Q sa 1 T 1 R 1 T s 1 R c 1 T w 1 R c 2 T w 2 R c 3 T s 2 = T oa ˙ Q i 1 Outside C 1 C w 1 C w 2 ˙ Air Q r 1 Zone Wall Q sa 1 : HVAC heat flow; ˙ ˙ Q i 1 : internal heat gain; ˙ Q r 1 : radiation heat gain Q sa 1 = ρ ˙ ˙ d T 1 d t = ˙ Q sa + ˙ V sa 1 C p ( T sa − T 1 ) Q i − 1 R 1 ( T 1 − T s 1 ) C 1 Q i 1 , ˙ ˙ Q r 1 : disturbance/prediction 0 = ˙ Q r + 1 R 1 ( T 1 − T s 1 ) − 1 R c 1 ( T s 1 − T w 1 ) T. Nghiem HVAC Modeling & Control 19

State-space Thermal Model of Zone Define variables: ◮ State variables x : all temperature variables. ◮ Disturbance variables w : internal gain, solar radiation, outside air temperature, etc. ◮ Input variables u : defined by application. ◮ Supply air flow rate u 1 = ˙ V sa 1 : ˙ Q sa 1 = ρ u 1 C p ( T sa − x 1 ) ◮ Blind control u 1 b ∈ [0 , 1]: ˙ Q r 1 = u 1 b w 1 ◮ Output variables y : e.g., y are all zone air temperatures. ◮ Parameters: capacitances and resistances. T. Nghiem HVAC Modeling & Control 20

State-space Thermal Model of Zone Gather all RC networks and all differential/algebraic equations: d d t x ( t ) = Ax ( t ) + Bu ( t ) + Kw ( t ) + ( L x x ( t ) + L w w ( t )) u ( t ) Discretize the state-space model: � � x ( k + 1) = ˆ Ax ( k ) + ˆ Bu ( k ) + ˆ ˆ L x x ( k ) + ˆ Kw ( k ) + L w w ( k ) u ( k ) y ( k ) = Cx ( k ) Linearize the model at some operating point: x ( k + 1) = ˜ Ax ( k ) + ˜ Bu ( k ) + ˜ Kw ( k ) y ( k ) = Cx ( k ) Model reduction techniques to reduce the dimension of the model. T. Nghiem HVAC Modeling & Control 21

Part II Advanced Control of HVAC Systems Overview Introduction to Model Predictive Control Model Predictive Control of HVAC Systems T. Nghiem HVAC Modeling & Control 22

Advanced Control of HVAC Systems In this lecture, advanced control = optimal supervisory control of HVAC system to minimize some objective function (e.g., energy consumption, energy cost). General optimization problem: minimize J = f ( x 0 ... N , u 0 ... N , w 0 ... N ) u 0 ... N subject to x k +1 = g ( x k , u k , w k ) constraints on x k , u k w k ∼ disturbance model T. Nghiem HVAC Modeling & Control 23

Ingredients of Optimal Control System model g Mathematical model of the HVAC system (Part 1). Disturbance model of w k ◮ Constrained in a bounded set w k ∈ W k . ◮ Stochastic model, e.g., T oa ∼ N ( ¯ T oa , σ 2 ) where ¯ T oa : predicted outside air temperature (weather forecast). T. Nghiem HVAC Modeling & Control 24