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**Modeling and Predictive Control Strategies in Buildings with Mixed-Mode Cooling**

Jianjun Hu, Panagiota Karava School of Civil Engineering (Architectural Engineering Group) Purdue University

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**Background - Mixed-Mode Cooling**

Hybrid approach for space conditioning; Combination of natural ventilation, driven by wind or thermal buoyancy forces, and mechanical systems; “Intelligent” controls to optimize mode switching minimize building energy use and maintain occupant thermal comfort.

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**Background - Mixed-Mode Strategies**

When outdoor conditions are appropriate: Corridor inlet grilles and atria connecting grilles open; Atrium mechanical air supply flow rate reduced to minimum value, corridor air supply units close; Atrium exhaust vent open; Institutional building located in Montreal - When should we open the windows ? - For how long? - Can we use MPC? Mixed-mode cooling concept (Karava et al., 2012)

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**Background – MPC for Mixed-Mode Buildings**

Modeling Complexity Pump and fan speed, opening position (inverse model identified from measurement data) - Spindler, 2004 Window opening schedule (rule extraction for real time application) - May-Ostendorp, 2011 Shading percentage, air change rate (look-up table for a single zone) – Coffey, 2011 Blind and window opening schedule (bi-linear state space model for a single zone) – Lehmann et al., 2012

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Objectives Develop model-predictive control strategies for multi-zone buildings with mixed-mode cooling, high solar gains, and exposed thermal mass. Switching modes of operation for space cooling (window schedule, fan assist, night cooling, HVAC) Coordinated shading control

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**MPC: Problem Formulation**

Offline MPC (deterministic); baseline simulation study for a mixed-mode building Thermal Dynamic Model: Nonlinear Linearized prediction models (state-space) Discrete Control Variables: Open/Close (1/0) Algorithms for discrete optimization On-line MPC (implementation, identification, uncertainty) Operable vents

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**MPC: Dynamic Model (Thermal & Airflow Network)**

Building section (9 thermal zones)

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**MPC: Dynamic Model (Thermal & Airflow Network)**

Heat balance for atrium air node is the air exchange flow rate between zones (obtained from the airflow network model) : pressure difference ΔP: Solved by FDM method and Newton-Raphson 𝐶 𝑎𝑡𝑟 𝑑 𝑇 𝑎𝑡𝑟 𝑑𝑡 = 𝑇 𝑤𝑎𝑙𝑙 𝑖 − 𝑇 𝑎𝑡𝑟 𝑅 𝑤𝑎𝑙𝑙_𝑎𝑡𝑟 𝑖 + 𝑄 𝑎𝑢𝑥 + 𝑚 𝑐 𝑝 𝑇 𝑐𝑜𝑟𝑟 − 𝑇 𝑎𝑡𝑟 𝑚 Thermal model 𝑚 =𝐶 𝐷 𝐴 2𝜌∆𝑃 ∆𝑃=𝑓 𝑃, 𝑇 𝑎𝑡𝑟 , 𝑇 𝑐𝑜𝑟 𝑚 =𝐶 𝐷 𝐴 2𝜌∆𝑃 ∆𝑃=𝑓 𝑃, 𝑇 𝑎𝑡𝑟 , 𝑇 𝑐𝑜𝑟

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**MPC: Dynamic Model (State-Space)**

State-space representation: is a nonlinear term, i.e.: heat transfer due to the air exchange. 𝑓 𝑿,𝑼, 𝑚 𝑿 =𝑨𝑿+𝑩𝑼+𝑓 𝑿,𝑼, 𝑚 𝒀=𝑪𝑿+𝑫𝑼 obtained from the airflow network model 𝑚 =𝑔 𝑿,𝑼 A, B, C, D: coefficient matrices X: state vector U: input vector Y: Output vector Linear time varying (LTV-SS) 𝑿 =𝑨 𝒕 𝑿+𝑩 𝒕 𝑼 𝒀=𝑪𝑿+𝑫𝑼

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**MPC: Dynamic Model (State-Space)**

States (X): X = [Ti , Tij , Tij,k]T i – zone index j – wall index k – mass node index Inputs (U): U = [Tout, Sij, Load]T Tout – outside air temperature; Sij – solar radiation on surfaces ij; Load – heating/cooling load; Outputs (Y): Y= [Ti , Tij , Tij,k]T Zone air temperature; Wall temperature; …………

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**MPC: Dynamic Model (LTV-SS)**

𝑿 =𝑨 𝒕 𝑿+𝑩 𝒕 𝑼 𝑇 𝑖 𝑇 𝑖𝑗 𝑇 𝑖𝑗,𝑘 ×1 = 𝐴 1,1 ⋯ 𝐴 1,280 ⋮ ⋱ ⋮ 𝐴 280,1 ⋯ 𝐴 280,280 ∙ 𝑇 𝑖 𝑇 𝑖𝑗 𝑇 𝑖𝑗,𝑘 ×1 + 𝐵 1,1 ⋯ 𝐵 1,52 ⋮ ⋱ ⋮ 𝐵 280,1 ⋯ 𝐵 280,52 ∙ 𝑇 𝑜𝑢𝑡 𝑆 𝑖𝑗 𝐿𝑜𝑎𝑑 𝑖 52×1 Find the matrices from the heat balance equations 𝐴 235,1 = 𝑚 𝑆𝐸1_𝑎𝑡𝑟 𝑐 𝑝 𝐶 𝑎𝑡𝑟_𝑏 𝐴 235,118 = 𝑚 𝑁𝑊1_𝑎𝑡𝑟 𝑐 𝑝 𝐶 𝑎𝑡𝑟_𝑏 e.g. atrium zone air node: 𝐶 𝑎𝑡𝑟_𝑏 𝑑 𝑇 𝑎𝑡𝑟_𝑏 𝑑𝑡 = 𝑇 11𝑤_𝑎𝑡 − 𝑇 𝑎𝑡𝑟_𝑏 𝑅 11𝑤_𝑎𝑖𝑟 + 𝑇 11𝑔_𝑎𝑡 − 𝑇 𝑎𝑡𝑟_𝑏 𝑅 11𝑔_𝑎𝑖𝑟 + 𝑇 31_𝑎𝑡 − 𝑇 𝑎𝑡𝑟_𝑏 𝑅 31_𝑎𝑖𝑟 + 𝑇 41_𝑎𝑡 − 𝑇 𝑎𝑡𝑟_𝑏 𝑅 41_𝑎𝑖𝑟 + 𝑇 51_𝑎𝑡 − 𝑇 𝑎𝑡𝑟_𝑏 𝑅 51_𝑎𝑖𝑟 + 𝑚 𝑆𝐸1_𝑎𝑡𝑟 𝑐 𝑝 𝑇 𝑆𝐸1 − 𝑇 𝑎𝑡𝑟_𝑏 + 𝑚 𝑁𝑊1_𝑎𝑡𝑟 𝑐 𝑝 𝑇 𝑁𝑊1 − 𝑇 𝑎𝑡𝑟_𝑏 + 𝑚 𝑎𝑡𝑟2_𝑎𝑡𝑟1 𝑐 𝑝 𝑇 𝑎𝑡𝑟_𝑚 − 𝑇 𝑎𝑡𝑟_𝑏 + 𝐿𝑜𝑎𝑑 𝑎𝑡𝑟_𝑏 𝐴 235,118 = 1 𝐶 𝑎𝑡𝑟_𝑏 𝑅 11𝑤_𝑎𝑖𝑟 𝐴 235,235 = −1 𝐴 𝐴 235,240 = 1 𝐶 𝑎𝑡𝑟_𝑏 𝑅 11𝑔_𝑎𝑖𝑟 𝐵 235,50 =1 𝐴 235,241 = 1 𝐶 𝑎𝑡𝑟_𝑏 𝑅 31_𝑎𝑖𝑟 𝐴 235,243 = 1 𝐶 𝑎𝑡𝑟_𝑏 𝑅 41_𝑎𝑖𝑟 𝐴 235,245 = 1 𝐶 𝑎𝑡𝑟_𝑏 𝑅 51_𝑎𝑖𝑟 𝐴 235,247 = 𝑚 𝑎𝑡𝑟2_𝑎𝑡𝑟1 𝑐 𝑝 𝐶 𝑎𝑡𝑟_𝑏

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**MPC: Control Variable, Cost Function, and Constraints**

Control variable: operation schedule Cost function: Min: where: E is the energy consumption; IOt is vector of binary (open/close) decisions for the motorized envelope openings 𝐽 𝐼𝑂 𝑡 =𝐸 𝐼𝑂 𝑡 = 0, 1 Constraints: Operative temperature within comfort range ( °C, which corresponds to PPD of 10%) during occupancy hours; Use minimal amount of energy: cooling/heating (set point during occupancy hours 8:00-18:00 is ˚C, during unoccupied hours is °C); Dew point temperature should be lower than 13.5 °C (ASHRAE 90.1); Wind speed should be lower than 7.5 m/s.

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**MPC: Optimization (PSO)**

“Offline” deterministic MPC: Assume future predictions are exact Planning horizon: 20: :00, decide operation status during each hour. find optimal sequence from 224 options; Wetter (2011)

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**MPC: Optimization (Progressive Refinement)**

Multi-level optimization Decide operation status for each two hours at night (20:00-5:00); Use simple rules (based on off-line MPC)

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**Simulation Study Assumptions: Cases:**

Local controllers were ideal such that all feedback controllers follow set-points exactly; Internal heat gains (occupancy, lighting) were not considered; An idealized mechanical cooling system with a COP value of 3.5 was modeled. TMW3 data (Montreal) Cases: Baseline: mechanical cooling with night set back Heuristic: Tamb ∈ [15℃, 25℃], Tdew ≤ 13.5 ℃, Wspeed < 7.5 m/s MPC

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**Results: Operation Schedule (Heuristic & MPC)**

Hours during which vents are open are illustrated by cells with grey background Heuristic strategy leads to higher risk of over-cooling during early morning (Day 1, Day 4, and Day 5);

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**Results: Energy Consumption & Operative Temperature (FDM & LTV-SS)**

Comfort Acceptability reduced from 80% to 60%

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**Results: MPC with PSO and Progressive Refinement (ProRe)**

Similar energy consumption and operative temperature; Much faster calculation with ProRe; 3 Days 3 Hours

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**Results: MPC with PSO and Progressive Refinement (ProRe)**

Fine-tune rules in Progressive Refinement method for different climate (LA)

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Conclusions For the simulation period considered in the present study, mixed-mode cooling strategies (MPC and heuristic) effectively reduced building energy consumption. The heuristic strategy can lead to a mean operative temperature deviation up to 0.7 °C, which may decrease the comfort acceptability from 80% to 60%. The predictive control strategy maintained the operative temperature in desired range. The linear time-variant state-space model can predict the thermal dynamics of the mixed-mode building with good accuracy. The progressive refinement optimization method can find similar optimal decisions with the PSO algorithm but with significantly lower computational effort.

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Acknowledgement This work is funded by the Purdue Research Foundation and the Energy Efficient Buildings Hub, an energy innovation HUB sponsored by the Department of Energy under Award Number DEEE In kind support is provided from Kawneer/Alcoa, FFI Inc., and Automated Logic Corporation

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