Comfort-Aware Building Climate Control Using Distributed ... - arXiv

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Comfort-Aware Building Climate Control Using Distributed-Parameter Models

arXiv:1708.08495v1 [math.OC] 28 Aug 2017

Runxin He, Student Member, IEEE, and Humberto Gonzalez, Member, IEEE

Abstract—Controlling Heating, Ventilation and Air Conditioning (HVAC) system to maintain occupant’s indoor thermal comfort is important to energy-efficient buldings and the development of smart cities. In this paper, we formulate a model predictive controller (MPC) system to estimate indoor climate and apartment’s geometric information based on only thermostats, and then make optimal control strategies to HVAC in order to maintain occupant’s comfort by predicted mean vote index. In order to have accurate spatial resolution and make the HVAC system focus on only a zoned area around the occupant, a convection-diffusion Computer Fluid Dynamics (CFD) model is used to describe the indoor air flow and temperature distribution. The MPC system generates corresponding PDE-contrained optimization problems, and we solve them by obtain the gradients of cost functions with respect to problems’ variables with the help of CFD model’s adjoint equations. We evaluate the performance of our method using simulations of a real apartment in the St. Louis area. Our results show our MPC system’s energy efficiency and the potential for its application in real-time operation of high-performance buildings. Index Terms—Model Predictive Control, Computational Fluid Dynamics, PDE Optimization, Thermal Comfort.

I. I NTRODUCTION

H

EATING, Ventilation, and Air Conditioning (HVAC) units are complex systems that coordinate incorporate mechanical and electrical components using data processing algorithms, designed to control the climate within buildings. For decades, the design of HVAC systems has been focused on minimizing energy waste by increasing the efficiency in heaters and coolers, or improving insulation of air ducts and exchangers. In this paper we focus on efficiently utilizing existing HVAC system technologies to maximize user comfort, rather than simply trying to maintain a constant temperature. To achieve our goal we develop estimation and control algorithms that consider spatio-temporal distributions of temperature and air flow, which take into account changes in floor-plan geometry (such as doors being opened or closed), outdoor weather, and the position of fans and portable heaters. All this information is fed into a distributed-parameter model and a multi-dimensional human comfort index to generate, and optimize, short-term predictions. Standard HVAC systems typically use a small number of thermostats, together with simple temperature regularization loops [1], to follow a temperature set point. However, human thermal comfort is dependent on many other variables besides temperature, such as humidity, air flow velocity, clothing, and metabolic rate. The multi-dimensional nature of human Runxin He is the with the Department of Electrical & Systems Engineering, Washington University in St. Louis, St. Louis, MO 63130. Email: [email protected].

comfort lead to the adoption of the Predicted Mean Vote (PMV) index [2] in the the ASHRAE 55 Standard [3] to quantify indoor thermal comfort. Our effort in this paper is motivated by our interest to explicitly incorporate PMV-based comfort measurements to the HVAC actuation loop. Hence, due to the important contribution of air flow convection in the dynamic behavior of indoor climate variables and the inability of concentrated-parameter models to describe this phenomena, we instead use a distributed-parameter computer fluid dynamic (CFD) model. This model is used within a large model predictive control (MPC) scheme, and a specialized optimization algorithm is developed to find its optimal solution to balance the predicted comfort and efficiency. Most results in HVAC control can be categorized within two main trends. The first corresponds to learning-based methods [4], [5], which result in the characterization of underlying complex phenomena, such as air convection, using low-dimensional formulas. While these data-driven methods have potential for energy efficiency, they are based on large quantities of data and training time, which are not suitable for residential apartments. The second trend corresponds to model-based predictive methods, which describe the dynamic evolution of building climate variables using first-principle physical models, or approximations of these models. Among model-based predictive methods, MPC stands out due to its flexible mathematical formulation and its robust performance in real-world implementations [6], [7], [8]. Indeed, MPC has become one of the standard methods for solving constrained multivariate control problems in process control applications [9]. MPC has been applied to zoned temperature control and temperature regularization in the past [10], [11], showing significant improvements in energy efficiency compared to other classical control methods [12]. Also, the flexibility of MPC allows for the stochastic nature of certain disturbance variables, such as weather forecast [13]. There exist many results regarding the use of indoor climate CFD models in optimal control schemes. For example, Doering and Gibbon [14] studied the existence and smoothness of laminar fluid models. Ito [15] found necessary conditions for the existence of solutions to optimal control problems with stationary Navier-Stokes and heat transfer constraints. In the specific case of HVAC control, Burns et al. [16], [17], [18] used linearized CFD models and quadratic cost functions, which resulted in high-dimensional, but theoretically tractable, Linear Quadratic Regulator problems. Their results show that these PDE models, usually considered too complex for online numerical calculations, can be effectively used for building control. However, the linearized approximations mean that their optimal solutions are accurate only if it is close to the

2

Heating/Cooling Units

Building

Control & Estimation Algorithm Humans Wearables/IoT

Thermostats

Fig. 1. Climate control solutions. Traditional solutions (shown in dashed gray lines) involve a single thermostat, one ventilation unit, and no human comfort input. Our new paradigm (shown in solid blue) considers multiple heterogeneous sensors, a controller and estimator based on distributed-parameter models, and multiple ventilation units.

original steady-state linearization point [11], which pose a serious limitation in practical applications. In this paper, our MPC scheme uses nonlinear CFD models, which guarantees the spatial accuracy of our predictions, significantly improving the energy efficiency of HVAC units [11]. Moreover, following our results in [19], we develop a first-order gradient-based optimization method to find optimal solutions using these nonlinear CFD models. Our paper is organized as follows. Sec. II describes the CFD model and the formulation of our MPC scheme. Sec. III presents our gradient-based algorithm to solve the PDEconstrained optimization problem, as well as our finite-element discretization used numerical computations. Sec. IV presents the results of our simulated experiments, based on a real apartment in the St. Louis area. II. PROBLEM STATEMENT We consider the problem of controlling the indoor temperature and air flow in a building equipped with a HVAC system consisting of one or several small units, with the goal of simultaneously improving both energy efficiency and human comfort. As illustrated in Fig. 1, traditional solutions to this problem rely on single thermostat measurements and a single-unit HVAC systems, thus equating human comfort to a single temperature value in the building as measured by the thermostat. Instead, we propose to use heterogeneous Internet-of-Things sensor data [20], [21], [22] together with first-principle fluid dynamics models to estimate and predict the spatial and temporal behavior of all the relevant climate variables, which in turn allows us to achieve our goals of efficiency and comfort. Our improved climate control scheme is also illustrated in Fig. 1. In this section we describe in detail all the components in our control scheme, including the CFD model, comfort evaluation model, and optimization algorithms for control and estimation. A. Computational Fluid Dynamic Model Before formally presenting our control scheme, we define the notation used throughout this paper. We will endow Rn

with the traditional 2-norm, denoted k·k, and we will denote the inner product between x1 and x2 in Rn by x1 · x2 . Our functional analysis notation closely follows that in [23]. Let S be a subset of a vector space, then we say that L2 (S) is the space of squared-integrable functions with domain in R S, endowed with the inner product hf , f i = f (x) · 1 2 1 S S p f2 (x) dx, and the norm kf kS = hf, f i. Moreover, H 1 (S) is the Sobolev space of functions in L2 (S) whose weakderivatives are also in L2 (S), and H01 (S) is the set of functions f ∈ H 1 (S) whose boundary is trivial,
i.e., f (x) = 0 for each x ∈ ∂S. We define L2 [t0 , tf ]; H 1 (S) as the set of functions

2

Rt 2

dt is u : [t0 , tf ] → H01 (S) such that t0f ku(t)kS + du(t) dx S finite. The kernel of our CFD model is the incompressible NavierStokes equation, which is a good approximation for the coupling of temperature with free flow convection at atmospheric conditions [24], [25]. Throughout the paper we make two major simplifications to this model. First, we assume that the air flow behaves as a laminar fluid which reaches steady-state behavior much faster than the temperature in the building. Theoretical [26] and experimental [27] results have shown that turbulent flows are present in residential building, such as in the area around HVAC vents, yet their overall effect in the temperature distribution is negligible. Hence, we consider a stationary laminar Navier-Stokes equation to describe the fluid behavior, and a time-dependent equation to describe the temperature behavior. Second, we consider only two-dimensional air flows moving parallel to the ground. Both assumptions reduce the accuracy of our model to some extent [28], yet they allow us to significantly simplify the computational complexity of our CFD-based control design. Following the notation presented above, let Ω ⊂ R2 be the region of interest, assumed to be compact and connected. Let u : Ω → R2 be the stationary air flow velocity, p : Ω → R be the stationary air pressure, and Te : Ω × [t0 , tf ] → R be the temperature. Then, the temperature convection-diffusion model [29] over the region Ω can be described by the following PDE:
∂ Te (x, t) − ∇x · κ(x) ∇x Te (x, t) + u(x) · ∇x Te (x, t) = ∂t = gTe (x, t), (1) where gTe : Ω × [t0 , tf ] → R represents the heat source in the room,
κ : Ω → R is the thermal diffusivity, and ∇x = T ∂ ∂ is the gradient operator. The initial condition of ∂x1 , ∂x2 the temperature is: Te (x, t0 ) = π0 (x),

∀x ∈ Ω.

(2)

Similarly, the air flow in Ω is governed by the following stationary incompressible Navier-Stokes PDEs:
1 − 4x u(x) + u(x) · ∇x u(x) + ∇x p(x) + Re + α(x) u(x) = gu (x), (3) ∇x · u(x) = 0,

(4)

where gu : Ω → R2 represents the external forces applied to the air (such as fans), Re is the Reynolds number of the air,

3

2

2

∂ ∂ and 4x = ∂x 2 + ∂x2 is the Laplacian operator. We introduce 1 2 the viscous friction coefficient α : Ω → R [30], to model the effect of solid materials in the air flow. Indeed, when the point x corresponds to a material that blocks air, we choose α(x)  u(x), which results in u(x) ≈ 0, and when the point x corresponds to air we choose α(x) = 0. We assume that the exterior walls of the building are solid and correspond to the boundary of Ω, denoted ∂Ω. Thus, the boundary condition for the temperature is:

Te (x) ≡ TA ,

∀x ∈ ∂Ω,

(5)

where TA is the atmospheric temperature, and the boundary condition for the air flow is: u(x) ≡ 0,

∀x ∈ ∂Ω.

(6)

Moreover, p(x) ≡ pA for each x ∈ ∂Ω, where pA is the atmospheric pressure. The existence and uniqueness of weak solution to the CFD model in equations (1) to (6) are given formalized below. Theorem 1 (Existence of weak solutions): Let α,
κ ∈ L2 (Ω) be bounded functions, gTe ∈ L2 [t0 , tf ]; H 1 (Ω) , and gu ∈ L2 (Ω) × L2 (Ω).
Then, there exists weak solutions Te ∈ L2 [t0 , tf ]; H 1 (Ω) , u ∈ L2 (Ω)×L2 (Ω) and p ∈ L2 (Ω) to the PDEs in eqs. (1) to (6). Theorem 2 (Uniqueness of weak solutions): Let |Ω| denote the area of Ω, and consider the same conditions as in Thm. 1. 2 , then the weak solutions Te , u, and p, If k∇x ukL2 < |Ω|1/2 Re defined in Thm. 1, are unique. The proofs of Thms. 1 and 2 are omitted from this section, and presented instead in full detail in Appendix B. Note that the condition in Thm. 2 is simply a sufficient condition, which corresponds to the state-of-the-art in terms of uniqueness conditions for stationary Navier-Stokes equations. We assume that there are nt temperature sensors in the building. The i-th sensor is located at xi ∈ Ω, and samples the temperature by averaging the values in a neighborhood using
2 the bump function Φi (x) = σ exp −(r2 − kx − xi k )−1 for kx − xi k < r, and Φi (x) = 0 Rotherwise, where σ > 0 is a normalization factor such that Ω Φi (x) dx = 1, and r > 0 models the sensitivity of the sensor. We also consider nf fan units distributed in different locations of the building, capable of blowing air while introducing or removing heat from the environment. Furthermore, we assume that there are nd doors in the building. We define θi ∈ {0, 1} as the configuration of the i-th door, i.e., θi = 1 when the i-th door is open, and θi = 0 when it is closed. Let Ωθi ⊂ Ω be the area occupied by the i-th door when it is closed. When the door configuration changes, so does the prediction generated by our CFD model. In particular, the parameters α and κ change for each x ∈ Ωθi as a function of θi . We n model this relation by defining α : Ω × {0, 1} d → R and nd κ : Ω × {0, 1} → R as follows: α(x, θ) = αw

nd X (1 − θi ) 1{x ∈ Ωθi },

κ(x, θ) = κ0 + (κw − κ0 )

i=1

-3

-2

-1 -0.5 0 0.5 1

hot

2

3

Fig. 2. Thermal sensations for several PMV values.

where κ0 is the open air thermal diffusivity, αw and κw are the solid wall viscosity and diffusivity, and 1{·} is the indicator function. Note that both α and κ are affine functions of θ ∈ Rn d . Now, using binary values for each θi means that our estimation algorithm will have to rely on mixed-integer methods, which tend to scale poorly in both computation time and computational resources [30]. We avoid this issue by relaxing the binary parameters θi ∈ {0, 1}, instead allowing them to belong to the unit interval [0, 1]. Although for each θi only the extreme values have meaningful physical interpretations, non-integer values can theoretically be interpreted as averaged observations over the optimization horizon [31], [32]. For example, if throughout the optimization horizon a door is open half the time, and closed half the time, it is likely that we will observe θi ≈ 0.5. B. Predicted Mean Vote Index Approximation The PMV index was proposed by Fanger [2] and recommended by ASHRAE [3] in order to predict the average vote of a large group of persons on the thermal sensation scale. It uses heat balance equation to relate six key factors to the average response of people on the thermal comfort. As shown in Fig. 2, the closer the PMV index to zero, the more comfortable the occupant feels. According to [3], the range of PMV index for an acceptable thermal environment of general comfort is from −0.5 to 0.5. For each person in the building, the PMV index is an implicit function of the following six factors: metabolic rate, clothing insulation, air temperature and humidity, air velocity, and mean radiant temperature. Due to its implicit formulation, and in order to reduce the computational effort in our optimization algorithms, we apply known approximations of the PMV index [33], [34]. A detailed presentation of the PMV index approximations is beyond the scope of this section, but it is presented in detail in Appendix A. After applying these approximations, the PMV index is influenced by both the resident’s personal information and variables in our CFD model. Given personal information, define the index value as a1 gu + a2 + pmv(Te (x, t; gTe , gu ), u(x; gu )) = a0 + a3 gu + a4 Te (x, t; gTe , gu ) + a5 Te (x, t; gTe , gu )2 + (8) a6 + a7 Te (x, t; gTe , gu ) + . a8 + a9 Te (x, t; gTe , gu ) + a10 Te (x, t; gTe , gu )2 Where {ai }10 i=0 are parameters related to residents’s personal information, they are presented in detail in Appendix A.

and

i=1

nd X

comfortable values

cold

(7) (1 − θi ) 1{x ∈ Ωθi },

C. Model Predictive Control and Estimation Model predictive control and estimation is a scheme where two optimization problems, one to estimate the initial condi-

4

tion of the system and its unknown parameters, the second to solve a finite-horizon optimal control, are iteratively applied using a receding horizon approach. Given a sequence of times, ∞ {ti }i=0 , at each ti we solve an optimal estimation problem over the past horizon [ti − T, ti ], and an optimal control problem over the future horizon [ti , ti + T 0 ]. Note that, in ∞ practice, the sequence {ti }i=0 is typically equidistant with ti − ti−1  min{T, T 0 }. The solution of the optimal control problem solved at ti is applied in the interval [ti , ti+1 ]. First, we formulate our optimal estimation problem to find the door configuration θ and initial temperature π0 , using the information from the nt thermostats in the building. Given the estimation time horizon as [ti − T, ti ], π0 : Ω → R, and θ ∈ Rnd , we define our optimal estimation objective as: nt  X

Je (π0 , θ) =

η0

Z Ω

k=1

Z

ti

+ ti −T

Z Ω

2 ∗ Φk (x) π0 (x) dx−Te,k (ti −T ) +

Φk (x) Te (x, t; π0 , θ) dx −

2  dt +

∗ Te,k (t)

2

+ η1 kπ0 kΩ . (9) where η0 , η1 > 0 are weight parameters, Te (x, t; π0 , θ) is the solution of eq. (1) with initial condition π0 (x) and door ∗ configuration θ, and Te,k (t) is the time signal obtained from the k-th temperature sensor. Using the objective function in eq. (9) we formulate our optimal estimation problem as follows: min Je (π0 , θ), π0 , θ

subject to: partial differential equations (1), (3), and (4), boundary and initial conditions (2), (5), and (6), θi ∈ [0, 1],

∀i ∈ {1, . . . , nd }.

(10) Second, we formulate the optimal control problem to find the fan and heater control signals, denoted gTe and gu in eqs. (1) and (3). The optimal control problems takes the optimal estimations computed by the problem in eq. (10), deˆ Abusing notation, let Tbe (x) = Te (x, ti ; π ˆ noted π ˆ0 and θ. ˆ0 , θ), i.e., the temperature distribution at time ti obtained from the optimal result of the problem in eq. (10). Given the optimal control horizon as [ti , ti + T 0 ], gTe : [ti , ti + T 0 ] × Ω → R, and gu : Ω → R2 , we define our optimal control objective as: Z Z ti +T 0
pmv Te (x, t; gTe , gu ), u(x; gu ) 2dt dx+ Jc (gTe , gu ) = Ωt ti 2

2

+ η00 kgTe kΩ×[ti ,ti +T 0 ] + η10 kgu kΩ , (11) where η00 , η10 > 0 are weight parameters, Ωt ⊂ Ω is the target area of interest within the building, and Te (x, t; gTe , gu ) and u(x; gu ) are the solutions of eqs. (1), (3), and (4) with control signals gTe and gu . Note that while in most applications the target area Ωt = Ω, our formulation enables us to consider the case of smart zoned climatization, for example by only considering the rooms currently being occupied. Using the objective function in eq. (11) and the results of the optimal

ˆ we formulate our optimal control problem estimator, Tbe and θ, as follows: min

g Te , g u

Jc (gTe , gu ),

subject to: partial differential equations (1), (3), and (4), boundary conditions (5) and (6), π0 (x) = Tbe (x), ∀x ∈ Ω,   gTe (x, t) ∈ gTe , gTe , ∀x ∈ Ω, t ∈ [ti , ti + T 0 ],   gu (x) ∈ gu , gu , ∀x ∈ Ω, (12) where gTe , gTe , gu and gu are minimum and maximum heater and fan power parameters, respectively.

III. NUMERICAL METHOD In this section we develop a numerical algorithm to solve the optimization problems defined in Sec. II-C. We develop a gradient-based optimization algorithm which is compatible with the PDE constraints in our optimization problems, where the gradients are computed using adjoint equations. Then, we numerically find solutions to both our CFD model and its set of adjoint equations using the finite element method (FEM), as described below.

A. Adjoint Variables and Fr´echet Derivatives We begin by formulating a Lagrangian function [35], [36] which is general enough to be valid for both our estimation and control optimization problems. In this section we will assume the existence of a general objective function J(Te , u, p, π0 , θ, gTe , gu ). Thus, the adjoint equations for each optimization problem can be found by using the particular functions Je or Jc , as defined in eqs. (9) and (11), respectively. 6 Let {λi }i=1 be the adjoint variables, or Lagrange multipliers, each associated to one of the eqs. (1) to (6) and defined in its respective dual space. Then, the Lagrangian function of our optimal estimation problem is: 6
L Te , u, p, π0 , θ, gTe , gu , {λi }i=1 =

 ∂ Te − ∇x · (κ ∇x Te ) + u · ∇x Te − gTe Ω×[t ,t ] + = λ1 , 0 f ∂t 

1 + λ2 , − 4x u + (u · ∇x )u + ∇x p + α u − gu Ω + Re + hλ3 , ∇x · uiΩ + hλ4 , Te i∂Ω×[t0 ,tf ] + hλ5 , ui∂Ω +

+ hλ6 , Te (·, t0 ) − π0 iΩ + J(Te , u, p, π0 , θ, gTe , gu ). (13)

A necessary condition for optimality is that the inner product of the partial derivatives of L with respect to each primal variable is equal to zero [36]. After manipulating the resulting partial derivatives, this necessary condition can be transformed into a new set of PDEs that the adjoint variables must satisfy.
Theorem 3: Let Te∗ , u∗ , p∗ , π0∗ , θ∗ , gT∗e , gu∗ be a minimizer of J subject to eqs. (1) to (6). Then, there exist adjoint 6 variables, {λi }i=1 , defined in their corresponding dual spaces

5

as shown in eq. (13), which are weak solutions of the following equations for x ∈ Ω and t ∈ [t0 , tf ]: ∂ λ1 + ∇x · (κ ∇x λ1 ) + u∗ · ∇x λ1 = 0, (14) ∂t Z tf 1 λ1 ∇x Te∗ dt + α λ2 − 4x λ2 − u∗ · ∇x λ2 + Re t0 +λ2 · ∇x u∗ − ∇x λ3 = 0, (15) −∇Te J +

∇x · λ2 = 0, (16)

with initial and final conditions λ6 (x) = λ1 (x, 0) and λ1 (x, tf ) = 0 for each x ∈ Ω, and boundary conditions λ1 (x, t) = 0 and λ2 (x) = 0 for each x ∈ ∂Ω and t ∈ [t0 , tf ]. The proof to Thm. 3 is an extension of [37, Theorem 1.17], and therefore omitted. Note that we also omitted the differential equations describing λ4 and λ5 , since they are irrelevant to our results in this paper. Using the result in Thm. 3 we obtain closed-form formulas for the Fr´echet derivatives of both objective functions, Je and Jc , as defined in eqs. (9) and (11), respectively. Theorem 4: Let J(Te , u, p, π0 , θ, gTe , gu ) be differentiable. Then, the Fr´echet derivatives of J with respect to π0 , θ, gTe , 1 nd and gu , exist. Moreover, for
each δπ0 ∈2H (Ω), 2δθ ∈ R , 2 1 δgTe ∈ L [t0 , tf ]; H (Ω) , and δgu ∈ L (Ω) × L (Ω):

 (17) Dπ0 J, δπ0 Ω = h∇π0 J − λ6 , δπ0 iΩ , D E ∂α Dθ J · δθ = λ2 · u, + ∂θ Ω Z tD  f ∂κE + ∇x λ1 · ∇x Te , dt · δθ, (18) ∂θ Ω t0



 DgTe J, δgTe Ω×[t0 ,t ] = ∇gTe J − λ1 , δgTe Ω×[t ,t ] , (19) 0 f f

 Dgu J, δgu Ω = h∇gu J − λ2 , δgu iΩ . (20) The formulas above result from applying standard linear variations to the objective functions, as shown in [19] and we omitt the detial derivatives in this paper.

B. Gradient-Based Optimization Using the closed-form formulas for the Fr´echet derivatives of Je and Jc , as defined in Thm. 4, we build a gradientbased optimization algorithm to solve the problems in eqs. (10) and (12) inspired in a project-gradient approach [38, Ch. 18.6]. Recall that projected gradient is a iterative method where a sequence of quadratic programming (QP) problems are solved. Thus, we find descent direction δπ0 and δθ as the unique solutions of the following QP with value Ve : Ve (π0 , θ) = min hDπ0 Je , δπ0 iΩ + Dθ Je · δθ+ δπ0 , δθ

+ Mπ0 (δπ0 , δπ0 ) + Mθ (δθ, δθ),

(21)

s.t.: θi + δθi ∈ [0, 1], ∀i ∈ {1, . . . , nd }, where Mπ0 and Mθ are positive-definite bilinear operators. Note that, typically, Mπ0 (δπ0 , δπ0 ) = 12 hπ0 , π0 iΩ and Mθ (δθ, δθ) = 21 δθ · δθ, although other operators such as BFGS [38, Ch. 6.1] can also be used. Similarly, we find

Require: i = 0, t0 = 0, ∆ > 0, T > ∆, T 0 >
∆, εtol ≥ 0, π0 ∈ H 1 (Ω), θ ∈ Rn , gT e ∈ L2 [t0 , tf ], H 1 (Ω) , gu ∈ L2 (Ω) × nt ∗ . L2 (Ω), Te,k |[−T,0] k=1 1: loop 2: loop . Solve estimation problem (10) for t ∈ [ti − T, ti ]. 3: Compute Te , u, and p by solving (1) to (6). 4: Compute λj , j ∈ {1, 2, 3, 6}, by solving (14) to (16). 5: Compute Dπ0 Je and Dθ Je in (17) and (18). 6: Compute Ve (π0 , θ), δπ0 , and δθ by solving (21). 7: if Ve (π0 , θ) ≥ −εtol then 8: Go to line 13. 9: end if 10: Compute βe in (23). 11: Update π0 ← π0 + βe δπ0 and θ ← θ + βe δθ. 12: end loop 13: Store π0 and θ. 14: loop . Solve control problem (12) for t ∈ [ti , ti + T 0 ]. 15: Compute Te , u, and p by solving (1) to (6). 16: Compute λj , j ∈ {1, 2, 3, 6}, by solving (14) to (16). 17: Compute DgTe Jc and Dgu Jc in (19) and (20). 18: Compute Vc (gTe , gu ), δgTe , and δgu by solving (22). 19: if Vc (gTe , gu ) ≥ −εtol then 20: Go to line 25. 21: end if 22: Compute βc in (24). 23: Update gTe ← gTe + βc δgTe and gu ← gu + βc δgu . 24: end loop 25: Apply gTe |[ti ,ti +∆] and gu |[ti ,ti +∆] . 26: Update ti ← ti + ∆ and i ← i + 1. nt ∗ 27: Store Te,k |[ti −T,ti ] k=1 . 28: end loop Fig. 3. Model predictive HVAC estimation and control algorithm.

descent directions δgTe and δgu as the unique solutions of the following QP with value Vc :

 Vc (gTe , gu ) = min DgTeJc , δgTe Ω×[t ,t ]+hDgu Jc , δgu iΩ+ δgTe , δgu

0

f

+ MTe (δgTe , δgTe ) + Mu (δgu , δgu ),   s.t.: (gTe + δgTe )(x, t) ∈ gTe , gTe ,   (gu + δgu )(x) ∈ gu , gu , ∀x ∀t, (22) and Mu are also positive-definite bilinear opera-

where MTe tors. Once we have found descent directions using the QP problems in eqs. (21) and (22), we use Armijo’s method [39] to compute step sizes that guarantee the convergence of our algorithms to points satisfying first-order necessary optimality conditions. That is, we find the largest scalars βe , βc ∈ [0, 1] such that, if Ve (π0 , θ) < 0 then: Je (π0 + βe δπ0 , θ + βe δθ)−Je (π0 , θ) ≤ a βe Ve (π0 , θ), (23) and if Vc (gTe , gu ) < 0 then: Jc (gTe + βc δgTe , gu + βc δgu ) − Jc (gTe , gu ) ≤

≤ a βc Vc (gTe , gu ), (24)

where a ∈ (0, 1) is a parameter. We summarize our HVAC estimation and control algorithm in Fig. 3. Note that the parameter εtol defines the convergence tolerance of our optimization algorithm. In practical applications it is recommended to choose εtol > 0, but the

6

d3 s2 h 4 d4

s3

h2 d2

d1 s1

h3

h1

changes in airflow due to different door configurations are significant enough to produce a sizable temperate estimation error, which in turns leads to uncomfortable conditions as shown in Fig. 6a. Fig. 6b shows that, after the change in door configuration at t = 50 s, it takes our door configuration estimator about 250 s to recalculate the new door configuration, which is compatible with the time constant of the system’s dynamics.

Fig. 4. Simulated apartment floor plan.

convergence of the optimization algorithm is guaranteed only when εtol = 0, as shown below. Theorem 5: Consider the algorithm in Fig. 3 and assume that εtol = 0. Then the sequences generated by the sub-methods in lines 2 to 12, and lines 14 to 24, accumulate in points nullifying Ve and Vc , respectively. The proof of Theorem 5 is a direct extension of the result in [40, Ch. 4] and we omitt it in this paper. IV. SIMULATED RESULTS We applied our method to a simulated St. Louis area apartment, whose floor plan is shown in Fig. 4. The apartment has an area of 7.6 × 16.8 m2 (1375 sq ft, approx.), has nd = 4 nd 3 doors, denoted {di }i=1 , nt = 3 thermostats, labeled {si }i=1 , 4 and is equipped with four HVAC vents, labeled {hi }i=1 . We assume that each vent works independently, endowed with a fan acting on an 1 × 0.5 m2 area. Our CFD model is governed by the constants Re = 102 , κ0 = 10−2 , αw = 103 , κw = 10−4 , and pA = 101.3 kPa. The sensor sensitivity is r = 1.0 m, and the parameters in (9) and (11) are η0 = 1.0, η1 = 0.1, η00 = 0.1, and η10 = 0.15. The bilinear operators in (21) and (22) are estimated using the BFGS method [38, Ch. 6.1]. Our simulations are implemented using Python, and the each PDE is solved using a FEM discretization computed with tools from the FEniCS project [41]. The floor plan was discretized into 6276 discrete elements. A. Variable door configuration and target location In this experiment we study the accuracy of our estimation method under changing door configurations, while our control methods maximizes efficient and comfort in 12 different target areas uniformly distributed across the apartment. We compare the performance of our method against a scheme consisting of a similar controller but no door configuration estimator. Thus, the comparison controller assumes that the doors are always closed. The atmospheric temperature is set to TA = 5 ◦ C, the clothing insulation to 0.155 ◦ C m2 /W, and the metabolic rate to 64.0 W/m2 , resulting in an outdoor PMV index of −4.1 (i.e., very cold as shown in Fig. 2). The simulation begins with all four doors closed, and at time 50 s all the doors are simultaneously open. Figs. 5a and 5b show how our method adapts to changing target areas when the door estimator is enabled, quickly reaching comfortable conditions after a few minutes while reducing energy consumption in unused areas. Fig. 5c shows the impact of our door configuration estimation method in the temperature distribution. Indeed, the

B. Simulation with different personal variables related to PMV In this simulation we show how our climate control scheme is capable of not only adapting to changes in building configuration, but also to changes in the occupancy patterns and behaviors. In particular, we consider the cases where metabolic rate and clothing insulation are known variables, and feed that information back to the controller. While this information is not typically known in a traditional climate control setup, it is possible to obtain it via wearable sensors or smartphone surveys [20], [21], [22]. Figs. 7a and 7b show the impact that changes in metabolic rate and clothing insulation have in the energy consumption when using our climate control scheme, respectively. We ran simulations over a period of 600 s, with atmospheric temperature set to 20 ◦ C. Our choice of atmospheric temperature is such that the PMV value is negative when occupants have low metabolic rate and clothing index, and is positive when occupants have high metabolic rate and clothing index. Thus, as we vary the values for metabolic rate and clothing insulation, the HVAC system automatically switches from heating to cooling. As shown in both figures, both variables have a significant impact in energy consumption, particularly when heating is required due to low metabolic rates or clothing insulation. Therefore, our scheme not only results in a more comfortable environment under different occupancy patterns, it also opens the door to the semi-autonomous operation of a building where the climate control system suggests, in realtime, changes in clothing insulation or occupant behavior that would help saving energy. C. Memory and time usage analysis Fig. 8a shows the maximum memory usage under different number of FEM elements, ne , resulting in a memory complexity roughly proportional to n1.14 . Similarly, Fig. 8b shows the e average CPU time of each iteration of the algorithm in Fig. 3 under different number of FEM elements, resulting in a time complexity roughly proportional to n1.59 . e While our scheme is still not fast enough to run in real-time, our CPU time is within less than an order of magnitude of the desired speed even in a non-optimized implementation using high-level languages and traditional CPU, rather than GPGPU, computation. Also, real-time implementations are a function of the estimation and control horizons, denoted T and T 0 in Fig. 3. As shown in Fig. 9, there is not a large impact when longer horizons are used, mostly due to the slow dynamic that climate variables have in general.

7

d3 h4 d4

d2

h3

2.0 1.5 1.0 0.5 0.0 −0.5 −1.0 −1.5 −2.0

h2 d1

target h1

(a) PMV distribution for specific target area.

d3 h4 d4

2.0 1.5 1.0 0.5 0.0 −0.5 −1.0 −1.5 −2.0

h2 d2

d1 target h1

h3

target h4 d4

d3

h2 d2

d1

h3

(b) PMV distribution for specific target area.

h1

7 6 5 4 3 2 1 0

(c) Temperature error distribution when no estimator is used for specific target area.

Fig. 5. Results of representative experiments in Sec. IV-A. Each plot was generated at t = 900 s using a different target area.

1.2

0.9

101

0.8

1.0

104

0.6

0.6

0.5

Memory usage

0.8

0.4

0.4

0.3

0.2

0.2

0.0

(M)

0.1 0 100 200 300 400 500 600 700 800 900 Time(s)

(F)

(a) Average absolute PMV value in target area. Left (M): using our estimator. Right (F): assuming closed doors.

Computation time

0.7

(b) Average estimation over all Pnerror d ˆ target areas, n1 i=1 θi − θi , d where θˆ is our estimator.

Fig. 6. Results of the simulations in Sec. IV-A using 12 different target areas.

100 10

103 3

4

103

10 Number of FEM elements

(a) Memory usage (GB) vs. number of FEM elements.

104 Number of FEM elements

(b) Computation time (sec) vs. number of FEM elements.

Fig. 8. Results of the experiments in Sec. IV-C. 1.5

0.7

0.7

0.6

0.0

0.5

−0.5

0.6 0.5

Energy

Energy

1.0

0.8

0.4 0.3

0.5

−1.0

0.4

−2.0 −2.5

0.2

0.2 0.1

0.1

0.0 50

0.0 0.05

60

70 80 90 100 110 120 Metabolic Rate

(a) Energy usage (kW h) vs. metabolic rate (W/m2 ) with clothing insulation 0.155 ◦ C m2 /W.

T = 150 (s) T = 100 (s) T = 200 (s)

−1.5

0.3

−3.0

0.10 0.15 0.20 0.25 Clothing Insulation

0

100

200

300 Time(s)

400

500

600

0.30

Fig. 9. Average PMV value in target area for different control horizons.

(b) Energy usage (kW h) vs. clothing insulation (◦ C m2 /W) with metabolic rate 81 W/m2 .

Fig. 7. Results of the simulations in Sec. IV-B. [6]

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Runxin He (S’16) received the B.S. degree in electrical engineering from Fudan University, Shanghai, China, in 2012, and the M.S. and Ph.D. degrees in electrical engineering from Washington University in Saint Louis, MO, USA, in 2017. His research focuses on the development and application of novel numerical methods to operation research and data science applications.

Humberto Gonzalez (S’03-M’12) received the B.S. and M.S. degrees in electrical engineering from Universidad de Chile, Santiago, Chile, in 2005, and the Ph.D. degree in electrical engineering and computer sciences from the University of California at Berkeley, CA, USA, in 2012. He is currently a Senior Data Scientist at Instart Logic. His research focuses on the development and application of novel numerical methods to data science applications.

9

A PPENDIX A P REDICTED M EAN VOTE F ORMULA According to Fanger [2] and ASHRAE [3], the PMV index is computed by  pmv = (0.303 exp(−0.036 M ) + 0.028) (M − W ) +
− 3.05 · 10−3 5733 − 6.99 (M − W ) − pa + − 0.42 (M − W − 58.15) − 1.7 · 10−5 M (5867 − pa ) +

− 0.0014 M (34 − Te ) − fcl hc (Tcl − Te ) +
 − 3.96 · 10−8 fcl (Tcl + 273)4 − (Tr + 273)4 . (25)

Where M and W are the metabolic rate and external work, both in W/m2 . The external work normally is around zero [26], [3], and the average value of human’s sedentary activity and standing activity is 70 W/m2 and 93 W/m2 respectively. The occupant’s metabolic rate can be obtained either from their wearable devices or some posterior estimation [20], [21], [22]. The term pa corresponds to the partial water vapor pressure, measured in Pascals. According to [42], [43], the specific humidity inside house, wi with unit kg · kg−1 , can be expressed by wo ρA gu + mg . (26) wi = ρA gu Where wo is the specific humidity comes out of the HVAC, ρ is the inside air’s density. mg is the rate of moisture generation within the building with unit kg · s−1 , A is the size of the fan with unit m2 . In the computation, we ignore the moisture diffusion through the fabric material [26]. Under ideal air condition, pa = 1.608 po wi , where the mixed air’s pressure po is the standard atmosphere, 1.013 · 105 Pa. Te and Tr are the air temperature and mean radiant temperature, both with unit ◦ C. Experiments results [44] show that their indoor distributions are close. In order to simplify the PMV index for optimization computation and noting that most buildings typically do not have sensors to continually measure the mean radiant temperature, we set the Tr equal to the air temperature, Te . Moreover, since the range of temperatures in the indoor environment is small, the difference of the fourth power terms can be adequately replaced by a lower-order difference [26]: 4 4
3.96 · 10−8 fcl (Tcl + 273) − (Tr + 273) ≈ ≈ 4.6 fcl (1 + 0.01Tr ) (Tcl − Tr ). (27)

The clothing surface temperature is approximated as [26], [2]:  Tcl = −0.155 Icl fcl 4.6 (1 + 0.01Tr ) (Tcl − Tr ) +  + hc (Tcl − Te ) − 0.028 (M − W ) + 35.7. (28) Where we approximate the radiation term with equation (27), then we can derive an explicit formula of Tcl and get rid of the iteration numerical solving process. hc is the convective heat transfer coefficient [2], [45] and is approximated as the natural convective heat transfer coefficient, hcn [33], [34]. The parameter fcl is equal to 1.0 + 1.29 Icl when Icl ≤ 0.078,

otherwise 1.05 + 0.645 Icl , where Icl is the clothing insulation index, in m2 ◦ C/W. After the simplications and derivatives above, the PMV index follows equation (8) where the parameters, {ai }10 i=0 , are  a0 = (0.303 exp(−0.036 M ) + 0.028) (M − W ) +
− 3.05 · 10−3 5733 − 6.99 (M − W ) − 0.0476 M +  − 0.42 (M − W − 58.15) − 9.974 · 10−2 M + , a1 = (0.303 exp(−0.036 M ) + 0.028) ·
· 3.05 · 10−3 + 1.7 · 10−5 M wo ρA,

a2 = (0.303 exp(−0.036 M ) + 0.028) ·
· 3.05 · 10−3 + 1.7 · 10−5 M mg , a3 = ρA,

a4 = (0.303 exp(−0.036 M ) + 0.028) · · (.0014 M + fcl hc + 4.6fcl ),

a5 = 0.046fcl (0.303 exp(−0.036 M ) + 0.028), a6 = (0.303 exp(−0.036 M ) + 0.028) ·

· (4.6 + hc )fcl (−0.028(M − W ) + 35.7),

a7 = (0.303 exp(−0.036 M ) + 0.028) ·
2 · Ic fcl 0.155h2c + 1.426hc + 3.45 , a8 = 1,

a9 = Ic fcl (()0.713 + hc ), and a10 = 0.007Ic fcl . (29) A PPENDIX B E XISTENCE AND U NIQUENESS OF S OLUTIONS TO THE CFD M ODEL In this section, we are going to show the solution’s existence and uniqueness to the CFD model containing equations (1), (2), (3), (4), (5) and (6) in Sec. II-A. First, let us describe our notation. The definitions and notations of L2 (Ω) and the Sobolev space H 1 (Ω) follow [46]. Let the subspace for H 1 (Ω) with trivial boundary be denoted:  H01 (Ω) = v ∈ H 1 (Ω) | v|∂Ω = 0 . (30) Let the divergence-free subspace of H01 (Ω) × H01 (Ω) be defined by:  V0 = v ∈ H01 (Ω) × H01 (Ω) | ∇x · v = 0 . (31) Let H0 be the completion of V0 w.r.t. L2 norm, and is given by:  H0 = v ∈ L2 (Ω) × L2 (Ω) | ∇x · v = 0, and v|∂Ω = 0 . (32) 2 The space H0 is equipped with the L -norm, denoted k·k 0. Let V1 = η ∈ H 1 (Ω) | η|∂Ω = 0 , endowed with the norm kf k1 = k∇x f k0 for each f ∈ V1 . Also, let:  L2 ([t0 , t0 + T ]; V1 ) = u : [t0 , t0 + T ] → V1 |  Z t0 +T 2 u is measurable and ku(t)k1 dt < ∞ , (33) t0

10

and: C 0 [t0 , t0 + T ]; L2 (Ω) = u : [t0 , t0 + T ] → L2 (Ω) | u(t) is measurable and continuous for a.e. t . (34)




Now we focus our attention on the existence and uniqueness of the CFD system in eqs. (1) to (6), stated in Thms. 1 and 2. In order to prove these theorems, we begin studying the weak solution PDE subsystem in eqs. (3), (4) and boundary condition (6). The techniques used in our proofs follow closely those previously published results [46], [47], [15]. According to the Hopf extension [46, Lemma I.4.2.3], for 1 1 each ε > 0 there exists a function u ¯ ∈ H (Ω) × H (Ω) such2 that ∇x · u ¯ = 0, u ¯|∂Ω = 0, and (v · ∇x ) u ¯, v Ω ≤ ε|v|1 for each v ∈ V0 . Then, any function u ∈ H 1 (Ω) × H 1 (Ω) satisfying the boundary condition (6) can be written as u = w+u ¯, for some w ∈ V0 . For simplification, define Σ as H 1 (Ω) × H 1 (Ω). We use a fixed-point theorem to show the existence of weak solutions of eq. (3). Consider the following weak formulation of (3) for each ϕ ∈ V0 : hα(x)w, ϕiΩ +

1 h∇x u, ∇x ϕiΩ + hˆ u · ∇x w, ϕiΩ + Re + hu · ∇x u ¯, ϕiΩ = hgu , ϕiΩ . (35)

Consider the mapping S such that given u ˆ, it returns u, the solution to the equation (35), as S(ˆ u) = u. Then, the fixed point of the mapping S is a weak solution to the equation (3). Let us define the following bilinear operator, for each w, ϕ ∈ V0 : 1 h∇x u, ∇x ϕiΩ + Re + hˆ u · ∇x w, ϕiΩ + hu · ∇x u ¯, ϕiΩ . (36)

σ0 (w, ϕ) = hα(x)w, ϕiΩ +

The proof of Lemmas 1 to 5 below can be found in [47]. Lemma 1: σ0 is bounded. Definition 1: A bilinear form a : V × V → R is called coercive if there exists a constant C > 0 such that for each 2 x ∈ V , |a(x, x)| ≥ CkxkV . Lemma 2: σ0 is coercive. Lemma 3: For each ϕ ∈ V0 , the equation (35) has a unique weak solution u. Moreover, there exists w ∈ V0 such that u can be separated as u = w + u ¯. Lemma 4: For each u ˆ ∈ V0 , the solution u = S(ˆ u) to equation (35) is always bounded. Lemma 5: The mapping S : Σ → Σ is compact. Based on these lemmas, we can derive the existence of weak solutions to eq. (3). Lemma 6: There exists at least one fixed point, say u, for the mapping S, such that u ∈ V0 + u ¯. The proof of Lemma 6 follows directly from Lemma 5, as shown in [48, Theorem 1.J]. We now turn our attention to the existence and uniqueness of weak solutions of eq. (1). Let us define bilinear operator σ1 for each t ∈ [t0 , t0 + T ] by: σ1 (Te , ξ) = hκ(x) ∇x Te , ∇x ξiΩ + hu · ∇x Te , ξiΩ .

(37)

Then, given t ∈ [t0 , t0 + T ], the weak formulation of eq. (1) can be written as follows, where ξ ∈ V1 :   dTe ,ξ + σ1 (Te , ξ) = hgTe , ξiΩ . (38) dt Ω Lemma 7: σ1 is bounded. Proof: Since κ(x) and u(x) are bounded, using H¨older’s inequality we can show that σ1 is bounded using the same argument as in Lemma 1. Lemma 8: σ1 is coercive. Proof: First, according to equation (4) and derivative by part, hu · ∇x Te , Te iΩ = 0. Then by Poincare-Friedrichs inequality [49], there exists C > 0 such that: |σ1 (Te , Te )| = hκ(x) ∇x Te , ∇x Te iΩ + hu · ∇x Te , Te iΩ = hκ(x) ∇x Te , ∇x Te i Ω

2

≥ C kTe k1 ,

(39) as desired. Proof of Thm. 1: The existence of weak solutions to eq. (1) follows by Lemmas 7 and 8, together with [50, Theorem 11.1.1]. The existence of weak solutions to eqs. (3) and (4) follows by Lemma 6. Finally, we prove a sufficient condition for the uniqueness of the weak solutions to our CFD system. Proof of Thm. 2: Note that the uniqueness of the weak solution to eq. (1) is guaranteed together with the existence result in [50, Theorem 11.1.1]. Suppose u1 and u2 are two different weak solutions to eq. (3) with the same boundary conditions. Let u ˜ = u1 − u2 , then for each ϕ ∈ V0 : hα(x) u ˜, ϕiΩ +

1 h∇x u ˜, ∇x ϕiΩ + hu1 · ∇x u ˜, ϕiΩ + Re + h˜ u · ∇x u2 , ϕiΩ = 0. (40)

Let us set ϕ = u ˜, then: hα(x) u ˜, u ˜iΩ +

1 h∇x u ˜, ∇x u ˜iΩ + hu1 · ∇x u ˜, u ˜iΩ + Re + h˜ u · ∇x u2 , u ˜iΩ = 0. (41)

Thus we can bound the norm of u ˜ as: 1 h∇x u ˜, ∇x u ˜iΩ = −hα(x) u ˜, u ˜iΩ − h˜ u · ∇x u2 , u ˜iΩ Re ≤ h˜ u · ∇x u2 , u ˜i Ω

(42)

2

≤ C ku2 k1 k˜ uk1 ,

1/2

9.1.2]. where C = |Ω|2 , as shown in [51, Lemma
2 1 Eq. (42) implies that Re − C ku2 k1 k˜ uk1 ≤ 0. Thus, if 1 uk1 = 0, as desired. Re − C ku2 k1 ≥ 0 then k˜