Naturally ventilated and mixed-mode buildings—Part I: Thermal modeling

Building and Environment 44 (2009) 736–749

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Building and Environment journal homepage: www.elsevier.com/locate/buildenv

Naturally ventilated and mixed-mode buildingsdPart I: Thermal modeling Henry C. Spindler a, Leslie K. Norford b, * a b

Optimal Energy Solutions LLC, Keene, NH, United States Massachusetts Institute of Technology, Cambridge, MA, United States

a r t i c l e i n f o

a b s t r a c t

Article history: Received 23 November 2007 Received in revised form 16 May 2008 Accepted 19 May 2008

Mixed-mode cooling strategies rely on several different means of delivering cooling to the occupied spaces of buildings. These different means, or modes, of cooling include different forms of natural ventilation through operable windows, ventilation assisted by low-power fans, and mechanical air conditioning. Control of mixed-mode cooling systems requires a thermal model tuned to accurately predict the dynamics of a specific building. This paper presents a flexible system-identification framework for linear thermal models that is well suited to accommodate the unique features of mixed-mode buildings. The effectiveness of this framework was demonstrated on a multi-zone, mixed-mode building, with model-prediction accuracy shown to exceed that published for other naturally ventilated or mixedmode buildings, none of which exhibited the complexity of this building. A companion paper employs the thermal model in an efficient algorithm to optimize control strategies over extended planning horizons. Ó 2008 Elsevier Ltd. All rights reserved.

Keywords: Natural ventilation Mixed-mode cooling Thermal models System identification

1. Introduction Mixed-mode cooling of buildings, which combines natural ventilation, fan-driven ventilation, and mechanical air conditioning, can reduce operating costs and CO2 emissions. Night ventilation, which makes use of building thermal mass, is an important element of mixed-mode cooling in many climates. In houses, experience guides decisions about mode switching: when to open up windows, when to run a fan, and when to close up the building and turn on the air conditioning. Commercial buildings are typically not occupied at times when such decisions need to be made and the penalties for decisions that in hindsight turn out to be wrong are typically larger, in electricity demand charges or thermal discomfort. Sound, automated control decisions must be made on the basis of predicted thermal response of a building to its internal loads and environmental conditions, which in turn requires an accurate thermal model of the building. We have developed a multi-zone, multi-control-mode thermal model of a test building, compared model predictions with measurements, and used the model to simulate the staged use of natural and fan-driven ventilation in a test building [1]. This paper concerns the thermal modeling and a companion paper [2] focuses on control. After a thorough review of the relevant literature, a data-driven model will be developed for a building with multiple thermal zones and multiple active operating modes. This paper will

* Corresponding author. E-mail address: [email protected] (L.K. Norford). 0360-1323/$ – see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.buildenv.2008.05.019

describe the test building, present the linear model, apply it first to the multi-zone building without regard for operating mode and then show that prediction accuracy is improved if the operating modes are also modeled. Methods to automate the selection of significant variables and operating modes for the models are also presented. 1.1. Literature review It is important to provide a context for the subject of the two papers, thermal modeling and automatic control of naturally ventilated and mixed-mode buildings. Substantial work has been done in these and related areas, including airflow models and occupant-based control. The second paper will review the relevant control literature. An investigation of natural ventilation cooling and control for an educational building [3] succinctly summarizes many key points.  Natural ventilation and exposed thermal mass are an effective form of cooling that requires careful attention, considering both technologies and user interactions.  Appropriate manual or automatic control is needed to prevent overcooling under light-load conditions.  Night ventilation in hot weather must be delayed until outdoor temperatures drop, which is often beyond the end of the occupied period and therefore requires automatic control.  Model-based control is an effective strategy but must be sufficiently robust to respond to external conditions and disturbances introduced by occupants.

H.C. Spindler, L.K. Norford / Building and Environment 44 (2009) 736–749

 Models used to control buildings are appropriately based on measurements, in contrast to first-principles models that are used for building design. Two types of models for control purposes are prevalent in the literature: physically based models and data-driven models. Physically based models rely on heat transfer, fluid mechanics and other engineering sciences to establish equations that model building temperatures, airflows and energy use; the equations contain coefficients to represent building geometry and the properties of building materials and equipment. Data-driven models, for example time series or neural networks, relate building measurements of interest, for example indoor temperature, to temperatures at earlier time steps and/or outdoor temperatures and internal or solar heat gains, through parameters that have no direct physical meaning. The principal benefit of physically based models is that they may be used to guide design. The shortcoming of physically based models is that their structure and the relevant physical parameters may not be in a form suitable for automatic control (this is particularly true of analytic models) or may require extensive calibration with measured data. Several physically based building ventilation models offer significant insights into buoyancy-driven and wind-driven flows and are useful for designing building apertures, but do not include thermal dynamics and therefore cannot be used for predictive control [4,5]. Analytic models have been extended to account for the dynamics associated with thermal mass but this approach is again intended for design, places restrictions on variations in exogenous variables and offers no guidance for estimation of relevant parameters in operating buildings [6]. A building-energy simulation model, used in conjunction with computational fluid dynamics simulation and modified to accurately account for natural ventilation and assess user behavior, was employed to design and test a control strategy for the naturally ventilated portions of a large office building [7]. There was no fanassisted ventilation or mechanical cooling and the controller therefore did not need to incorporate a dynamic thermal model. A multi-zone, coupled thermal-airflow analysis program (not publicly available) was incorporated in a design method for natural and hybrid ventilation systems [8] and was also used to assess the performance of such systems in a simulated office building [9,10]. The simulation tool is not well suited for on-line control of ventilation systems; it was intended to be used for design and analysis and a building model would be difficult to calibrate with measured data. Data-driven models, by their very nature, do not exist before the building is built. They cannot be used for design purposes because they are developed for a particular building, perhaps under particular conditions. However, data-driven models may achieve higher levels of accuracy without requiring of the user painstaking identification, estimation and entry of relevant building parameters. For purposes of assessing model accuracy, this paper distinguishes two types of simulation: step-ahead estimates of independent variables at the next time step, at which point the estimate is replaced with a measured value; and pure simulation over an extended period, with no corrections from measurements. 1.1.1. Physically based models of naturally ventilated buildings A model based on single-zone analytical airflow calculations [11] and a three-time-constant thermal model [12] were developed for the purpose of evaluating night-cooling strategies using natural ventilation [13]. The design-stage tool was applied to a portion of an educational building that was naturally ventilated using stack, single-sided and cross-ventilation. Inside temperature predictions

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were compared with measured data; maximum and minimum temperatures were over- and under-predicted by approximately 1  C and 2.5  C. A school in Norway was modeled with 21 zones during the heating season [14] using ESP-r [15]. Measured exhaust-air temperatures were compared with predictions; discrepancies over a single day were as great as 3–4  C. There was no way to incorporate into the simulation tool the CO2-concentration-driven damper control governing room airflow rates. This shortcoming was thought to contribute to the poor predictive performance of the model. Furthermore, the model assumed well-mixed conditions, despite the fact that the building used displacement ventilation. An analytical airflow model was coupled with TRNSYS [16] to predict the thermal behavior of a single room in Athens cooled by natural ventilation through a single operable window [17]. The model was developed to investigate the use of fuzzy control strategies to regulate indoor air quality. To calibrate the model to experiments, a correction factor was added to the calculated flow rates. The authors reported an average prediction error for the room air temperature of 0.29  0.38  C for a simulation over a single day.1 An equation-based thermal model was developed for a single room with an operable window and adjustable blind to evaluate the impact of visual-comfort control on thermal comfort within the room [18]. No comparison with experimental data was provided. Individual mixed-mode test cells in Lyon were modeled in TRNSYS to design a fuzzy controller for thermal and air-quality control [19]. Cooling was provided by a combination of fan-driven ventilation air (at three speeds) and a mechanical air conditioner. Over three test periods, average prediction errors varied from 1.59  1.38  C to 1.95  1.36  C. The authors suggested that the large bias in the predictions may have been due to the difference between the air-inlet temperature and its (remotely) measured value. Temperatures within a small test chamber were predicted by an analytical model and a Takagi–Sugeno fuzzy model (see, e.g., Refs. [20,21]). The chamber consisted of a fixed window with an operable shade, which was moved during experiments. Errors for the analytical model over several one-day periods were reported to be in the range of 5–20%. Error figures were not reported for two types of fuzzy models, although single-day simulations showed errors of up to 10  C (approximately 50%). 1.1.2. Data-driven models of naturally ventilated buildings Several authors have applied neural networks to the localized modeling of airflow and temperatures within rooms. In one study, a test chamber with four fixed openings was equipped with temperature and velocity sensors in multiple locations [22]. Steadystate models were developed to map outside temperature, humidity, pressure, wind speed and direction, and the air velocity and direction through the openings to temperatures and velocities at the various locations in the room. Temperature predictions were within 2.3  C of measured values. Others have used neural networks to model the effect of room and aperture geometry on interior air velocities [23,24]. In these cases, simulations with computational fluid dynamics rather than experiments provided data for model training. In another study, experiments designed to inform PI controller design for temperature control in a single naturally ventilated room found that flow through the upper and lower openings of the test room’s one window was independent of local wind speed, wind direction and room temperature [25]. Flow was linearly related to the size of the (tilted) window openings, or in other words, proportional

1

Note that this is an average error rather than an RMS error.

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to the total window area multiplied by the sine of the opening angle. Such a result would not have been predicted from analytical methods. The control strategy was used in conjunction with a thermal model to assess the heat-removal potential of the system. An ARMAX model was constructed using data from measurements in a naturally ventilated test cell in Delft [26]. The single-room zone was equipped with a moveable awning for shade, an automatic window with two openings and a heater. All three components influenced the temperature of the cell, for which a linear predictive control strategy was developed. Nonlinear trigonometric equations were used to calculate the solar radiation striking the window based on solar and awning positions. The ventilation contribution to the heat balance at the interior temperature node was proportional to the window-opening area multiplied by the inside–outside temperature difference. The authors hypothesized that the significant model-prediction errors observed at times of large window openings were related to the linear approximation made of the ventilation cooling effect and the fact that the window was minimally open for the bulk of the training data. The model was used primarily to make one-step-ahead predictions, but in one instance it was shown to make accurate predictions for four time steps (15 min each) into the future. Further details of the modeling and control strategies are available [27]. Considerable effort has been devoted to modeling greenhouses for climate control. Neural networks were shown to provide an accurate mapping of wind speed, wind direction, greenhouse opening area (10 levels) and fan speed (32 levels) to the ventilation rate of a greenhouse [28]. In this instance, CO2 was used as a tracer gas for flow rate measurements. A simple first-order thermal model of the greenhouse incorporated the flow rates as predicted by the neural network. The flow model was used in a robust-control framework to control both temperature and CO2 concentration in the greenhouse. Radial basis function neural networks (RBFNNs) have been used to predict greenhouse temperatures. RBFNNs were used to map inside temperature, outside temperature, inside humidity and outside solar radiation to the inside temperature [29]. A variety of time lags on the different inputs were investigated in an attempt to establish the most effective set of inputs to use. The best model was found to employ radiation inputs at times k and k  21, humidity inputs at times k and k  3, outside temperature inputs at times k and k  2 and inside temperature inputs at times k  1 and k  2. The time step was 5 min. Over a 3257-point test set, the pure-simulation RMS error was 1.48  C. A variety of on-line and off-line training methods was discussed for the same greenhouse problem [30]. It was found that a Levenberg–Marquardt training algorithm outperformed all others investigated for on-line or off-line training in terms of error, model size and generalization capability. Finally, multi-objective genetic algorithms were employed to assist in the selection of inputs (and the best number of lag terms to use) as well as the RBFNN architecture [31]. For the same greenhousedsingle zone with no operable componentsdmodel-prediction RMS error was improved to 1.30  C for the full test set. In almost all models formed, the wind speed and direction were not identified by the genetic algorithm as useful inputs. One additional input used in this last study was the solar radiation measured at an inside location. For a greenhouse in Avignon, France, a neural-network model (with no dynamics) mapped outside solar radiation, outside drybulb temperature, outside wet-bulb temperature, wind speed, heater heat flux, aperture opening angle, water misting time fraction, leaf area, Julian date and hour of day to the inside drybulb and wet-bulb temperatures, the inside solar radiation and the soil temperature [32]. For a Silsoe, France greenhouse, neuralnetwork inputs were outside solar radiation, outside long-wave sky radiation, outside dry-bulb temperature, outside dew-point temperature, wind speed and direction, and aperture opening

angle; outputs were inside solar radiation, inside dry-bulb and dew-point temperatures, crop canopy temperature and soil temperature. By examining weights in the input layer of the neural nets, the authors argued that certain inputs were less useful than others and could be discarded. In an additional study of ventilation rates, the authors found that wind direction and inside-outside temperature difference were not relevant inputs, as indicated by the weights of these model inputs and confirmed by the training of neural networks without those inputs. The best training errors reported for the inside dry-bulb temperature predictions at Avignon and Silsoe were 0.93  C and 1.16  C. No errors on a test data set were reported. Neural-network modeling for greenhouse climate control was assessed in Ref. [33], which discussed breaking a data set into odd and even points for training and validation to prevent overtraining and to capture the system’s behavior under all observed conditions. The odd and even sets are not statistically independent, however. The alternative approachdbreaking the data set into two distinct groups (time 1 to time 2, then time 2 to time 3)dmay be problematic for nonlinear models such as neural networks because they do not extrapolate well to conditions not observed in the training data. Bottleneck neural networks [34,35] were suggested in Ref. [33] to reduce model order and, consequently, the number of parameters to train. A nonlinear, data-driven model was used to predict air-change rates in a single-zone test room ventilated by wind-driven airflows through a variety of window-opening configurations [36]. The authors also generated air-change predictions from a nodal numerical model and corrected the predictions via a nonlinear, empirical relationship with air-change measurements. One-hourahead predictions were used in simulation to adjust window openings in order to control room temperature in response to varying outdoor conditions. 1.1.3. Data-driven models of conventional buildings and building systems As summarized in several reviews [22,37,38], neural networks have been used to model buildings and building systems for the purpose of load and temperature estimation as well as process variable estimation. In the ‘‘Great Energy Predictor Shootout’’ [39], many contestants used some form of neural network to predict electricity, chiller water energy and hot water energy consumption of two commercial buildings. Neural networks predicted the power consumption of a central plant (chiller compressor, condenser fans, supply air fan and chilled water pumps) given set points and uncontrolled variables such as outdoor temperature [40–42]. Chiller power consumption and ice storage tank charge/discharge rates were also modeled for a central plant with ice storage capabilities [43]. These models were utilized by a neural-network supervisory controller to determine process set points that led to minimal energy consumption. The only instance known to the authors where neural networks were used to predict room temperatures in occupied buildings was presented in Ref. [44], where a functional link neural network was trained to predict room temperatures as generated by a simulation. The functional link neural network eliminates connections between nodes if the associated weights are smaller than a certain tolerance. The inputs to the network are those associated with a second-order polynomial kernel (see, e.g., Ref. [45]). The authors suggested that it was beneficial to train the model to predict the temperature change occurring over the next time step rather than the temperature at the end of the next time step.2

2 This recommendation may be based, in part, on the authors’ experience training a model on noise-free data generated by simulation.

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An inverse grey-box approach based on a 2-C/3-R nodal thermal model was applied to commercial-building loads and temperatures [46–48]. The method was developed to reduce the amount of training data required to estimate the parameters in a transfer function, by relating those parameters to resistances and capacitances via a state-space representation of a single thermal zone. The physical parameters were assigned a range of potential values. Within the range so defined, a global direct search [49] selected values for the physical parameters to minimize the difference between measured loads and those predicted by the transfer function model. A local, nonlinear regression was used to optimize the values of the physical parameters and therefore the coefficients of the transfer function. The authors reported that the predicted loads of a simulated building (with well-known inputs) were within 2% of the actual loads. A two-week training period was used, followed by a four-week test period. For a real building, the loads were predicted within about 10%. Temperatures were predicted within 0.6  C (approximately 8% of full range). Polynomial models for power consumption of the air handling unit and the total cooling plant were used to investigate optimal control of thermal storage inherent in building mass [46]. Recursive least squares applied to data collected during building operation was used to estimate the parameters of a 2-C/3-R model [50]. Errors were provided only on training data and were 1  C RMS. The information required to develop the model consisted of inside and outside temperatures and all heat gains acting on the inside temperature. A 2-C/2-R model was proposed for a simple building for which inside temperature was influenced by solar radiation, electrical heaters and outside temperature [51]. The authors argued that a continuous-time-domain modeling approach was appropriate and provided a method for parameter estimation using maximum-likelihood estimation. Error statistics were not provided for the pure simulation performed. A first-order ARMAX model for use in self-tuning control of HVAC systems implemented a series of ‘‘jacketing’’ rules to ensure that the parameters identified on-line did not acquire unreasonable values, [52]. Model identification performance degraded over time, leading to a reduction of control quality. The authors suggested that this shortcoming was the consequence of continuously updating the parameters. A fourth-order time-series model was applied to a test chamber heated by a radiant floor panel and by solar radiation [53,54]. The model, developed for optimal predictive control of space temperature, was generated and updated continuously using recursive least squares coupled with supervisory rules for robust estimation. The RMS prediction error over a 24-h simulation was 0.27  C. It was not stated whether the simulation was over training or validation data. Thermodynamic constraints were developed for a linear timeseries model of a single thermal zone to ensure that parameter identification via least-squares regression yielded physically valid results [55,56]. Parameters were constrained to satisfy a steadystate energy balance and to provide thermal time constants that were positive real numbers. The RMS prediction error over a 24-h test period in a 60 m2 test room was 0.02  C. Model-free reinforcement learning was used in an investigation of optimal supervisory control of heat flows into and out of building thermal mass [57]. Machine learning substituted for a physical model in the dynamic-programming optimization method. Hourly zone thermostat set points were found to be qualitatively similar to those produced by a direct-search optimization algorithm. 1.1.4. Literature review summary With the exception of an ARMAX model [25–27], data-driven models for naturally ventilated and mixed-mode buildings were based on neural networks or fuzzy inference systems. The systems examined were all single-zone rooms or buildings. The flexibility of

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the nonlinear models was used to incorporate the thermal behavior of the space when it was operated in different modes. The ARMAX model did not explicitly account for the different operational modes of the one-room test chamber. The issue of model input (or feature) selection was addressed, directly [29–31] and to a lesser degree [32,33]. Data-driven modeling of conventional buildings was also restricted to single-zone buildings. In the work reviewed, the issue of mode-input selection was not addressed and the issue of multiple operating modes (associated with different airflow patterns) was irrelevant. Operational mode switching did exist (for example, in a room cooled by a VAV-system), but the only impact on the zone was a potentially variable amount of heat delivered to the room. Broadly speaking, the test building in our study is much more complex than those examined in the literature, particularly those for which a model’s predictions have been compared with measured data. As will be shown, the accuracy of the predictions for this building is as good as or better than any reported above. 2. Test building 2.1. Building description The test building, shown in Figs. 1–3, is a nature center housed in a barn that was renovated to be largely passively heated from solar radiation and cooled entirely by natural ventilation. The six thermal zones considered in this study were the basement, the first-floor assembly room and the sunspace, a lower attic above the first-floor rooms (attic 1), an upper attic under the roof (attic 2), and the structure or thermal mass of the building. Fig. 2 labels a number of the openings used to control wind-driven and buoyancy-driven airflows: louvered openings 1 and 2 and doors 3 and 4 at the east and west ends of the first floor, and openings 6 and 7 between the first floor and the lower attic. Note also the fan, near openings 6 and 7, installed as part of this study to exhaust air from the first floor into the lower attic. Thermocouples were used to measure the outside dry-bulb temperature and temperatures in each of the thermal zones, in the east and west louvered openings 1 and 2, and in attic opening 6. Special low-error thermocouple wire was used for all measurements; the batch in use had a deviation of 0.2  C at 93.5  C. Variance data were not provided by the manufacturer; in practice variance was found to be negligible, as observed with measurements of steady indoor temperatures. A weather station 30 m from the

Fig. 1. A view of the test building from the southwest.

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Fig. 2. The first-floor plan. Numeric labels (with #) denote controlled apertures, numbered arrows in black show thermocouples on the first floor and numbered arrows in grey designate thermocouple locations on other levels as projected onto the first-floor plan. Specifically, opening 6 is in the lower attic, 7 is in the upper attic, 9 is in the basement and 10 is outside. (Floor plan courtesy of G. Ives.)

building recorded wind direction and speed and solar radiation on the horizontal and on each of the cardinal directions. Electrical measurements were taken at the circuit level, in sufficient detail to assign loads to individual zones. All measurement data were sampled at intervals of 60 s. 2.2. Features, targets and control modes The term ‘‘feature,’’ borrowed from the machine-learning vocabulary, describes an input to a model. A feature may be a raw measurement, such as a temperature, or may be a composite term such as one related to heat input to a space when the fan is running: (fan speed)  (Tout  Tassembly). The full set of inputs will be referred to as the ‘‘feature vector.’’ All features and outputs were scaled to the range of approximately [0, 1]. When a model is trained, the model parameters are adjusted so that the errorsdthe difference between the model outputs and the ‘‘targets’’ provideddare minimized. The targets for the model outputs are temperatures for the six zones identified above. The assembly temperature is representative of the occupied space temperature to be controlled; the other zones are needed to varying degrees (as will be shown) to improve predictions of these temperatures. The structure of a linear model is given in Eq. (1), where T1 refers to the assembly temperature, T2 to the mass temperature, and Tn to the basement temperature. Some exogenous inputs (Isun, horiz and Isun, south) are given as examples of the many exogenous system inputs. The influence of controlled elements is shown (e.g., the fan term). The prediction of T1 at time t is a function of T1 at previous times. If one lag term is important for accurate predictions, then the term T1(t  1) would be included in the model and the associated coefficient a11 would be determined through model training. If five lag terms are important, then T1(t  1), T1(t  2), T1(t  3), T1(t  4)

and T1(t  5) would all be included, along with the associated coefficients: a11, a12, a13, a14 and a15

T1 ðtÞ ¼ a11 T1 ðt  1Þ þ a12 T1 ðt  2Þ þ / þa21 T2 ðt  1Þ þ a22 T2 ðt  2Þ þ / . þan1 Tn ðt  1Þ þ an2 Tn ðt  2Þ þ / þb11 Isun;horiz ðt  1Þ þ b12 Isun;horiz ðt  2Þ þ / þb21 Isun;south ðt  1Þ þ b22 Isun;south ðt  2Þ þ / . þc11 Fanðt  1ÞðTout ðt  1Þ  T1 ðt  1ÞÞ þ / T2 ðtÞ ¼ similar . Tn ðtÞ ¼ similar

(1)

Control modes are the combinations of openings and fan operation shown in Fig. 4. The fan, controlled to run only at night or in early morning hours before occupants arrived, was operated in conjunction with louvered apertures 1 and 2; its operation is denoted as mode 1. When the fan was off, the state of the openings between the first floor and the lower attic determined whether buoyancy flow up through the attic was present (mode 2) or not (mode 3). These two choices each split as a function of whether first-floor doors were open and, if so, whether the upper-attic windows were open. Mode 4 means that the building was entirely shut, mode 5 denotes cross-ventilation only, and mode 8 allows combined buoyancy- and wind-driven flows. Mode 7 consisted of negligibly few data points. 3. Model development A model to predict temperatures in each thermal zone of a building should account for the heat transfer processes that

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3.1. Multi-zone models Six linear zonal models were constructed using the features listed in Table 1. Note that there are no lag terms for wind direction. The heat input due to wind-driven flow through an aperture is shown in Eq. (2). Thermocouple locations

7

6 10 8 1

9

Fig. 3. Sectional view, showing the location of selected thermocouples. (Courtesy of G. Ives.)

affect these temperatures. Linear processes are readily modeled. Steady-state heat flow through a wall is linear in indoor–outdoor temperature difference and transient heat conduction can be approximated as linear processes over discrete time steps. Solar heat gains are directly proportional to solar radiation incident on a window. However, heat flows associated with natural ventilation are nonlinear functions of wind speed and indoor and outdoor temperatures. Control modes may create different flow regimes, which in turn cause changes in relations between features and predicted variables. These issues prompted our initial investigation of modeling methods designed to account for nonlinear behavior. The Principal Hessian Direction Regression Tree (PHDRT) method cuts a data space into regions where linear regressions may be accurately applied [58,59]. The method uses the direction of greatest curvature of a data set to separate the data into two regions; this process continues iteratively until the root-mean-square error for a linear regression in a region falls below a user-defined threshold. A somewhat related method, developed after our work was completed, divided simulated whole-building-energy consumption into a tree of binary regions by use of the Fisher Discriminant [60]. The PHDRT method successfully identified buoyancy-driven flow regimes (upward or downward flow) for a two-zone thermal model. However, when applied to measured data from the sunspace in the test building the PHDRT method was no better than a simple linear regression using all the data. Two forms of black-box models, neural nets and a kernel recursive least squares (KRLS) method [61,62] that provided a compact model representation without the need to specify a neural-net architecture, were also investigated. Both were less successful than linear models, which were subsequently used for this investigation. Two types of simulations were performed: onestep-ahead predictions and pure simulation over an extended time period, with no input of zone-temperature measurements. For numerical stability, regressions employed singular-value decomposition [63]. Data were divided into three portions: a training set to directly adjust model parameters, a test set to select model features (i.e., model order) and a separate validation set to check the performance of the models.

Heat inputwind fðTout  Tin Þ Cp ðq; geometryÞ


1=2

Vwind

(2)

where Cp is the wind pressure coefficient, a function of wind angle q and building geometry. The relation between wind angles and the wind pressure coefficient is highly nonlinear; further, estimating a value for Cp would require time-averaged values of wind speed and indoor–outdoor temperature difference. For these reasons, wind direction was not considered in linear models. A graphical representation of the model parameters selected via regression is given in Fig. 5. These parameters are the a, b and c coefficients in Eq. (1). Note the dominance of the each zone’s temperature parameters in the plot of that zone’s parameters: assembly-temperature parameters dominate in the first plot, mass temperature parameters dominate in the second, etc. The plotting technique deliberately exaggerates the smaller parameters. From the first plot, it is clear that the assembly and mass temperatures are strongly coupled, and from the second plot, it is clear that the direction of influence is not balanced. Attic 1 can be seen to be coupled to the mass. The mass may serve as a smoothed surrogate for the assembly temperature, to which it is directly coupled via airflow. Curiously, the sunspace appears to be coupled to the mass and attic 1. The mass connection may be related to conduction heat transfer through the building/sunspace wall. The attic 1 connection may merely indicate a strong correlation (but no causal relationship) between the two spaces. Fig. 6 illustrates the performance of the models composed of the above parameters over one week of the validation period. Qualitatively, model predictions were good, with no extreme deviations. The model captured the average behavior of the building but did not fully mimic building behavior in response to particular inputs. For example, the model over-predicted temperatures in the afternoons of August 20 and August 21, when the building was completely shut. Having control settings as features does not guarantee that the model will make effective use of them. Fig. 7 shows errors associated with prediction of assembly-room temperatures by the multi-zone model. Table 2 lists pure-simulation errors for all zones, under the multi-zone model and, to be discussed next, the multi-zone and multi-mode model. 3.2. Multi-mode models In an attempt to develop a model more capable of predicting building behavior, the data were manually divided into modes as illustrated in Fig. 4. Shown below in Fig. 8 are the (adjusted) model parameters for each mode of each zone. Note the distinct difference between the parameter envelope for mode 1 versus the other modes in Fig. 8. This distinction is somewhat artificial, because some features3 were deemed irrelevant during fan operation due to the dominant effect of the fan on the thermal behavior of the building. For the other modes, it is difficult to detect a regular pattern distinguishing the parameter weights from one mode to another. The multi-mode model showed a dramatic improvement in performance. The pure-simulation RMS error on the validation set for the assembly temperature dropped from 0.72  C to 0.34  C,

3 For the assembly and mass zones, when the fan was on, features [3–5, 13–15, 30–31] were excluded; for the attic 1 zone, [4–6, 13–15, 17, 31, 32] were excluded; and for the attic 2 zone, [1, 2, 5, 6, 13–15, 17, 30, 32] were excluded.

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Table 1 Features used in generating models. ID

Feature

Number of lag terms

Scaling rangea

ID

Feature

Number of lag terms

Scaling range

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Tassembly Tmass Tattic 1 Tattic 2 Tsunspace Tbasement Tout Isun, north Isun, south Isun, east Isun, west Isun, horiz Sin(wind-dir) Cos(wind-dir) Wind speed Qbsmt Qfirst

5 5 5 5 5 5 5 5 5 5 5 5 0 0 5 5b 5

275, 330 275, 300 275, 300 275, 300 275, 300 275, 300 275, 300 0, 300 0, 800 0, 950 0, 1100 0, 1100 1, 1 1, 1 0, 6 0, 3 0, 7

18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

E louver door W louver door Conf/assembly door Office windows S Office windows W Attic slider Fan slider Fan speed Sunspace door Attic 2 windows Assembly windows Basement door (Tassembly  Tattic 1)  fan speed (Tattic 2  Tattic 1)  fan speed (Tout  Tassembly)  fan speed

2c 2 2 2 2 2 2 2 2 2 2 2 5d 5 5

0, 1 0, 1 0, 2 0, 1 0, 1 0, 1 0, 1 0, 5000 0, 2 0, 1 0, 1 0, 1 10 000, 11 500 135 000, 3000 42 000, 0

a

Temperatures in K, solar radiation in W/m2, wind speed in m/s, electrical power in kW, and fan-input voltage in mV. Electrical power Qbsmt was used only for the model of the basement zone; Qfirst was used for all zones except the sunspace and basement. c Controls were used as features only for multi-mode modeling. d For multi-zone modeling, five lag terms of features 30–32 were available for models of all zones except the sunspace and basement. For multi-mode models, five lag terms of feature 30 were available for attic 1, five lag terms of feature 31 for attic 2, and five lag terms of feature 32 for the assembly and mass zones. In all cases, these features were available only when the fan was on at time t  1 (for making a prediction at time t). For automated feature selection, the same conditions holding for multi-mode models were in effect. b

while the RMS error on the combined training and validation sets dropped from 0.64  C to 0.40  C. Bear in mind that the simulations were run much longer than required in practice; the RMS error for 24-h predictions would be even smaller than reported. More important than the global error measures is that the model predictions were adjusted depending on the mode of operation of the building; there is a better likelihood that trial control strategies may be investigated with some confidence. The improvement in performance is visually apparent in the differences between Figs. 6 and 9. An exception is the last day, August 22, when the operating modes may have been incorrectly recorded. Note in Fig. 5 and 8 that model parameters for wind speed, feature 15, are essentially zero. Put more directly, the linear models

Root Mode (12)

Fan On? Y

N Attic or Fan Slider Open?

Mode 1

Y

N

Mode 2

Mode 3

Door 1, 2 or 3 Open?

Door 1, 2 or 3 Open?

Y

N

Mode 6

Y Mode 7

Attic 2 Windows Open? Y Mode 8

Mode 4

Attic 2 Windows Open? Y

N Mode 9

N

Mode 5

Mode 10

Fig. 4. Tree representation of building modes.

N Mode 11

showed no significant thermal effect due to wind speed. Further analysis showed that wind speed appeared to influence fluctuations in indoor temperature, showing that there was wind-generated indoor air movement that might improve occupant comfort. As will be shown in the companion paper, the response of the building under the cross-ventilation mode (mode 5) is intuitively reasonable: stronger than with all windows closed and not as strong under either combined buoyancy- and wind-driven flows (mode 8) or fan operation (mode 1). However, the response could not be correlated to wind speeds measured during the test period. 4. Automated feature selection When many elements exist that may influence the calculation of a cost function (e.g., the minimization of model-prediction errors) it may not be clear which elements are most important. A method is needed to identify those elements. The method outlined in this section is appropriate for any model structure, not just for linear models. The general concept has been called greedy optimization or forward/backward selection [64]. In the case at hand, there are 32 types of features identified as possible model inputs. Others could be identified, such as time of day, day of year, etc. Furthermore, the optimal number of lag terms for each feature is not known. The best possible model will utilize only those features that are relevant and that improve model performance. The feature-selection technique described below is suboptimal since local minima exist in the performance-versus-feature-vector space. Another approach to the problem would be via genetic algorithms (see, e.g., Ref. [31]). Note that the term ‘‘feature’’ refers to any input to a model, and that Tout(t  1) and Tout(t  2) are two distinct features. The use of the term ‘‘greedy optimization’’ becomes apparent after review of the algorithmdone always selects the feature associated with the biggest performance gain at every step of the process. 1. Assume an initial number of lag terms common to all feature types (2 in this case). 2. Initial feature elimination (backward selection) a. Identify the next (or first) feature type with at least one lag term available. b. For this feature type, remove one lag term. In other words, if T(t  1) and T(t  2) are available features, eliminate the feature T(t  2).

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Fig. 5. Representation of model parameters. To reveal the minor components, all parameters were converted as follows: Plotted parameteri ¼ sign(parameteri)jparameterij1/2. The axis numbering is identical to the numbering in Table 1 (e.g., in the plot of basement parameters, the five parameters corresponding to the sixth feature, Tbasement, are located at 6.0, 6.2, ., 6.8). The parameter indicated by the arrow in each plot corresponds to the coefficient of that zone temperature at time t  1. The constant term is plotted at 0.4 on the abscissa.

c. Using a training set, perform regression to compute all model parameters without the identified feature. d. Compute the cost function: RMS error on a separate test set. e. Replace the eliminated feature, and return to Step 2a until all feature types have been identified. f. Identify the feature whose removal led to the largest reduction of RMS error on the test set. If no error reduction was found, proceed to Step 3. Otherwise, eliminate the identified feature from the feature vector and return to Step 2a using the new feature vector as the starting point. 3. Forward/backward selection (repeat this loop a fixed number of times, or until no performance benefit obtained) a. Feature addition (forward selection) Perform the same operations as outlined in Step 2, except rather than removing a feature from every eligible feature type, add one (perhaps up to a maximum). In essence, those features whose addition most benefits the model are added. Repeat this forward selection a set number of times or until no further benefit is achieved. b. Feature elimination (backward selection) Perform the identical operations outlined in Step 2. Repeat this backward selection a set number of times or until no further benefit is achieved. Then return to repeat Step 3. Again, Step 3 may be repeated a set number of times, or until no further benefit is obtained. The structure of Step 3 of the procedure is intended to help the algorithm maneuver past local minima in the cost function surface. Note the underlying assumption that if T(t  3) is relevant, then T(t  2) and T(t  1) are as well. This assumption dramatically reduces the search space of the problem. If it is important to relax this constraint, the same basic algorithm could be used, with only a few minor modifications. (Essentially, every place ‘‘feature type’’ exists, replace it with ‘‘feature.’’ Also, the maximum number of lag terms becomes 1.) This unconstrained problem may be more efficiently addressed by a genetic algorithm solution. Finally, it should be noted that the set of features produced by this algorithm is optimized for use in a particular model structure

(linear, in this case) and for performance on a particular measure. The suboptimal set of features could change if either the cost function or the model structure was altered. Greedy-feature optimization was implemented into the modeling process, using the data divided into control modes. An initial population of two lag terms was given to each of the feature types used (refer back to Table 1 for a listing). Each of the feature types was permitted a maximum of five lag terms. The initial feature elimination (Step 2, above) was performed, followed by seven repetitions of the forward/backward selection (each consisting of two loops of Step 3a and one of Step 3b). If no improvement was found before the seven loops were completed, the algorithm terminated. Several divisions of the full training/test period into training data sets and test data sets were employed. The optimization process produced a sparse set of features but the predictive power of the models in each case was not distinctly better than that of models without feature optimization. 5. Automated mode selection Because feature optimization yielded relatively little performance benefit compared with the dramatic benefit associated with dividing the data set up into modes, an algorithm was developed to automatically group, or cluster, data points associated with each mode (defined by a binary string of control settings), eliminate sparsely populated modes, and further reduce the number of modes if desired by clustering modes. Merger of clusters is accomplished in two steps. Candidates for clustering are identified on the basis of proximity, which is expressed by the number of shared bits in the binary control string, but are merged on the basis of performance, which is expressed by step-ahead RMS errors achieved with the trial cluster. Fifty-one distinct modes were generated from the data, nine of which were associated with the fan and clustered together. The method was used to generate 25, 15 and 5 mode clusters. Some of the clusters contained very different combinations of openings; for example, one of the 25 mode clusters contained modes with the attic and assembly zones fully isolated and fully connected.

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Fig. 6. Multi-zone model: measured (black) and predicted (grey) temperatures over the first part of the validation data set.

The performance of the models created for the automatically clustered modes was very close to that obtained using the original mode definitions, and is much better than the case where no modes were used at all (see Table 2). Therefore, it is possible to generate mode clusters automatically (if imperfectly) and obtain much of the performance achieved when ‘‘expert knowledge’’ is used to cluster the modes. The pure-simulation RMS errors did not increase monotonically as the number of mode clusters decreased. The key to understanding this observation is the notion of generalization ability. Assuming that modes belong together in a cluster, the greater number of data points in that cluster the more likely the model built for that cluster will be able to incorporate a wide range of operating temperatures, solar radiation, etc. Consequently, the model will be able to generalize better and perform well in conditions not used for model training. Returning to the observation, as the number of mode clusters dropped from 15 to 5, the number of points in each cluster increased, and the resulting models were better able to make predictions on the full set of conditions experienced during the validation set. In this case, the improved generalization outweighed the loss of information associated with cluster merging.

6. Performance comparison with a physical model A previous study of the building [65] included the development of a physically based model of the building. The model consisted of two mass nodes (air and thermal mass) and contained inputs such as outside temperature, wind speed and direction, estimated solar radiation (based on a clear-sky radiation model coupled with cloud cover information from a nearby airport) and building usage. Measured electrical and solar radiation data were not available. Windows and doors, as well as the opening to the attic, were all fixed. For three week-long data sets taken from this previous study, the RMS error was 0.74  C and the maximum magnitude error was 3.32  C. For comparison, the pure-simulation errors for the assembly zone (complete set) in the linear data-driven models were 0.42  C and 1.62  C, a significant error reduction for a building operated in a more complex fashion, with variable openings. 7. Application to a second building A thermal model was established for a second building to assess the broader applicability of the model structure and parameterestimation process. The three-story office building, located in the

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Fig. 7. Errors associated with the multi-zone model, assembly-room temperatures ( C): left, histogram of pure-simulation errors over the complete data set; right, histogram of one-step-ahead errors over the complete data set.

UK, featured a central atrium and operable windows [1,66]. Ventilation fans and associated dampers at the top of the atrium were operated independently and as necessary to reduce temperatures in the atrium; the open-plan offices were otherwise naturally ventilated. Four operating modes were selected: fans on and fans off, in each case for occupied and unoccupied periods. Using these modes and automated feature selection (number of lagged terms in the time series), RMS errors varied from 0.5  C to 0.8  C for the seven modeled thermal zones. There were relatively few high-outdoortemperature data points when the fan was off and the model could not establish the benefit of fan operation in hot weather. As a result, the model as used with available data was considered to provide reasonably good fit but more data would be needed to establish a model adequate for ventilation control. 8. Conclusion Mixed-mode cooling systems can reduce building operating costs and carbon emissions. Controlling mixed-mode cooling systems in a way that maximizes their energy-savings benefits while preserving occupant comfort is a challenging problem. Control requires an accurate, building-specific thermal model.

First-principles models for building airflow were considered but not selected due to the large number of variables that would require measurement, including local wind speed and direction and the size of window openings, and the complexity of the models for multi-zone buildings. Data-driven models were selected, with the recognition that such models must be tuned to the performance of each building in which they are used, that a model structure should be sufficiently general to apply to buildings other than a single test building, and that the calibration process will necessarily require a substantial set of measurements of building temperatures, internal loads, and environmental conditions. The nonlinearities associated with natural ventilation prompted an initial exploration of data-driven models that explicitly account for nonlinear behavior, including a flexible regression, the Principle Hessian Direction Regression Tree (PHDRT) [51] that partitioned data sets into regions where individual linear models could be applied, and two black-box methods, a kernel recursive least squares (KRLS) method [54] and neural networks. These models worked well with simulated data sets but, when applied to data from an occupied, multi-zone test building, gave temperature predictions that were no more accurate and in some cases for the black-box models, less stable than those from the linear models. The acceptance of the data

Table 2 Root-mean-square and maximum absolute errors ( C), for pure simulation with the validation data sets. Zone

Assembly Mass Attic 1 Attic 2 Sunspace Basement

Multi-zone

Multi-zone and multi-mode Manual feature selection

Automated feature selection

Automated mode selection (number of mode clusters) 43

25

15

5

RMS error

Max jerrorj

RMS error

Max jerrorj

RMS error

Max jerrorj

RMS error

RMS error

RMS error

RMS error

0.72 0.67 0.45 0.50 1.22 0.48

2.36 2.11 1.61 1.67 4.84 1.15

0.34 0.34 0.41 0.38 1.21 0.33

1.49 1.50 1.41 1.25 4.78 1.21

0.35 0.34 0.42 0.38 1.24 0.59

1.54 1.40 1.31 1.46 4.85 1.41

0.34 0.30 0.38 0.48 1.19 0.32

0.35 0.32 0.37 0.47 1.19 0.32

0.41 0.39 0.35 0.47 1.18 0.32

0.37 0.36 0.50 0.44 1.19 0.36

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Fig. 8. Model parameters (adjusted as before) for each mode. Left column: assembly (modes 1, 4, 5, 8 and 9), attic 1 (modes 1, 8, 9, 10 and 11) and sunspace (mode 12). Right column: mass (modes 1, 4, 5, 8 and 9), attic 2 (modes 1, 8, 9, 10 and 11) and basement (mode 12).

as opposed to reliance on ‘‘correct analytical predictions’’ permitted the creation of very flexible and very accurate linear models, which relied on measurements of indoor air and mass temperatures, electrical loads, and outdoor conditions. Many elements influence the thermal behavior of a building. Due to the difficulty of assessing the relevance of these elements a priori, an automated method for selecting model features (or inputs) was developed and demonstrated. Features were judged

strictly on how their inclusion in a model impacted the accuracy of model predictions. The algorithm led to some interesting findings, such as that wind did not play an important role in the thermal behavior of the test building. The performance benefit obtained from automated feature selection was less important for the test building than that obtained from the incorporation of building modes into the modeling approach. The control modes were determined manually, based on knowledge of how the building was

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Fig. 9. Multi-mode model: measured (black) and predicted (grey) temperatures over the first part of the validation data set.

operated. A procedure to automate the process of defining the key building modes was proposed. Dedicated models were created to learn the operation of the building in each mode. Therefore, the building ‘‘model’’ actually consisted of a set of models, each appropriate for a given control mode. Model predictions in multi-day (and multi-week) pure simulations differed from measured temperatures by an RMS error of 0.3–0.4  C for most zones. This modeling accuracy exceeded that reported in the literature for much simpler buildings and much shorter prediction horizons. As long as the first stage of the modeling process is begun with a test of the nonlinearity of the system, the modeling framework elucidated here is expected to perform well in other buildings. Had data from a strongly nonlinear building been available, nonlinear models could have been developed for those particular operational modes requiring them. The simplest, anddfrom the positive results found in the flow-regime–change modeling experimentsdthe most effective tool to begin with would be the PHDRT algorithm. However, any other tool, such as KRLS, could be substituted in its place. The key

is the focus on the essential operational modes of the building and the selection of the correct features (or model inputs). Finally, is the modeling approach applicable to other buildings? The premise of the work is that modeling is intended to serve automatic control. Given that purpose and the need for sensors, data logging equipment and computer-based control, the model is not intended for residential buildings, where occupants can make reasonable control decisions on the basis of their perceived thermal comfort and an overnight weather forecast. The structure of the model separates control modes and accepts measurements of temperatures and wind speed; it should therefore account for fan-, buoyancy- and wind-driven airflows. While mode separation improved the accuracy of the model (as shown in this paper) and will be shown to give distinct temperature profiles (in the companion paper), the lack of sensitivity of the model to wind speed is puzzling. The linear structure of the model precluded the use of wind direction as a model feature and the model could not be used to assess the impact of variation of wind direction for a given wind speed.

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