Technology. Sirisha. Ritesh. A. Khire. on the. Thermal Control

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1 Impact of Weather Uncertainties on Active Building Envelopes (ABE): An Emerging Thermal Control Technology Sirisha Rangavajhala Ritesh A. Khire Achille Messac Corresponding Author Achille Messac, Ph.D. Distinguished Professor and Department Chair Mechanical and Aerospace Engineering Syracuse University, 263 Link Hall Syracuse, New York 13244, USA Tel: (315) Fax: (315) Bibliographical Informationn Rangavajhala, S., Khire, R. A., and Messac, A., Impact of Weather Uncertainties on the Design of Active Building Envelope (ABE): An Emerging Thermal Control Technology, 11h Multidisciplinary Analysis and Optimization Conference, Portsmouth, Virginia, Paper No. AIAA , Sep. 6-8, 2006.

2 11th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference 6-8 September 2006, Portsmouth, Virginia AIAA th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, 6-8 Sep 2006, Portsmouth, Virginia Impact of Weather Uncertainties on Active Building Envelopes (ABE): An Emerging Thermal Control Technology Sirisha Rangavajhala, Ritesh A. Khire, and Achille Messac Rensselaer Polytechnic Institute, Troy, NY 12180, U.S.A. In this paper, we investigate the impact of weather uncertainties on the optimal design of Active Building Envelopes (ABE). Recently, ABE systems have been proposed as a technology that uses solar energy to maintain a comfortable indoor environment. ABE systems are multidisciplinary in nature, where solar radiation energy is converted into electrical energy by means of a photovoltaic unit (PV unit) that powers a thermoelectric heat pump unit (TE unit). Our earlier work focused on a deterministic optimization of ABE systems. In this paper, we account for critical uncertainties in the outside temperature and solar radiation. Two conflicting issues of interest are: (1) objective function minimization, and (2) minimization of constraint violation under uncertainty. The main design constraint in ABE systems is to ensure that (i) the heat absorbing capacity of the TE unit and (ii) the amount of heat entering the house (also called cooling load) are as close to each other as possible under uncertainty. Also, we must ensure that the underlying physics of the interacting subsystems is not violated. The above requirements, which are imposed as equality constraints in the deterministic problem, need to be appropriately formulated to ensure feasibility under uncertain conditions. In this paper, we use an equality constraint formulation to address the above issues, which results in a multiobjective optimization problem. This yields several designs of the ABE system, each offering a different tradeoff between the objective function value and the ability to match the heat absorbing capacity with the cooling load. As we show in the paper, this uncertainty based design approach gives the designer a realistic understanding of the tradeoff involved in selecting an appropriate design of ABE systems. I. Introduction With the rising costs of conventional fuels, maintaining a comfortable indoor temperature using conventional technologies has become an expensive endeavor, summer and winter alike. Active Building Envelope (ABE) systems are proposed to alleviate this problem by minimizing our dependence on conventional fuels. ABE system is a new technology that actively uses solar energy to maintain a comfortable indoor environment. In these systems, solar radiation energy is converted into electrical energy by means of a photovoltaic unit (PV unit) that powers a thermoelectric heat pump unit (TE unit). In this paper, we investigate the impact of weather uncertainties on the optimal design of ABE systems. Through a systematic robust optimization study, we show that these uncertainties play a critical role in the optimization of ABE systems, and significantly affect the decision making process. A. Background of ABE Systems A brief description of the ABE system is provided in this subsection, for more details, see references. 1 4 As shown in Fig. 1, the ABE system comprises a photovoltaic unit (PV unit) and a thermoelectric heat pump PhD Candidate, Department of Mechanical Engineering, and AIAA student member. PhD Candidate, Department of Mechanical Engineering, and AIAA student member. Professor, Department of Mechanical Engineering, and AIAA Fellow, Corresponding author, messac@rpi.edu Copyright c 2006 by Achille Messac. Published by the, Inc. with permission. 1of16 Copyright 2006 by Achille Messac. Published by the American American Institute Institute of Aeronautics of Aeronautics and Astronautics, andinc., Astronautics with permission.

3 unit (TE unit). The PV unit consists of photovoltaic cells (solar cells), which are solid-state devices that convert solar radiation energy into electrical energy. The TE unit consists of thermoelectric heaters/coolers (referred to here onwards as TE coolers), which are solid-state devices that convert electrical energy into thermal energy, or vice-versa. The gap between the PV unit and the external wall acts as an external heat dissipation zone for the TE unit (see Fig. 1). The external walls of the ABE system consist of two layers. The external layer facing the PV unit is made of a good thermal insulating material, and the internal layer is made of a material with high heat storage capacity. Thermoelectric Cooler Internal Heat Sink External Heat Sink Heat Dissipation Zone Thermal Mass Photovoltaic System Thermal Insulation Air Flow Figure 1. Active Building Envelope (ABE) System In Fig. 1, the words Thermal Insulation and Thermal Mass pertain to the external and internal layers of the ABE wall, respectively. TE coolers are dispersed inside the openings provided in the insulating layer. Each TE cooler consists of two heat sinks. The internal heat sink either absorbs or dissipates heat to the thermal mass layer. The external heat sink either absorbs heat from, or dissipates heat to, the surrounding air. The internal and external heat sinks act in opposing ways. In the next subsection, we discuss the issues that are critical in the design of ABE systems. B. Design Issues in ABE Systems We divide this subsection in two parts. In the first part, we review the ABE system related research published in the literature. As we shall see, so far in the literature, the primary focus has been on the deterministic design of ABE systems, not addressing the need to account for weather uncertainties in its design. In the second part, we discuss the role of weather uncertainties in the design of ABE systems. 1. Deterministic Design/Optimization of ABE Systems Since ABE systems are relatively new, only limited information is available in the literature. Van Dessel et al. 4 performed a preliminary investigation to explore the feasibility of ABE systems for thermal conditioning of indoor spaces. Khire et al. 1, 2, 5 performed extensive optimization studies to uncover critical issues in the design/optimization of the TE unit. Khire and Messac 6 used the Selection-Integrated Optimization (SIO) methodology to design ABE systems as adaptive systems. Rivas et al. 3, 7 performed an economic evaluation of ABE systems to explore their potential to outperform conventional air-conditioning technologies from a life-cycle cost perspective. In the ABE related research discussed above, it was assumed that the operating conditions (governed by the weather) are deterministic entities, which is not the case in practice. To perform a realistic design, it is important to account for the inherent weather uncertainties in the design/optimization of ABE systems. This paper presents our first step in this direction. We note that other sources of uncertainties, such as those from material properties, may also be present in ABE systems. However, in this paper, we focus our attention on weather uncertainties. 2of16

4 2. Role of Weather Conditions in ABE Systems In ABE systems, weather uncertainties are likely to affect: (1) the electric power generated by the PV unit, (2) the heat absorption capacity of the TE unit, and (3) the amount of heat entering the house, also called the cooling load. PV unit: Solar radiation is the primary source of energy for ABE systems. Any variation in solar radiation affects the power generated by the PV unit. Additionally, the output of the PV unit (electrical power) also depends on the outside air temperature. As the outside temperature increases, so does the PV unit temperature. An increase in the PV unit temperature adversely affects its power output. Solar radiation and the outside temperature therefore influence the performance of the PV unit. Hence, we consider the uncertainties in these two quantities in the optimization of the ABE system. TE unit: In the TE unit, TE coolers absorb heat from the house and dissipate it to the surroundings. This dissipation of heat is facilitated by heat sinks shown in Fig. 1. The heat dissipation capacity of heat sinks is adversely affected by an increase in the outside temperature. Any worsening of the heat sink dissipation capacity negatively affects the performance of the TE unit. Hence, to design a robust TE unit, and thereby a robust ABE system, it is important to account for the uncertainties in the outside temperature. House: The amount of heat entering the house, or cooling load, depends on the outside and inside temperatures, and solar radiation. The outside and inside temperatures affect the cooling load because of conduction heat transfer through the walls. Solar radiation, on the other hand, affects the cooling load because of radiation heat transfer through transparent and open areas such as windows and doors. In this paper, we assume that the house contains no transparent or open areas. Hence, we account for the uncertainties in the outside and inside temperatures only, and not for the uncertainty in solar radiation. Based on the above discussion, three quantities solar radiation, the outside temperature, and the inside temperature are expected to play an important role in the performance of ABE systems. Therefore, to design a realistic ABE system, it is critical to account for the uncertainties in these three quantities. We note that there are other weather parameters that are likely to affect the performance of ABE systems, such as cloudiness and humidity. Considering the introductory scope of this paper, we do not account for the uncertainties in these parameters. C. Motivation for This Paper The variation in weather can take place in different time scales, long-term or short-term. The long-term variation, for example, is caused by change in seasons. On the other hand, short-term variation can arise because of the change in the hour of the day. In air-conditioning applications, the long-term variability can be tackled by: (1) either using a different system in each season, for example, a heater in the winter and a cooler in the summer, or (2) by designing an adaptive system that can change its configuration to accommodate the change in weather patterns. 6 The short-term variability in weather patterns is likely to have a significant impact on factors such as the cooling load and the demand of electricity. For example, the maximum temperature on a particular day of the year, when observed over several years, is not constant thus introducing an uncertainty in its actual value. In the case of an ABE system design, if the short-term variability in weather is not accounted for, the desired thermal conditioning of the space may not be reliably accomplished. In this paper, we use robust design optimization to include weather uncertainties in the optimization of ABE systems. We quantify weather uncertainties in terms of means and standard deviations using the data available in the literature. We modify the deterministic optimization problem according to a previously developed constraint formulation approach to account for these uncertainties. To accomplish an effective constraint handling under uncertainty, this approach formulates the robust design optimization problem as a multiobjective problem. This multiobjective problem yields several ABE system design alternatives, each representing a different tradeoff between objective function minimization and constraint satisfaction. The design selected following this approach is expected to be more reliable than a deterministic design, which does not consider weather uncertainties. The paper is organized as follows. In Section II, the engineering models of the overall ABE system and the individual subsystems are provided. The deterministic optimization formulation is also presented. In 3of16

5 Section III, uncertainties in weather conditions are introduced in the ABE design. The impact of weather uncertainties on each of the subsystems is discussed. In Section IV, the interactions between the subsystems under uncertainty are discussed, and the robust optimization formulation is presented. Section V presents the results and discussion for the robust optimization of the ABE systems. Concluding remarks are presented in Section VI. II. A Brief Introduction to the Model of Active Building Envelopes (ABE) In this section, we provide the details of the overall model of ABE systems used in this study. The engineering models for the PV unit, TE unit, and the house are explained, followed by the deterministic optimization problem formulation. A. Overall Model of ABE System TE Unit (Heat Pump) Collection of TE coolers, H H s H p : Parallel H s : Series I V H H H : Voltage p I V H : Current H V H = V S I H = I S PV Unit (Generator) Collection of solar cells, S S s S p : Parallel S V s : Series S S V I p S : Voltage S I S : Current Cooling Load, Q load House Generates cooling load Q load = f (T - o T ) i T i = Inside temperature T o T o and E affects I S Changing with season T o : Outside Temperature E: Radiation Figure 2. Model of Active Building Envelope (ABE) System Figure 2 shows a schematic of the model of the ABE system that integrates the models of the PV unit, the TE unit, and the house. As shown in Fig. 2, the model of the house determines the cooling load based on the outside temperature (internal heat sources are not included in this preliminary study). The TE unit is designed to absorb the cooling load. The TE unit consists of TE coolers placed in a grid formation, see Fig. 2. We determine the input voltage and the input current required to operate the TE unit from the number of TE coolers connected in series, H S, and in parallel, H P, as shown in Fig. 2. The TE unit is powered by the PV unit, which is also a grid of solar cells (see Fig. 2). By using the appropriate number of solar cells in series, S S, and in parallel, S P, we satisfy the input voltage and current requirements of the TE unit. Next, we briefly describe the three models shown in Fig Model of TE Unit The TE unit is a collection of TE coolers. The basic element of a TE cooler is called a thermocouple. 8 The amount of heat absorbed by one TE cooler, 9 Q pc, that contains n thermocouples is given as [ Q pc = n Si t T c 1 ] 2 i2 t R K (T o +(Q pc + v t i t )R h T c ) (1) where v t = S (T o +(Q pc + v t i t )R h T c )+i t R (2) In Eqs. 1 and 2; S, K, andr are the Seebeck coefficient, the thermal conductance, and the electrical resistance of the thermocouple, respectively; T c is the temperature of the cold junctions of the TE cooler; T o is the outside temperature; v t and i t are the input voltage and the input current for one TE cooler, 4of16

6 respectively; and R h is the thermal resistance of the heat sink attached to each TE cooler. To determine the heat absorbed by the entire TE unit, we multiply Eq. 1 by the product of H S and H P. We use CP type off-the-shelf TE coolers in this example. The properties of this type of TE coolers is available in the manufacturer s catalog. 10 The input voltage, V TE, and the input current, I TE, required to operate the TE unit are determined as follows. 2. Model of PV Unit V TE = H S v t and I TE = H P i t (3) For the PV unit that contains S S solar cells in series and S P solar cells in parallel, the governing equation is given as follows. 11 I PV S P = I ph I o [ ( q exp Ak b T pv ( VPV S S + I )) ] PVR s 1 1 ( VPV + I ) PVR s S P R sh S S S P where V PV and I PV are the voltage and the current generated by the PV unit, respectively; I ph, I o, R s, R sh, and A are the photocurrent, the reverse saturated current, the series resistance, the shunt resistance, and the ideality factor of the solar cell, respectively; q is the electron charge; k b is the Boltzmann s constant; and T PV is the solar cell temperature. In Eq. 4, the photocurrent and the solar cell temperature depend on the weather conditions. To determine the photocurrent and the solar cell temperature, we use the following models given in Ref. 11 I ph = E e I sc (5) T PV = (E e 300) (T o 273) (6) where E e is the solar radiation per unit area, and I sc is the short circuit current. We note that in this paper, we use the solar cell data (such as I o,r s,r sh,a,andi sc ) provided in Ref Model of House We assume that all the walls of the house considered in this example are installed with an ABE system, and the conduction through the ABE walls is the only mode of heat transfer (no windows or doors in the house). The amount of heat conducted through the ABE wall, or the cooling load, is determined by Fourier s law as T o T i Q load = k ABE A ABE (7) t ABE where k ABE, A ABE,andt ABE are the thermal conductivity, the surface area, and the thickness of the ABE wall, respectively; and T i is the temperature inside the house. The above individual models of the PV unit, the TE unit, and the house are now integrated, and the design of the ABE system is posed as an optimization problem, which is discussed next. B. Deterministic Optimization Problem for ABE Systems In the deterministic optimization of ABE systems, we minimize the total number of TE coolers and solar cells. We assume that the cost of an ABE system depends on the number of TE coolers and solar cells included in it. By minimizing their total number, we expect to design a cost effective ABE system. The optimization problem formulation at a specific weather condition in a given day is as follows: (4) subject to min x=[s S,S P,H S,H P] N =(S S S P )+(H S H P ) (8) h 1 (H S H P ) Q pc = Q load (9) h 2 V TE = V PV (10) h 3 I TE = I PV (11) 1 S S,S P,H S,H P 50 (12) 5of16

7 In the above formulation, the objective function, N, represents the total number of TE coolers and solar cells used in the ABE design. In the above formulation, the equality constraint given in Eq. 9 ensures that the ABE system absorbs the total cooling load. On the other hand, the equality constraints given in Eqs. 10 and 11 ensure the physics based compatibility of the TE and PV units (i.e., the TE unit operates according to the voltage and current supplied by the PV unit alone). As we shall see next, these equality constraints play a critical role in the optimization of ABE systems when weather uncertainties are introduced. III. Handling Uncertainties In ABE Systems So far, we discussed the deterministic design and optimization of ABE systems. In this section, we begin discussing the pertinent issues in the uncertainty based design of ABE systems. We begin by discussing how uncertainties in outside and inside temperatures and solar radiation are incorporated in each of the subsystems of the ABE model. We present the assumptions relevant to uncertainties made in this study. We discuss each subsystem separately, in terms of its inputs and outputs, and the associated impact of uncertainty. The robust design formulation for the ABE system, where the subsystem interactions are incorporated, is then presented in the following section. A. Assumptions Made in This Study We use a robust design optimization (RDO) approach to evaluate the impact of weather uncertainty on the ABE design. Two conflicting issues are generally of interest to the designer in most robust design problems: (1) optimizing the mean performance of the design, often referred to as optimality, and (2) minimizing the performance variation of the design, often referred to as robustness. Robust optimization approaches attempt to achieve a tradeoff between optimality and robustness We parenthetically note that the definition of performance is often subjective. We define the performance measure that we use for the ABE system later in the paper. In RDO problems, probability theory is generally used to quantify the uncertainties in the design variables by modeling them as random variables. For the ABE system design, three uncertain quantities are considered: (1) outside temperature, (2) inside temperature, and (3) solar radiation. We represent the uncertainties in the above quantities by their means and standard deviations, and observe how these uncertainties propagate through each subsystem. We make the following assumptions in this study. 1. As the design point for the ABE system; we select Albany NY, and August 1st as it is a date of high temperature. 2. The maximum outside temperature for the chosen design point is modeled as a normal random variable, 15 with a mean of 27.8 o C and a standard deviation of 2 o C. This maximum temperature is chosen as the outside temperature, T o, for the house model. The desired mean inside temperature is 18 o C, and the desired standard deviation is 1 o C. 3. Based on the solar radiation data available for Albany, NY, 15, 16 a mean solar radiation value of W/m 2, and a standard deviation of 9 W/m 2 is used. 4. Out of the three uncertain inputs to the ABE model, outside temperature and solar radiation are not independent of each other. Based on the historical data available for Albany NY, 15 we estimate the correlation coefficient to be The generic procedure to estimate the correlation coefficient of a sample of data can be found in reference The outside temperature depends on various quantities, such as the time of the day, solar radiation, and the day of the year. On the other hand, the desired inside temperature is governed by human comfort, which is independent of the outside temperature. We therefore make a reasonable assumption that the maximum outside temperature and the desired inside temperature are independent of each other. We now proceed to the next step of incorporating the above uncertainties into the TE unit, the PV unit, and the house. We discuss the effects of weather uncertainties on the inputs and the outputs of each of the subsystems. 6of16

8 B. Uncertainty Propagation in the Subsystems Figure 3 shows the inputs and the outputs of each of the three subsystems in the ABE model, namely, the TE unit, the PV unit, and the house. To facilitate our understanding, a normal distribution symbol is shown next to those quantities that are random in nature (however, they need not necessarily be normal). We note that Fig. 3 shows the effect of weather uncertainties on the individual subsystems only; the effects of interactions between the subsystems are discussed in the next Section. 1. PV Unit: Solar radiation, the outside temperature, solar cell properties, and the number of solar cells in series and parallel are the inputs to the PV unit, as shown in Fig. 3. Because of the random nature of solar radiation and outside temperature, the outputs of the PV unit model PV current and PV voltage are both random as well. Inputs Subsystem Outputs 1. Maximum air Temperature 2. Desired inside Temperature 3. Material prop -erties and area House 1. Heat load 1. TE current 2. TE voltage 3. No. of TE coolers (series) 4. No. of TE coolers (parallel) 5. Material properties TE unit 1. Heat absorbed 1. Solar Radiation 2. Maximum air Temperature 3. No. of solar cells (series) 4. No. of solar cells (parallel) 5. Solar cell properties PV unit 1. PV current 2. PV voltage indicates that the corresponding quantity is random, not necessarily normally distributed Figure 3. Uncertainty Inputs and Outputs of Each Subsystem 2. TE Unit: As shown in Fig. 3, the inputs to the TE unit are PV current, PV voltage, material properties, and the number of TE coolers in series and in parallel. The output of the TE unit is the amount of heat absorbed, which is a random quantity because of the uncertain inputs to the TE unit (PV voltage and PV current). 3. House: The outside temperature, material properties, the enclosed area, and the inside temperature are the inputs to the house model (see Fig. 3). Because of the uncertainties in the outside and inside temperatures, the output of the house model, the cooling load, is also a random quantity. It is desired that the cooling load be absorbed by the TE unit. This desire is imposed as an equality constraint in the deterministic optimization, as shown 7of16

9 in Eq. 9. As we shall see later, satisfying this equality constraint under uncertainty greatly influences the performance of the ABE system. Having presented the uncertainty discussion for each of the subsystems, we now combine them to form an overall ABE model under uncertainty. We study the interactions between the subsystems, which are imposed as equality constraints in Eqs. 9, 10, and 11 in the deterministic formulation. We also study how they are appropriately formulated to account for uncertainty. The robust design optimization is presented at the end of the next section. IV. Robust Design Optimization Formulation In this section, we discuss how the deterministic optimization problem presented in Section II-B is reformulated to account for weather uncertainties. Our interest is in evaluating the effects of weather uncertainties on the number of solar cells and TE coolers required by the ABE system, which is the objective of the deterministic optimization. We first discuss the design variables and the equality constraint formulations under uncertainty, followed by the presentation of the RDO formulation. A. Design Variables For the ABE system, the design variables in the deterministic optimization problem are the number of TE coolers and solar cells in series and parallel (see Eq. 8). The same design variables are used in the RDO formulation as well. Since these variables do not have a direct functional dependence on the uncertain weather conditions, we do not consider them as random variables. For the same reason, the side constraint shown in Eq. 9 is left as-is in the robust design formulation. However, the uncertainty in the weather conditions does affect the optimum values of the number of solar cells and TE coolers indirectly because of the uncertainty propagation between the interacting subsystems. These interactions are given in the form of three equality constraints in the deterministic problem, shown in Eqs. 9, 10, and 11. In addition to the above design variables, the equality constraint formulation presented next introduces additional design variables in the RDO formulation. We discuss the pertinent details next. B. Equality Constraints Three equality constraints are imposed in the deterministic ABE optimization problem (see Eqs. 9, 10, and 11). The first equality constraint in Eq. 9 is a designer-imposed constraint, which ensures that the cooling load is completely absorbed by the TE unit. The second and third equality constraints are physics-based constraints that govern the operations of the PV and TE units, and must be satisfied regardless of the uncertainty. Equality constraints impose strict limitations on the design variables, thereby greatly influencing the optimal design. 18 For example, as shown in Fig. 4(a), the feasible region of a generic equality constraint, h(x), lies strictly at h(x) = 0, where x is the set of design variables. When uncertainty is introduced in the problem, the feasible region of the constraint still remains at h(x) = 0 (shown in Fig. 4(b)), where X is the set of random design variables. However, the left hand side of the equality constraint, h(x), now becomes a random function with a nonzero standard deviation, which cannot be exactly equal to a constant value. From the above discussion, one can readily see that satisfying equality constraints under uncertainty can bring about glaring difficulties. Equality constraints h 2 and h 3 given in Eqs. 10 and 11 must be satisfied, because violation of these physics-based constraints under uncertainty would essentially result in a design failure. Under uncertainty, a careful formulation of the above equality constraints is needed to ensure feasibility of the ABE design. 18 We use a previously developed equality constraint formulation approach 18 to appropriately formulate the equality constraints present in Eqs. 9, 10, and 11. We begin by discussing the physics-based constraints. 1. Physics-Based Equality Constraints Rangavajhala et al. 18 discuss that physics based equality constraints can be potentially eliminated by substituting for the dependently distributed random variables (explained next) in the constraint. For the set of assumptions made in this study, we observe that the V TE and I TE depend on V PV and I PV. We 8of16

10 Deterministic RDO feasible h(x) =0 Possibly large constraint violation h(x) =0 feasible infeasible infeasible infeasible infeasible 0 h(x) (a) 0 (b) h(x) Figure 4. Equality Constraints in Deterministic Versus RDO Problems assume that the statistical dependence of V TE and I TE on V PV and I PV, respectively, is given by h 2 and h 3 in Eqs. 10 and 11. In order to eliminate the constraints given in Eqs. 10 and 11, we substitute Eq. 4 in Eq. 1 to obtain an expression for the heat absorbed, Q pc. In other words, we eliminate the two equality constraints given by Eqs. 10 and 11 by substituting for the variables V TE and I TE. Doing this ensures that the amount of heat absorbed by the TE unit never violates the underlying physics given by Eqs. 10 and 11 under uncertain conditions. 2. Designer Imposed Equality Constraints The remaining equality constraint, h 1, in Eq. 9 presents an interesting challenge. This is a designer imposed constraint, and violation of this constraint does not violate any laws of physics. An exact satisfaction of this constraint requires that two independently distributed random variables, cooling load, Q load (Eq. 9), and the heat absorbed by the TE unit, Q pc (Eq. 9), must be equal to each other which is a challenge 18 under uncertainty. The violation of Eq. 9 implies that the ABE system is unable to maintain the desired inside temperature, which may not be desirable. Therefore, we give emphasis to satisfying this constraint as closely as possible even under uncertainty. To accomplish this task, we use the approximate moment matching approach developed by Rangavajhala et al, 18 which is illustrated in see Fig. 5. In this approach, the original deterministic equality constraint is replaced by two inequality constraints (see Eqs. 14 and 15). In these inequality constraints, the following are done: (1) the mean of the designer-imposed constraint, h 1, is restricted to be as close to zero as possible through a parameter, δ µ (see Eq. 14), and (2) the standard deviation of constraint h 1 is required to be as small as possible, which is enforced through a parameter, δ σ (see Eq. 15). The two parameters, δ µ and δ σ, that control the constraint satisfaction of h 1 under uncertainty are introduced in the optimization problem as: (1) objective functions to be minimized, and (2) additional design variables, as discussed in Section IV-A. The smaller the δ µ and δ σ values, the smaller is the constraint violation under uncertainty. 18 However, there is a tradeoff associated with minimizing constraint violation under uncertainty. This tradeoff is discussed later in this section. Figure 6 summarizes the equality constraint formulation presented in this subsection. The highlighted boxes in Fig. 6 show the interactions between the subsystems, and how they are formulated to account for uncertainty. A comparison of Fig. 6 with Fig. 2 helps gain an understanding of the addition and propagation of uncertainty in the overall ABE model. We discuss the objective function formulation next. 3. Objective Function The deterministic optimization is a single objective problem (see Eq. 8), where the total number of TE coolers and solar cells is minimized. As discussed in Section IV-A, the value of the design variables, and hence the total number of TE coolers and solar cells, N, is not directly influenced by the considered uncertainties. The objective N is considered a deterministic quantity in this study. However, using the approximate moment matching method requires the equality constraint parameters, δ µ and δ σ, to be minimized in addition to the 9of16

11 Traditional Method µ h =0 possibly high σ h 0 h(x) Approximate Moment Matching Method mean close to zero low standard deviation σ h <δ σ 0 h(x) δ µ δ µ Figure 5. Approximate Moment Matching Formulation for Designer Imposed Constraints deterministic objective function, N. This approach of formulating the designer-imposed equality constraints results in a multiobjective formulation that involves minimizing three objectives N, δ µ and δ σ. C. Robust Design Optimization Formulation Having discussed the individual components of the optimization problem, viz., the design variables, the constraints, and the objective function, the approximate moment matching formulation 18 for the ABE system becomes subject to min x=[s S,S P,H S,H P,δ µ,δ σ] {N,δ µ,δ σ } (13) δ µ µ h1 δ µ (14) 0 σ h1 δ σ (15) 1 S S,S P,H S,H P 50 (16) where µ h1 and σ h1 are the mean and the standard deviation of h 1, respectively, which can be determined using a first order Taylor series approximation. The expressions to compute an estimate of the mean and the standard deviation of a generic function, g(x), are given as 19 µ g = g(µ X ) (17) [ n x 2 g n x n x Var[g] = g X i σ Xi] + g X=µX X i X j Cov(X i,x j ) X=µX (18) i=1 i=1 j=1,i j X=µX where σ X denotes the vector of the standard deviations of X, Cov(X i,x j ) denotes the covariance between the variables X i and X j,andi, j = {1,..., n x }. The standard deviation of g can then be computed as σ g = Var[g]. For the present problem, the explicit analytical forms for the partial derivatives required in the above equation cannot be obtained for the TE and the PV units, because the underlying equations are nonlinear. We use the finite difference method to obtain the partial derivatives. 10 of 16

12 TE unit (Heat Pump) Q absorbed = function(i S, V S,H S, H P ) Eliminate TE current TE voltage PV unit (Generator) I S = function(i S, V S, T o, E, S S, S P ) Q load = Q absorbed Satisfy as closely as desired T o E House Q load = function(to, Ti) T o T i Figure 6. Equality Constraints in the ABE System RDO Formulation As discussed earlier, the focus in RDO problems is to minimize the effects of uncertainty on the performance of a design. Next, we define a measure that quantifies the performance of the ABE system under uncertainty. 1. A Measure of Performance of the ABE system Under Uncertainty It is desired that the ABE system maintains a desired inside temperature irrespective of the outside temperature, which means that the cooling load should be equal to the heat absorbed by the TE unit. As discussed earlier, this constraint (Eq. 9 in the deterministic optimization) cannot be exactly satisfied under uncertainty. That is, the heat absorption of the TE unit and the cooling load do not exactly match. This can result in (i) over-cooling, (ii) under-cooling, or (iii) frequent fluctuations between over-cooling and under-cooling of the house. We observe that the violation of the equality constraint h 1 under uncertainty can be a good performance measure for the ABE system. The desired performance of the ABE system is obtained when the cooling load and the heat absorbed are exactly equal to each other, which means that the constraint violation of h 1 under uncertainty is exactly zero. When uncertainty is considered, the actual performance of an ABE system can be considered good if the cooling load is as close as possible to the heat absorbed by the TE unit. The constraint violation of h 1 under uncertainty readily tells the designer the deviation between the desired performance and the actual performance. The violation of constraint h 1 is itself a random variable, whose mean and standard deviations are restricted by δ µ and δ σ, respectively (see Eqs. 14 and 15). In the approximate moment matching formulation, δ µ can be viewed as a measure of the mean performance of the ABE system. A high value of δ µ indicates that the mean cooling load and the mean heat absorbed are significantly different from each other, which results either in over-cooling or under-cooling of the house. On the other hand, the quantity δ σ can be viewed as a measure of the robustness of the performance of the ABE system design. A high value of δ σ indicates that there is a large variation in the constraint violation of h 1 under uncertainty. For example, each instance of the constraint violation as a random variable represents a case of over-cooling or under-cooling. A large variation in the constraint violation implies that the ABE system is fluctuating between a large range of cooling conditions. This means that the mean performance of the ABE system is not robust. Small values of δ µ and δ σ potentially yield better mean performance and robustness for the ABE system. The uniqueness of the approximate moment matching method is that three important design aspects of the ABE system, viz., (1) the number of TE coolers and solar cells, (2) the mean performance of the design, and (3) the robustness of performance of the design, are combined together in a single optimization formulation. This provides the designer the means to explore the design space systematically with an emphasis on uncertainty. The multiobjective nature of the approximate moment matching formulation has interesting implications. As we shall see in the next section, a tradeoff exists among the mean performance, the robustness of performance, and the objective function minimization under uncertainty. In other words, a better mean 11 of 16

13 performance and robustness can be obtained only at the expense of a deteriorated objective function value. This situation poses a complicated decision making challenge for the designer, which we study in the next section. V. Results and Discussion In this section, we discuss the results obtained by solving the approximate moment matching problem presented in the previous section. We present the Pareto frontier for the multiobjective problem. We also discuss the tradeoff between the equality constraint satisfaction and the objective function minimization, and provide insights into the decision making process. A. Pareto Frontier 20, 21 The multiobjective problem presented in the previous section can be solved using two approaches: (1) Construct an aggregate objective function (AOF) that adequately reflects the designer s preferences for the three objectives, and optimize the AOF to obtain a single optimum design. This approach is known as the Integrated Generating and Choosing (IGC) approach, or (2) Generate several Pareto optimal designs first, and choose the most desirable solution later. This approach is known as Generate First Choose Later (GFCL) approach. In this paper, we use the GFCL approach, as it complements the exploratory nature of this work. We solve the three-objective problem given in Eqs. 13 through 16, using the normalized normal constraint method 22 to obtain the Pareto solutions. A Pareto filter 22 is then employed to filter all non-pareto and locally Pareto solutions. The resulting Pareto frontier, consisting of only globally Pareto solutions, is shown in Fig δ σ 4 3 B 2 A δ µ N Figure 7. Pareto Frontier for the Robust Formulation The Pareto frontier in Fig. 7 clearly shows that there is a tradeoff between the three objectives N, δ µ (measure of mean performance), and δ σ (measure of robustness), which can be discussed as follows. To minimize the impact of uncertainties on the ABE system performance, we wish that the constraint h 1 be satisfied as closely as possible. Accordingly, if we select a design with a small δ µ and δ σ values, it results in a high value of N, which results in high cost (see point A in Fig. 7). This implies that an ABE design that has a desirable mean performance and high robustness may not be cost-effective. However, in order to obtain a better mean performance, increasing the value of N alone may not yield desirable results. A high value of N causes the robustness of performance to deteriorate (see high δ σ value for point B in Fig. 7). This shows that increasing the number of TE coolers and solar cells does not necessarily yield a robust design. A design that fully minimizes all the three objectives simultaneously is not possible, and a design that achieves a compromise or a tradeoff among the objectives must be selected. In the next 12 of 16

14 subsection, we discuss the procedure we adopted to select the most desirable ABE design. B. Decision Making under Uncertainty From the above discussion, one can readily agree that the decision making criterion for the robust design of the ABE system is not simply the minimization of the objective function, N. The equality constraint satisfaction under uncertainty (for h 1 ), which translates to the mean performance and the robustness of performance of the ABE system, trades-off with the objective function. The final decision regarding the desirable design for the ABE system can be considered highly subjective. Different regions in the Pareto frontier entail different preferences among the objectives. Plotting a complete representation of the Pareto frontier provides the designer a visual representation of the associated tradeoffs, enabling him/her to make an informed final design selection. The final design selection can be made using a number of decision making techniques such as visualization of the Pareto frontier (shown in Fig. 7). In this paper, we use a sequential filtering approach that separates the Pareto solutions with desirable objective function values. This process is discussed in greater detail in ref. 23 The approach to select the final ABE design is discussed next. From the Pareto frontier in Fig. 7, we observe that the values of N range from 2 to 10, δ µ ranges from 0 to 6 W/m 2,andδ σ ranges from 1.6 to 6 W/m We rank the objectives in their order of importance (which is subjective): (1) N, (2) δ µ, and (3) δ σ. 2. We employ the following filters to the Pareto points in a decreasing order of the rank of the objectives. (a) Filter N: We retain those designs from the Pareto frontier that satisfy the criteria N 7. This step results in 132 points shown by + in Fig. 8. (b) Filter δ µ : We now proceed to the next important objective, δ µ. The range of δ µ values after the previous filter is to We retain those Pareto points with δ µ < 2. This results in 12 Pareto points, shown by o in Fig. 8. (c) Filter δ σ : Out of the 12 designs resulting from filters 1 and 2, the range of δ σ values is to We choose two designs with the least δ σ values shown by * in Fig One of the two points has a lower δ µ, which is chosen as the final design, as shown by the solid circle in Fig. 8. The final design configuration is given by the number of TE coolers in series and parallel and the number of solar cells in series and parallel: S S =1,S P =2,H S =5,H P =1. The order of the above filters and their definitions can be changed to suit the designer s choice. For example, if the robustness of the performance is more important for the designer, Filter δ σ canbeemployed before the other two. Such filtering schemes can prove to be useful in visualization and decision making when the robust design problems at hand consist of more than three objectives. 23 We note that the selection criteria (filters) could change when other sources of uncertainty are included. This process of generating a large number of Pareto solutions first, and choosing the most appropriate one later helps the designer obtain a better understanding of the design space. If the designer has reasonable prior knowledge of the desirable ranges for each of the objectives, more sophisticated methods such as Physical Programming 24, 25 can be used to obtain a single Pareto solution that adequately reflects the designer s preferences. C. Comparison of Results In this paper, we used the approximate moment matching approach, where the designer can exercise control on both the mean and the standard deviation of the equality constraint h 1. This is unlike what is typically done in the literature. 18 Satisfying the constraint at its mean value only 13, is one of the approaches typically followed to formulate equality constraints in RDO problems. This formulation can be viewed as being equivalent to deterministically satisfying the constraint h 1 if using a first order Taylor series approximation shown in Eq. 17. In this subsection, we compare the results obtained for ABE system using the approximate moment matching and the traditional RDO formulations. For the traditional approach, we assume that the equality constraints in Eqs. 10 and 11 are eliminated, as done in the approximate moment matching approach. 13 of 16

15 δ σ δ µ 2 1 Complete Pareto set 4<N<7 δ µ <2 Designs with least δ σ Final Design N Figure 8. Choice of Final ABE Design The robust formulation for the ABE system optimization, using the traditional approach of formulating equality constraints, is given as subject to min x=[s S,S P,H S,H P] N (19) µ h1 = 0 (20) 1 S S,S P,H S,H P 50 (21) To compare the results obtained using the traditional and the approximate moment matching approaches for the ABE system design, we use a metric to quantify the term equality constraint satisfaction. A metric called probability of constraint satisfaction, PCS, is adopted from the literature, 18 which provides a measure of the closeness of the equality constraint satisfaction. The metric is calculated based on a Monte Carlo simulation, and is defined as PCS = N s (22) N where N s is the number of simulation cycles for which the equality constraint, h 1, lies within ±1 W/m 2, i.e., 1 h i 1, and N = is the total number of simulation cycles. We now illustrate how the traditional and approximate moment matching methods compare in terms of the PCS value for the ABE systems. We solve the traditional formulation given in Eq. 19 through 21. We assume that the inside and outside temperatures and solar radiation are normal random variables with means, standard deviations and correlation as defined earlier in Section III. For both the traditional and the approximate moment matching robust formulations, we perform a Monte Carlo simulation using their respective optimal designs, and compute the PCS values. The solution given by the traditional approach yields a PCS of The design configuration is S S =1,S P =2,H S =6,andH P = 1, with N =8. Out of the several ABE designs obtained using the approximate moment matching approach, we consider the final design chosen in the previous subsection for the PCS comparison. The PCS value of the final design 14 of 16

16 chosen (S S =1,S P =2,H S =5,H P = 1, which yields N = 7) is , which is greater than obtained by the traditional approach. In fact, there could be several other ABE designs in the Pareto frontier that could potentially yield comparable or better PCS values than the traditional method, each with a different tradeoff among the objectives. An important advantage of the approximate moment matching approach discussed in Section IV-C is that it yields a multiobjective problem. If we were to satisfy the equality constraint h 1 only about its mean value, we would obtain only one ABE design, since the traditional formulation presented above minimizes N only. The equality constraint satisfaction, and in turn the mean performance and the robustness of the ABE system, using such an approach largely depend on the standard deviation of the input uncertainties. In such cases, the designer has very limited control over the robustness of the ABE design for a given standard deviation of the input uncertainties. Using the approximate moment matching approach on the other hand, the designer can exercise his/her discretion to, say to obtain a lower N when compared to that possible using the above traditional approach, even if it comes at the expense of poor robustness of the ABE system. For example, the proposed robust formulation can not only yield the solution given by the traditional approach (N = 8), but also many other solutions with N < 8, each with different robustness characteristics, as shown by the Pareto frontier in Fig. 8. The approximate moment matching approach provides the designer a realistic understanding of the complicated decision making problem at hand, and provides the flexibility to explore the design space with an added emphasis on equality constraint satisfaction under uncertainty. VI. Concluding Remarks An Active Building Envelope (ABE) system is an emerging thermal conditioning technology that has promising features such as lowering our dependence on expensive conventional fuels. In this paper, we presented our first step towards performing a realistic optimization of ABE systems. Specifically, we investigated the impact of weather uncertainties on the optimal design of ABE systems. We examined how the variations in solar radiation, the atmospheric temperature, and the indoor temperature are expected to influence the performance of ABE systems. Accordingly, we quantified the uncertainties in these quantities in terms of their means and standard deviation values, which are estimated from weather data recorded over the past several years. Incorporating these uncertainties in the ABE system optimization posed two strict requirements: (a) to ensure that the physics governing the interactions between two subsystems, called the TE unit and the PV unit, is not violated under uncertainty, and (2) to minimize the deviation between the desired performance and the actual performance (i.e., the deviation between the cooling load for the house and the heat absorbed by the ABE system). These requirements are formulated as three equality constraints. Under uncertain conditions, these equality constraints needs a careful formulation. We used an equality constraint approach that formulates the robust design optimization problem as a multiobjective problem. This formulation allows the designer to express preferences over three entities: (1) the mean performance of the design, (2) the objective function minimization, and (3) the robustness of the design, in terms of constraint violation under uncertainty. We explored the Pareto frontier of the ABE system to understand the tradeoff among the above three entities, and presented a systematic procedure that is expected to help the designer select an appropriate robust design of the ABE system. VII. Acknowledgements Support from the National Science Foundation, Award numbers CMS , CMS , and DMI , is much appreciated. References 1 Khire, R., Messac, A., and Dessel, S. V., Design of Thermoelectric Heat Pump Unit for Active Building Envelope Systems, International Journal of Heat and Mass Transfer, Vol. 48, No , September 2005, pp Khire, R., Van Dessel, S., and Messac, A., Active Building Envelopes: A New Solar Driven Heat Transfer Mechanism, 19th European PV Solar Energy Conference, Paris, France, June Rivas, F., Khire, R., Messac, A., and Desel, S. V., Economic Viability Assessment of Active Building Envelope 15 of 16