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Building energy modeling: A systematic approach to zoning and model reduction using Koopman Mode Analysis
Accepted Manuscript Title: Building Energy Modeling: A Systematic Approach to Zoning and Model Reduction using Koopman Mode Analysis Author: Michael Georgescu Igor Mezi´c PII: DOI: Reference:
S0378-7788(14)00891-3 http://dx.doi.org/doi:10.1016/j.enbuild.2014.10.046 ENB 5429
To appear in:
ENB
Received date: Revised date: Accepted date:
15-5-2014 5-10-2014 21-10-2014
Please cite this article as: Michael Georgescu, Igor Mezi´c, Building Energy Modeling: A Systematic Approach to Zoning and Model Reduction using Koopman Mode Analysis, Energy & Buildings (2014), http://dx.doi.org/10.1016/j.enbuild.2014.10.046 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights (for review)
Building Energy Modeling: A Systematic Approach to Zoning and Model Reduction using Koopman Mode Analysis Highlights:
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Koopman modes are used to analyze building models and create zoning approximations Koopman modes reveal spatial structures of building response at multiple time scales Model complexity is reduced while preserving main features of building response
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*Manuscript
Building Energy Modeling: A Systematic Approach to Zoning and Model Reduction using Koopman Mode Analysis Michael Georgescua,b,∗, Igor Mezi´ca,b a Department
of Mechanical Engineering, University of California, Santa Barbara, CA, 93106, United States Institute for Energy Efficiency, University of California, Santa Barbara, CA, 93106, United States
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Abstract
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As the scope of building design and construction increases and building systems become more integrated, the use of building energy models has become increasingly widespread in evaluating and predicting building performance. Despite the growing sophistication of building modeling tools, errors can arise from approximations that are made by a practitioner during model creation. This paper examines the process of model zoning, i.e., how the volume of a building is divided into regions where properties are assumed to be uniform. Zoning is performed during model creation to decrease model complexity. However, accuracy reduces when dissimilar regions of a building are defined by a single zone. In this paper, a systematic approach to creating zoning approximations is introduced. Utilizing the Koopman operator, the time-series output produced by a building simulation can be decomposed into spatial modes which capture the thermal behavior of a building at different time-scales. Identification of spatial structures within these modes forms a framework for the creation of simplified models of varying levels of granularity. In this paper, a detailed model is analyzed, and model accuracy is studied as coarser building representations are created using the introduced method.
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Keywords: Koopman Operator, Building Energy Models, Lumped Models, Model Zoning, Model Order Reduction, Data Visualization
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1. Introduction
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Development of models for use in predicting building energy consumption and comfort has been ongoing for several decades. Model-based analysis has found use in multiple areas of the building systems community. Applications of model-based analysis during the design and construction phase include proof of concept feasibility studies [1, 2, 3], sensitivity analysis [4], and calculating benchmarks energy efficiency such as that which is performed in obtaining LEED certification [5]. After a building is constructed, models are used for control or retrofit studies [6, 7] and predictive or optimal control [8, 9, 10] which can be implemented to create efficient control sequences and building schedules based off of forecasts of weather and building usage. Building models are indeed finding a wide variety of uses, and as a result, the field is becoming increasingly interdisciplinary with interests from architects to engineers, from both industry and academia, with varying technical backgrounds. Fundamentally, the relevant dynamics of a building can be described by a system of partial differential equations (PDEs). This system is often too time consuming to fully ∗ Corresponding
author. Tel.: +1 8058935095. Email addresses: [email protected] (Michael Georgescu), [email protected] (Igor Mezi´ c) URL: http://www.engr.ucsb.edu/~mgroup/ (Igor Mezi´ c) Preprint submitted to Energy and Buildings
simulate over a long duration of time (e.g., a yearlong period), therefore approximations are normally defined to reduce simulation time. Examples of approximations include one-dimensional heat transfer through walls and floors and uniform material properties. Thus the original PDEs are transformed into a lumped parameter system which takes less time to solve numerically. Building simulation environments such as EnergyPlus, TRNSYS, and ESP-r use these assumptions when predicting building performance, and numerous studies exist that reinforce the validity of these simulation environments as well as their assumptions for predicting building heat transfer [11, 12]. Within the building simulation environments mentioned, a detailed model, generally, treats each room of a building as an individual thermal zone with uniform properties. At this level of detail, inputs are defined for each room separately and the temperature history of each room is individually calculated. However in practice, a model where each room corresponds to a unique thermal zone can be time consuming to develop or analyze. To reduce the time required to create and analyze a model, a practitioner defines zoning approximations where the properties of multiple rooms are combined into a single thermal zone. Since the number of parameters becomes increasingly difficult to manage as model complexity grows, and such models require more time to fully simulate, zoning approximations are made to mitigate these issues. Using this simplification, a model of a building contains fewer thermal zones October 5, 2014
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than there are physical rooms. Currently, the zoning of rooms is often performed heuristically based on experiences of the practitioner, but is usually based on some similarity between the regions being combined (e.g. similar internal loads, shared HVAC components, etc.). The creation of zoning approximations for building energy models is within the realm of model order reduction (MOR) [13]. MOR studies the properties of dynamical systems with the goal of systematically finding approximations which best preserves the behavior, within some tolerance, of the original system. When first-principles modeling of a system is performed, often high-dimensional mathematical models result. MOR is often applied to reduce the computation complexity of such systems. Techniques for MOR have been applied in other fields where systems of large dimension are numerically simulated including fluid mechanics [14] and circuit design [15]. Several existing studies describe the effects that zoning approximations have on the accuracy of building models. In [16], the sensitivity to zoning of a 5 room model is explored, and the author shows that models with too few thermal zones can under-predict energy usage and comfort by over-predicting the rate of air mixing caused when rooms are grouped together. In [17], zoning of a building model is performed in order to optimize the performance of a HVAC system utilizing decentralized control. A technique for model order reduction of building models is presented in [18] where zoning approximations are created based off of the time-constants of heat transfer between zones through inter-zone surfaces. These approximations preserve the physical interpretation of the building model bringing insight to the behavior of zones through control of the aggressiveness of the reduction scheme. However this approach requires knowledge of the equations describing the temperature evolution of the building model. When using building simulation environments, such as EnergyPlus, the thermal balance of the model is not explicitly available. In the case of EnergyPlus, this is a result of tight coupling between the numerical integration schemes of the program and the thermal balance equations which represent the building model. This paper presents a timeseries based approach for creating zoning approximations based off of observations of temperature with no explicit knowledge of the equations describing the thermal balance of the building model. In this paper, zoning approximations of building models are systematically created by utilizing properties of the Koopman operator and the notion of Koopman modes: the Koopman operator is a linear, infinite-dimensional operator that captures properties of nonlinear, finite dimensional dynamics without linearization. Using properties of this operator, modes which capture spatial relationships in building thermal behavior can be extracted from the temperature history of zones produced by a building simulation. By creating zoning approximations based on the identification of spatial structures within these modes, simplified realizations of the building plan are created which
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attempt to preserve the behavior of the original system. In the following sections of this paper, a detailed building model is created and model accuracy is studied as coarser building representations are created using this method. The notion of Koopman modes was first introduced in [13] and applied in the study of fluid dynamics [19] and power grid instabilities [20]. The strategy in this paper of identifying spatial structures of modes is similar to identifying coherency between power generators in [21]. The Koopman operator approach has also previously been used to study the thermal behavior of other building features; in [22], an EnergyPlus model of a building was partially calibrated to measured data by comparing the modes produced by the model to that from measured temperature data, and in [23], Koopman modes are used to validate modeling practices used in the ASHRAE 90.1 modeling appendix [24]. This paper is an expansion of those previous works. The remainder of this paper is organized as follows: in the next section, an overview of the Koopman operator is given. In section 3, the case study building, and associated building model, are introduced. Then in section 4 spatial-temporal modes of a detailed building model are analyzed, and using characteristics of these modes, zoning approximations of varying levels of coarseness are systematically created. The paper is concluded with a summary of the zoning approximations created using this method and a comparison of simulation performance using increasing levels of zonal approximation. 2. The Koopman Operator In the analysis of dynamical systems, understanding the structure of systems plays a considerable role. Typically, a set of initial in the state-space of a system of ODEs are evolved in time into trajectories, and with some additional conditions on the system being examined, the behavior of trajectories are analyzed in relation to the stability of fixed points and their associated manifolds in the system. Because the equations describing a building model can be of a high dimension and are often not accessible analytically, methods are required which are measurement, or time-series based, to study such systems. In this context, the Koopman operator can be applied to building models for the visualization and analysis of these systems. By projecting the time-evolution of outputs from a building simulation onto eigenfunctions of the operator, spatial features of the system being studied can be extracted. To introduce the Koopman operator, consider the evolution of a nonlinear dynamical system given by
x(t + 1) = F (x(t))
(1)
where x ∈ M are the state space variables belonging to a finite, but multi-dimensional space M , and F : M → M 2
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maps the variables at time t to time t + 1. The Koopman operator U is a linear operator that acts on M in the following manner: for g : M → R, where g is a function describing observations of the state space variables, U maps g to a new function U g given by
U g(x) = g(F (x(t))) = g(x(t + 1)).
the relationship between Fourier analysis and the Koopman operator is first identified in [25]. There are several methods available for calculating Koopman modes such as using the Arnoldi algorithm [20], or by taking Fourier averages over a spatial field [13]. When observables are periodic, the decomposition can be computed using discrete Fourier transformation [19]. For more information about model decomposition using the Koopman operator, refer to the references above and the review paper [26]. In this paper, the Koopman operator is utilized to create zoning approximations by analyzing spatial features of a subset of Koopman modes that reflect the primary features of a building model’s temperature behavior. The calculated modes are observed at frequencies corresponding to physically significant time-scales of the building’s temperature behavior. By creating zoning approximations that preserve spatial characteristics of these modes, simplified models are created that aim to best preserve the thermal behavior of the original model for the simulated conditions.
(2)
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k = 1, ..., n.
3. The Case Study Building
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The Student Resources Building (SRB) was constructed in 2007 at the University of California, Santa Barbara. The building is a hub for campus organizations and primarily serves student needs. The 68,000 sqft. floor plan contains three above ground floors and is separated into northern and southern halves by a 4 story 5500 sqft. atrium acting as a divider between these halves. The building includes several areas of different space utilization including offices, classrooms, child daycare, and auditorium / theater, as well as common areas were student groups can assemble. Energy saving features of the building design include extensive use of natural ventilation throughout the building floor space. The atrium and offices in the perimeter of the building are naturally ventilated with baseboard heaters while rooms in the core of the buildings are conditioned by one of six air handling units. Mechanically ventilated rooms are conditioned by either variable air volume units with terminal reheat or constant air volume units. Fan coil units condition unoccupied rooms that contain electrical or telecommunications equipment which require continuous space cooling. HVAC chilled water is provided by district cooling from a campus chilled water loop while hot water for heating is generated by a gas fired boiler located in the plant of the building. Because of the coastal setting, cool offshore winds flow perpendicular to the length of the atrium. Using motors along the perimeter of the atrium, vents can be opened allowing hot air to advect from the atrium ceiling while cool outdoor air is drawn through vents located on the ground floor of the building.
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U ψk (x) = λk ψk (x).
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Although the dynamical system may be nonlinear and evolve on a finite-dimensional space, the Koopman operator is linear, but infinite-dimensional. In this work, the function g is a time-series of space temperature measurements. The function g can be any time-series measurable quantity. There are no requirements on the smoothness of the function g, but in numerical implementation, it is convenient that measurements exist at equal time intervals. The Koopman operator describes the evolution of an observable one step in time, and iterative application of the operator describes the trajectory of the observable. Because the Koopman operator is linear, its eigenfunctions and eigenvalues are defined as follows: for eigenfunctions ψk : M → C and constant eigenvalues λk ∈ C
n X
λk ψk (x)vk .
k=1
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Vector-valued observables, G : M → Rm , that are in the span of eigenfunctions, ψk , k = 1, ..., n, where n may be infinity, can be expressed as projections onto the eigenfunctions of the Koopman operator by
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In Eq.4, {vk }nk=1 are a set of vectors called Koopman modes, and are coefficients of the projections of observables onto the eigenfunctions of the operator. Koopman modes describe the dynamics of observables at different eigenvalue frequencies and growth rates (proportional to θk and rk where λk = rk eiθk ), and will be the basis for creating zoning approximations described later in this paper. For dynamics on an attractor, Koopman modes do not grow (i.e. rk = 1), and Koopman eigenvalues are on the unit circle. When observables are bounded, G(x) then can be decomposed by calculating Fourier averages, G∗ω (x) ∈ Cm , of the form: n−1 1 X i2πjω e G(x(j)). n→∞ n j=0
G∗ω (x) = lim
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If G∗ω is nonzero, for a given ω, the observable, G(x), has a nonzero component at the Koopman mode of frequency ω. The notion of Koopman modes was introduced in [13], and 3
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Figure 1: (Color Online) Exterior views of the university building studied. Northern entrance of the SRB (left), and southern entrance (right).
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Figure 2: Exterior views of the EnergyPlus model of the university building studied. Northern exterior (top), and southern exterior (bottom).
3.1. The EnergyPlus Model
building walkthroughs. Usage profiles were modeled after the operating hours of the building and observations of occupant activity. The HVAC system of the building model is defined based on design documents.
The whole building energy model of the SRB was created using DesignBuilder [27] and simulated using EnergyPlus v6.0.0.023. Weather input for the simulation was gathered from TMY3 weather data for the local area. In the model studied, each room is represented as its own thermal zone. A rendering of the building model is shown in Figure 2. The model contains 215 zones. With a timestep duration of 30 minutes, the model takes 3 hours to simulate one year of operation on a computer with 2.53 GHz CPU. Architectural data for the model was obtained from CAD drawings, and thermal properties of materials used in construction were estimated using information from literature. Internal heat gains include occupancy, lighting, and equipment operation. In the case of lighting, heat generation was calculated from specifications available in building drawings. Heat from equipment operation and occupancy were estimated by an audit conducted during
4. Koopman Mode Analysis of the EnergyPlus Model Applying properties of the Koopman operator, and the concept of Koopman modes, the thermal behavior of the detailed EnergyPlus model is studied. Analysis of Koopman modes allows identification of spatial structures of a system’s observables which exhibit similar temporal dynamics. For the task of creating zoning approximations, the observable of interest is zone temperature. Using the EnergyPlus model of the case study building, a yearlong simulation is performed, and from the simulation, a set of Koopman modes are calculated from model 4
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predictions of zone air temperature. The Koopman spectrum calculated from the EnergyPlus model is shown in Figure 3.
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Figure 4: Magnitude of Koopman modes ordered by normalized magnitude. Zoning approximations are based off of spatial structures of highlighted modes.
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Figure 3: (Color Online) Koopman spectrum for the zones of the detailed EnergyPlus model. Color reflects amplitude of temperature response. Period (Hours) of largest modes indicated: A(8760Hr), B(4380Hr), C(168Hr), D(24Hr), E(12Hr).
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Horizontal bands in the spectrum reveal that the temperature behavior of a particular zone has spectral content across multiple Koopman modes, while vertical bands indicate that a large group of zones is influential to a particular Koopman mode which indicates spatial coherency. Within the spectrum, the five modes that are largest in magnitude correspond to the time-scale periods of 8760 hours (year), 24 hours (day), 4380 hours (half-year), 168 hours (week), and 12 hours (half-day). The periods reflect time-scales that are physically significant and thus, in many cases, only several modes are required to describe important characteristics of the temperature response of the building. Figure 4 compares the magnitude of modes illustrating how their magnitude quickly decreases beyond the first few. Because Koopman modes are defined on the complex set of numbers, each mode has an associated amplitude and phase. The temperature response of building zones is related to the amplitude and phase of the calculated Koopman modes in the following manner:
Tm (t) =
∞ X
kVk,m ksin(ωk t + ∠Vk,m ),
Figure 5: (Color Online) Amplitude and phase of zones for the five modes considered. Adjacent zones with similar amplitudes and phases are grouped together when creating zoning approximations.
the yearlong and daily Koopman mode indicating that the largest oscillations in zone temperature occur at these two frequencies. Additionally, the daily mode has 2 clusters which are concentrated at a phase of roughly 0 and π radians. Considering Eq.6 for the 24 hour Koopman mode, mechanically ventilated rooms have a phase near 0 radians because their temperature generally rises between 00:00 - 12:00 (HVAC mainly off) and lowers between 12:00 24:00 (HVAC mainly on). Naturally ventilated rooms on the other hand have a phase near π radians because their temperature lowers between 00:00 - 12:00 (morning heat loss) and rises between 12:00 - 24:00 (afternoon heat gain). In Figures 6 - 8, the amplitudes and phases of zones from the five considered Koopman modes are shown spatially
(6)
k=1
where Vk = (vk,1 , ..., vk,m )T and ωk is the frequency corresponding to the k-th Koopman mode. In Figure 5, the distribution of amplitudes and phases of the 5 largest modes are shown. Each point in Figure 5 corresponds to a particular zone for a particular mode. Clusters in the distribution are informative of primary attributes of the temperature response of building zones. The clusters with the largest amplitude belong to 5
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against a plan view of the building to illustrate these observations. Koopman modes calculated from the temperature response of a building simulation are a result of oscillatory heat loads the building model is subject to. The modes are influenced by model parameters, simulated weather, internal heat loads, and HVAC heating and cooling for zones with mechanical conditioning. The modes capture the dynamics of a particular simulation, and changes to the simulation, such as the use of alternate weather conditions or different internal loads, will produce a different temperature response resulting in a different set of Koopman modes. Analysis of the modes requires some understanding of the heat flows the building is subject to, and some conclusions can be drawn from prior knowledge of the model inputs. In a physical context, the magnitude and phase of each mode is influenced by the inputs (heat loads) that the building is subject to from internal and external sources. In the model simulation performed, sources of heat input consist of weather, HVAC heating and cooling, occupancy, lighting, and equipment. Of these inputs, weather primarily contributes to building thermal behavior at the 8760 hour and 24 hour time-scale due to changes in building heat transfer between night and day as well as the different seasons. Internal heat generation (from occupants, HVAC heating and cooling, lighting, equipment) influences thermal behavior at the 168 hour and 24 hour time-scale due to defined set-points and operation schedules. The 4380 hour and 12 hour modes are sub-harmonics of the 8760 hour and 24 hour Koopman modes respectively and adds further delineation between the behavior of building zones. In Figure 6, the yearlong mode possesses a nearly uniform phase (between 2-3 radians), but has large changes in amplitude. The phase of zones are similar in magnitude since heat transfer at this time scale occurs at a slower rate compared to the time constant of the thermal mass of the building. The amplitudes of zones in this mode however vary greatly due to differences in construction throughout the building plan; the zones corresponding to more heavily insulated areas within the building core (not including the central atrium), have a lower amplitude of temperature oscillation compared to more heavily glazed, naturally ventilated, offices along the building perimeter. An exception to this is shown in a handful of zones which are representative of server rooms of the building modeled. These zones contain continuously operating HVAC equipment which condition the zone to a constant temperature. Therefore, these zones contain no oscillatory components in the modes considered. Modes at shorter time scales (168 hours and lower) contain large variations in phase. Compared to the yearlong mode, which is primarily influenced by weather, the modes at shorter time-scales are additionally affected by internal building loads. Different configurations of weather, defined internal loads, and spatial location for modeled zones
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can be seen in the Koopman mode phase at these scales. For example, the phase of zones within the 24 hour mode, shown in Figure 7, can be used to identify zones which are mechanically ventilated (phase ≈ 0 radians) and naturally ventilated (phase ≈ π radians). This is due to naturally ventilated zones experiencing maximum temperature during the day (due to solar radiation and higher outdoor air temperature), and mechanically ventilated zones experiencing minimum temperature during the day (due to HVAC cooling). Patterns between zones among the amplitude and phase of Koopman modes reveal building areas which are similarly influenced by simulated weather and internal loads. By identifying spatial structures of adjacent zones occurring across multiple modes, a zoning approximation can be made that best preserves the thermal behavior of the original model. 4.1. Creating Zoning Approximations
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From the calculated Koopman modes of the case study building model, adjacent zones are grouped together if their amplitudes and phases within the Koopman modes considered are within some tolerance. Under this criteria, the temperature behavior of the adjacent zones are coherent. From the discussion of Koopman modes presented, combining zones using this approximation physically corresponds to combining zones which behave similarly due to internal heat generation and environmental heat transfer. The following definition of coherency is used for comparing the amplitudes and phases of zones and creating zoning approximations: Definition 1. Coherency of Koopman Modes: consider a vector-valued observable G : M → Rm , where G = (g1 , ..., gm )T and gj : M → R for j = 1, ..., m. Let V1 (G), ..., Vk (G), k ≤ n be a collection of Koopman modes of interest. Note that Vk (G) = (vk,1 , ..., vk,m )T ∈ Rm . Fix k,1 , k,2 ≥ 0 and consider j1 , j2 ∈ 1, ..., m. Then, gj1 and gj2 are (k,1 , k,2 )-coherent (with respect to the k-th chosen Koopman mode) if |kvk,j1 k − kvk,j2 k| < k,1
(7)
and |∠(vk,j1 ) − ∠(vk,j2 )| < k,2
(8)
for all j1 , j2 = 1, ..., m where m is the number of zones. Note that in Eq.8, ∠(•) is the angle of a complex number and the equation is the shortest distance between two points on a circle. When creating zoning approximations, adjacent zones are combined if they satisfy this definition of coherency given the tolerances k,1 and k,2 . Because the amplitude and phase of zones can be distributed differently across the Koopman modes considered, the values of k,1 and k,2 can be chosen independently for each mode. The values of for each mode determine the aggressiveness of the algorithm 6
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ip t cr us an M ed pt Ac ce Figure 6: (Color Online) 8760 hour and 4380 hour Koopman mode amplitudes and phases plotted against a plan view of the building. (Note differences in magnitude of color scale between images)
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ip t cr us an M ed pt Ac ce Figure 7: (Color Online) 168 hour and 24 hour Koopman mode amplitudes and phases plotted against a plan view of the building. (Note differences in magnitude of color scale between images)
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Figure 8: (Color Online) 12 hour Koopman mode amplitude and phase plotted against a plan view of the building. (Note differences in magnitude of color scale between images)
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at combining adjacent zones. If = 0, only zones with identical temperature responses will be combined, but if a larger value for is chosen, a greater number of zones will be considered to be similarly behaving causing a coarser zoning approximation to be calculated. To summarize, the procedure of creating zoning approximations from an EnergyPlus model using Koopman modes is: (i) Simulate full-order EnergyPlus mode outputting observables of interest (in this case zone temperature). (ii) Calculate Koopman modes by projecting observables onto eigenfunctions of the Koopman operator. (iii) Combine adjacent zones which have coherent temperature responses. Using this procedure zoning approximations,with varying degrees of coarseness, are created by changing the values of k,1 and k,2 based on the number of desired unique zones. When zones are combined, shared walls become converted into internal zone objects acting as thermal mass for the combined zones, and internal loads from equipment, lighting, and occupancy are combined for the aggregated zone. The performance of these reduced-order models will now be discussed.
4.2. Comparison of Zoning Approximations Measuring the difference between zoning approximations can be difficult due to a lack of homogeneity present in building plans. It would make little sense to compare the temperature behavior of rooms supplied by an HVAC system since their temperature can be controlled to a preset value. Nor would it be relevant to measure the effect on HVAC energy consumption that naturally ventilated rooms have since they are not directly heated or cooled. Because there are no constraints in Eqs. 7 and 8 on the combining of zones, naturally conditioned and mechanically conditioned spaces can be merged. This is especially the case as simplified model realizations become coarse and the metric of Eqs. 7 and 8 become less stringent in comparing the temperature responses between adjacent zones. To measure the impact of the generated zoning approximations on model accuracy under a uniform criterion, the methodology for comparing a baseline model to a proposed energy model from ASHRAE Standard 90.1 Appendix G [24] is adopted. For the comparison between model realizations to be analogous, all zones of the evaluated models are simulated with heating and cooling. This methodology calculates the performance of a baseline, and compares that performance to those of proposed models (i.e. simplified models) under identical operating conditions. The approach eliminates errors caused by differences in conditioned floor area between model realizations when condi-
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Figure 9: Simplified model accuracy (relative to initial detailed model) as the values of 1,2 are increased.
tioned spaces of different types (e.g HVAC vs. naturally ventilated) are merged. The accuracy of a simplified model is measured by comparing the total HVAC energy needed to control the building to a uniform temperature. The accuracy of the simplified models with varying level of zoning refinement are shown in Figure 9 and Table 1. The baseline (non-simplified) model contains 215 zones. As different values of k,1 and k,2 are used to calculate coherency of Koopman modes, more adjacent zones are determined to be sufficiently similar to each other, reducing the total number of zones. As the number of zones reduces, the calculated HVAC energy decreases as shown in Table 1. This occurs when zones are combined due to increased air mixing within the building leading to a reduction of inter-zone heat transfer [16]. As the building model becomes increasing simplified and more zones are combined, the building is better able to resist changes in temperature causing a decrease in the cooling and heating load of the simplified building model. In the reduced models, zoning approximations using Koopman modes preserve features of the building floor plan as shown in Figures 6-8. For intuition on how this approach simplifies the building plan, the floor plans of the 215 zone, 103 zone, and 57 zone models are illustrated in Figure 10. For this particular building, when the number of zones is reduced by half, created models reach a point of oversimplification where accuracy greatly diminishes. Essentially, as the number of zones is reduced, ”clumps” are created in the representation of the building plan. Tracking the progress of plan layout with respect to the amount of model simplification, zones with similar loads, orientation, and spatial location (core vs. perimeter) are the first to combine. Error greatly increases when these guidelines begin to be violated. It should be noted that the reduction of zones is with respect to the metric shown in Eqs. 7 and 8 which is based on temperature behavior. With the metric used, some small zones remain in the reduced floorplans which correspond to areas such as vacant spaces and server rooms. Although these spaces have unique temper-
Table 1: Change in model predictive capability of heating and cooling loads as the number of zones is reduced.
1,2 0 0.01 0.06 0.1 0.15 0.20 0.25 0.30 0.35 0.45 0.70 0.92
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# of Zones 215 212 195 173 137 103 79 69 57 40 16 7
Htg. (% Error) 0 0.1 0.1 1 1.6 9.9 18.3 23.5 31.2 36.2 56.4 66.8
Clg. (% Error) 0 0.1 0.2 0.8 0.5 2.2 3.1 4.3 6.2 9.1 13.6 15.6
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ature responses compared to adjacent areas, they are small enough in volume that they could potentially be merged earlier in the reduction process with minimal impact to total HVAC energy consumption. Depending on the intent of the modeling practitioner, other metrics can be created which preserve additional aspects of building performance besides temperature during the reduction process. 5. Discussion and Conclusions In this paper, a detailed building energy model is analyzed using the Koopman operator in order to identify, and systematically develop, zoning approximations based off of observations of zone temperature. The goal of these approximations is to reduce model complexity while minimally impacting model accuracy. Through calculating Koopman modes, influences of the simulated heat loads on zones of the building model can be identified at different physically significant time-scales. By identifying spatial structures across these time-scales, models containing 10
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Figure 10: Boundaries of two zoning approximations compared to the original model zone layout (plan view of the building shown). Left plan is original model, middle plan is 103 zone simplified model (1,2 = 0.20), and right plan is 57 zone simplified model (1,2 = 0.35).
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a reduced number of zones are created. While reducing the number of zones of the building model studied, several observations were made on types of simplifications that had little impact on accuracy. The following simplifications can potentially be applied if a simplified model is initially being created, and this would be an approach to take if one is unable to first create, and simulate, a detailed building model. Some guidelines to observe which can help maintain model accuracy coming from the presented approach are: (i) The space use classification is the same throughout the thermal zone (ii) All rooms in a thermal zone that are adjacent to glazed exterior walls face the same orientation or their orientations vary by less than 45 degrees. (iii) Separate zones should be assumed for interior and perimeter rooms.
The guidelines mentioned are not definitive for every model, but instead serve as a rule of thumb and oversimplification should always be a concern to the practitioner. In the building model studied, there exists a balance point at which accuracy greatly diminishes due to over consolidation of zones from the original detailed building model. In this model, the balance point occurs when there are
roughly half as many zones as there are rooms. This balance point though depends on the layout of the building. Future work may be to investigate approaches of determining a bound for this balance point based off of characteristics of a building layout. To apply the method proposed in this paper, a detailed building model must first be created so that a simplified model can be produced. Although creating a detailed model can be time-consuming for a practitioner, software packages to create such models are evolving and allow buildings with complex geometries and constructions to be created in shorter amounts of time [28]. As software becomes more developed (e.g. integration of building information models (BIM) into the building energy modeling workflow), a potential burden to the practitioner may soon become the amount of data that becomes available for analysis rather than the time necessary to create a model. By creating zoning approximations for a detailed model, less parametric information is needed to describe the building plan. This approach can be of aid to building energy modeling in the future when such difficulties with model complexity are realized. 6. Acknowledgments The authors would like to acknowledge Bryan Eisenhower, Valerie Eacret, and Erika Eskenazi for their con-
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tributions, as well as Kazimir Gasljevic and UC Santa Barbara Facilities Management. This work was partially funded by Army Research Office Grant W911NF-11-1-0511, with Program Manager Dr. Sam Stanton.
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nonlinear function state space variables i-th state space variable high dimension manifold set of real numbers Koopman operator observation function vector-valued observation function (G = (g1 , ..., gm )T ) building temperature response k-th Koopman eigenfunction k-th Koopman eigenvalue k-th Koopman mode of observable k-th Koopman mode of vectorvalued observable (Vk (G) = (vk,1 , ..., vk,m )T ) eigenvalue/eigenfunction index set of complex number coherency tolerance
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[7] K. Kabele, J. Bartonova, M. Patakiova, K. Vavrinova, Simulation in control system sensor location design, in: Building Simulation 2011. Proceedings. Twelth International Conference of Internation Building Performance Simulation Association, 2011, pp. 2573 – 2574, http://www.ibpsa.org/proceedings/BS2011/ P_1798.pdf Accessed: 7/15/2013. [8] F. Oldewurtel, A. Parisio, C. N. Jones, D. Gyalistras, M. Gwerder, V. Stauch, B. Lehmann, M. Morari, Use of model predictive control and weather forecasts for energy efficient building climate control, Energy and Buildings 45 (0) (2012) 15 – 27. doi:http://dx.doi.org/10.1016/j.enbuild.2011.09.022. URL http://www.sciencedirect.com/science/article/pii/ S0378778811004105 [9] P. May-Ostendorp, G. P. Henze, C. D. Corbin, B. Rajagopalan, C. Felsmann, Model-predictive control of mixed-mode buildings with rule extraction, Building and Environment 46 (2) (2011) 428 – 437. doi:http://dx.doi.org/10.1016/j.buildenv.2010.08.004. URL http://www.sciencedirect.com/science/article/pii/ S0360132310002477 [10] S. Royer, S. Thil, T. Talbert, M. Polit, A procedure for modeling buildings and their thermal zones using co-simulation and system identification, Energy and Buildings 78 (0) (2014) 231 – 237. doi:http://dx.doi.org/10.1016/j.enbuild.2014.04.013. URL http://www.sciencedirect.com/science/article/pii/ S0378778814003132 [11] D. Bradley, M. Kummert, T. McDowell, Experiences with and interpretation of standard test methods of building energy analysis tools, in: Proceedings of the International Building Performance Simulation Association-Canada Bi-Annual Conference, eSIM 2004, 2004, http://strathprints.strath.ac.uk/6582/ 1/strathprints006582.pdf Accessed: 7/15/2013. [12] US:DOE, Energyplus simulation software: Testing and validation, http://apps1.eere.energy.gov/buildings/energyplus/ energyplus_testing.cfm?utm_source=EnergyPlus&utm_ medium=redirect&utm_campaign=EnergyPlus%2Bredirect%2B1 Accessed: 7/15/2013 (2013). [13] I. Mezic, Spectral properties of dynamical systems, model reduction and decompositions., Nonlinear Dynamics 41 (2005) 309–325. [14] C. Rowley, Model reduction for fluids, using balanced proper orthogonal decomposition, International Journal of Bifurcation and Chaos 15 (03) (2005) 997–1013. [15] R. W. Freund, Krylov-subspace methods for reduced-order modeling in circuit simulation, Journal of Computational and Applied Mathematics 123 (1) (2000) 395–421. [16] W. O’Brien, A. Athienitis, T. Kesik, Thermal zoning and interzonal airflow in the design and simulation of solar houses: a sensitivity analysis, Journal of Building Performance Simulation 4(3) (2011) 239–256. [17] V. Chandan, A. Alleyne, Optimal partitioning for the decentralized thermal control of buildings, Control Systems Technology, IEEE Transactions onAccessed: 7/15/2013. [18] S. Goyal, P. Barooah, A method for model-reduction of non-linear thermal dynamics of multi-zone buildings, Energy and Buildings 47 (0) (2012) 332 – 340. doi:http://dx.doi.org/10.1016/j.enbuild.2011.12.005. URL http://www.sciencedirect.com/science/article/pii/ S0378778811006001 [19] C. Rowley, I. Mezic, S. Bagheri, P. Schlatter, D. Henningson, Spectral analysis of nonlinear flows, Journal of Fluid Mechanics 641 (2009) 115–127. [20] Y. Susuki, I. Mezic, Nonlinear koopman modes of coupled swing dynamics and coherency identification, in: Power and Energy Society General Meeting, 2010 IEEE, 2010, pp. 1 –8. [21] Y. Susuki, I. Mezic, Nonlinear koopman modes and coherency identification of coupled swing dynamics, Power Systems, IEEE Transactions on 26 (4) (2011) 1894–1904. doi:10.1109/TPWRS.2010.2103369. [22] B. Eisenhower, T. Maile, M. Fischer, I. Mezic, Decomposing building systems data for model validation and anal-
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[1] J. Binks, Closing the loop. office tower simulation assumptions vs reality, in: Building Simulation 2011. Proceedings. Twelth International Conference of Internation Building Performance Simulation Association, 2011, pp. 941 – 948, http://www.ibpsa.org/proceedings/BS2011/P_1362.pdf Accessed: 7/15/2013. [2] S. Gross, H. Hu, A feasibility study on natural ventilation in a midrise student dormitory building, in: Building Simulation 2011. Proceedings. Twelth International Conference of Internation Building Performance Simulation Association, 2011, pp. 878 – 885, http://www.ibpsa.org/proceedings/bs2011/p_ 1350.pdf Accessed: 7/15/2013. [3] D. Hes, A. Dawkins, C. Jensen, L. Aye, A modelling method to assess the effect of tree shading for building performance simulation, in: Building Simulation 2011. Proceedings. Twelth International Conference of Internation Building Performance Simulation Association, 2011, pp. 161 – 168, http://www.ibpsa. org/proceedings/BS2011/P_1176.pdf Accessed: 7/15/2013. [4] B. Eisenhower, Z. O’Neill, V. A. Fonoberov, I. Mezi, Uncertainty and sensitivity decomposition of building energy models, Journal of Building Performance Simulation 5 (3) (2012) 171– 184. arXiv:http://dx.doi.org/10.1080/19401493.2010.549964, doi:10.1080/19401493.2010.549964. URL http://dx.doi.org/10.1080/19401493.2010.549964 [5] UCSGBC, Leed-nc (leadership in energy and environmental design) version 2.2, www.usgbc.org/leed/nc/ Accessed: 2/13/2012 (2011). [6] L. Chahwane, L. Stephan, E. Wurtz, B. Zuber, On a novel approach to control natural and mechanical night ventilation, in: Building Simulation 2011. Proceedings. Twelth International Conference of Internation Building Performance Simulation Association, 2011, pp. 1800 – 1807, http://www.ibpsa. org/proceedings/BS2011/P_1584.pdf Accessed: 7/15/2013.
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ysis using the koopman operator, in: SimBuild 2010. Proceedings. Fourth National Conference of International Building Performance Simulation Association-USA, 2010, pp. 434 – 441, http://www.ibpsa.us/pub/simbuild2010/papers/ SB10-DOC-TS08B-03-Eisenhower.pdf Accessed: 7/15/2013. M. Georgescu, B. Eisenhower, I. Mezi´ c, Creating zoning approximations to building energy models using the koopman operator, in: SimBuild 2012. Proceedings. Fifth National Conference of International Building Performance Simulation AssociationUSA, 2012, pp. 40 – 47, http://www.ibpsa.us/simbuild2012/ Papers/SB12_TS01b_3_Georgescu.pdf Accessed: 7/15/2013. ASHRAE, ASHRAE Standard 90.1:Energy Standard for Buildings Except Low-Rise Residential Buildings, 2010, https://www.ashrae.org/standards-research--technology/ standards--guidelines Accessed: 7/15/2013. I. Mezic, A. Banaszuk, Comparison of systems with complex behavior, Physica D 197 (2004) 101–133. M. Budisic, R. Mohr, I. Mezic, Applied koopmanism, Chaos: An Interdisciplinary Journal of Nonlinear Science 22 (4) (2012) 047510. doi:10.1063/1.4772195. URL http://link.aip.org/link/?CHA/22/047510/1 DesignBuilder, http://designbuilderusa.com/, accessed: 1/15/2013 (2012). N. Jones, B. McCrone, B. Walter, K. Pratt, D. Greenberg, Automated translation and thermal zoning of digitalbuilding models for energy analysis, in: Building Simulation 2013. Proceedings. Thirteenth International Conference of Internation Building Performance Simulation Association, 2013, pp. 202 – 209. URL http://www.ibpsa.org/proceedings/BS2013/p_1027.pdf
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S0378-7788(14)00891-3 http://dx.doi.org/doi:10.1016/j.enbuild.2014.10.046 ENB 5429
To appear in:
ENB
Received date: Revised date: Accepted date:
15-5-2014 5-10-2014 21-10-2014
Please cite this article as: Michael Georgescu, Igor Mezi´c, Building Energy Modeling: A Systematic Approach to Zoning and Model Reduction using Koopman Mode Analysis, Energy & Buildings (2014), http://dx.doi.org/10.1016/j.enbuild.2014.10.046 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights (for review)
Building Energy Modeling: A Systematic Approach to Zoning and Model Reduction using Koopman Mode Analysis Highlights:
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Koopman modes are used to analyze building models and create zoning approximations Koopman modes reveal spatial structures of building response at multiple time scales Model complexity is reduced while preserving main features of building response
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*Manuscript
Building Energy Modeling: A Systematic Approach to Zoning and Model Reduction using Koopman Mode Analysis Michael Georgescua,b,∗, Igor Mezi´ca,b a Department
of Mechanical Engineering, University of California, Santa Barbara, CA, 93106, United States Institute for Energy Efficiency, University of California, Santa Barbara, CA, 93106, United States
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Abstract
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As the scope of building design and construction increases and building systems become more integrated, the use of building energy models has become increasingly widespread in evaluating and predicting building performance. Despite the growing sophistication of building modeling tools, errors can arise from approximations that are made by a practitioner during model creation. This paper examines the process of model zoning, i.e., how the volume of a building is divided into regions where properties are assumed to be uniform. Zoning is performed during model creation to decrease model complexity. However, accuracy reduces when dissimilar regions of a building are defined by a single zone. In this paper, a systematic approach to creating zoning approximations is introduced. Utilizing the Koopman operator, the time-series output produced by a building simulation can be decomposed into spatial modes which capture the thermal behavior of a building at different time-scales. Identification of spatial structures within these modes forms a framework for the creation of simplified models of varying levels of granularity. In this paper, a detailed model is analyzed, and model accuracy is studied as coarser building representations are created using the introduced method.
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Keywords: Koopman Operator, Building Energy Models, Lumped Models, Model Zoning, Model Order Reduction, Data Visualization
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1. Introduction
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Development of models for use in predicting building energy consumption and comfort has been ongoing for several decades. Model-based analysis has found use in multiple areas of the building systems community. Applications of model-based analysis during the design and construction phase include proof of concept feasibility studies [1, 2, 3], sensitivity analysis [4], and calculating benchmarks energy efficiency such as that which is performed in obtaining LEED certification [5]. After a building is constructed, models are used for control or retrofit studies [6, 7] and predictive or optimal control [8, 9, 10] which can be implemented to create efficient control sequences and building schedules based off of forecasts of weather and building usage. Building models are indeed finding a wide variety of uses, and as a result, the field is becoming increasingly interdisciplinary with interests from architects to engineers, from both industry and academia, with varying technical backgrounds. Fundamentally, the relevant dynamics of a building can be described by a system of partial differential equations (PDEs). This system is often too time consuming to fully ∗ Corresponding
author. Tel.: +1 8058935095. Email addresses: [email protected] (Michael Georgescu), [email protected] (Igor Mezi´ c) URL: http://www.engr.ucsb.edu/~mgroup/ (Igor Mezi´ c) Preprint submitted to Energy and Buildings
simulate over a long duration of time (e.g., a yearlong period), therefore approximations are normally defined to reduce simulation time. Examples of approximations include one-dimensional heat transfer through walls and floors and uniform material properties. Thus the original PDEs are transformed into a lumped parameter system which takes less time to solve numerically. Building simulation environments such as EnergyPlus, TRNSYS, and ESP-r use these assumptions when predicting building performance, and numerous studies exist that reinforce the validity of these simulation environments as well as their assumptions for predicting building heat transfer [11, 12]. Within the building simulation environments mentioned, a detailed model, generally, treats each room of a building as an individual thermal zone with uniform properties. At this level of detail, inputs are defined for each room separately and the temperature history of each room is individually calculated. However in practice, a model where each room corresponds to a unique thermal zone can be time consuming to develop or analyze. To reduce the time required to create and analyze a model, a practitioner defines zoning approximations where the properties of multiple rooms are combined into a single thermal zone. Since the number of parameters becomes increasingly difficult to manage as model complexity grows, and such models require more time to fully simulate, zoning approximations are made to mitigate these issues. Using this simplification, a model of a building contains fewer thermal zones October 5, 2014
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than there are physical rooms. Currently, the zoning of rooms is often performed heuristically based on experiences of the practitioner, but is usually based on some similarity between the regions being combined (e.g. similar internal loads, shared HVAC components, etc.). The creation of zoning approximations for building energy models is within the realm of model order reduction (MOR) [13]. MOR studies the properties of dynamical systems with the goal of systematically finding approximations which best preserves the behavior, within some tolerance, of the original system. When first-principles modeling of a system is performed, often high-dimensional mathematical models result. MOR is often applied to reduce the computation complexity of such systems. Techniques for MOR have been applied in other fields where systems of large dimension are numerically simulated including fluid mechanics [14] and circuit design [15]. Several existing studies describe the effects that zoning approximations have on the accuracy of building models. In [16], the sensitivity to zoning of a 5 room model is explored, and the author shows that models with too few thermal zones can under-predict energy usage and comfort by over-predicting the rate of air mixing caused when rooms are grouped together. In [17], zoning of a building model is performed in order to optimize the performance of a HVAC system utilizing decentralized control. A technique for model order reduction of building models is presented in [18] where zoning approximations are created based off of the time-constants of heat transfer between zones through inter-zone surfaces. These approximations preserve the physical interpretation of the building model bringing insight to the behavior of zones through control of the aggressiveness of the reduction scheme. However this approach requires knowledge of the equations describing the temperature evolution of the building model. When using building simulation environments, such as EnergyPlus, the thermal balance of the model is not explicitly available. In the case of EnergyPlus, this is a result of tight coupling between the numerical integration schemes of the program and the thermal balance equations which represent the building model. This paper presents a timeseries based approach for creating zoning approximations based off of observations of temperature with no explicit knowledge of the equations describing the thermal balance of the building model. In this paper, zoning approximations of building models are systematically created by utilizing properties of the Koopman operator and the notion of Koopman modes: the Koopman operator is a linear, infinite-dimensional operator that captures properties of nonlinear, finite dimensional dynamics without linearization. Using properties of this operator, modes which capture spatial relationships in building thermal behavior can be extracted from the temperature history of zones produced by a building simulation. By creating zoning approximations based on the identification of spatial structures within these modes, simplified realizations of the building plan are created which
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attempt to preserve the behavior of the original system. In the following sections of this paper, a detailed building model is created and model accuracy is studied as coarser building representations are created using this method. The notion of Koopman modes was first introduced in [13] and applied in the study of fluid dynamics [19] and power grid instabilities [20]. The strategy in this paper of identifying spatial structures of modes is similar to identifying coherency between power generators in [21]. The Koopman operator approach has also previously been used to study the thermal behavior of other building features; in [22], an EnergyPlus model of a building was partially calibrated to measured data by comparing the modes produced by the model to that from measured temperature data, and in [23], Koopman modes are used to validate modeling practices used in the ASHRAE 90.1 modeling appendix [24]. This paper is an expansion of those previous works. The remainder of this paper is organized as follows: in the next section, an overview of the Koopman operator is given. In section 3, the case study building, and associated building model, are introduced. Then in section 4 spatial-temporal modes of a detailed building model are analyzed, and using characteristics of these modes, zoning approximations of varying levels of coarseness are systematically created. The paper is concluded with a summary of the zoning approximations created using this method and a comparison of simulation performance using increasing levels of zonal approximation. 2. The Koopman Operator In the analysis of dynamical systems, understanding the structure of systems plays a considerable role. Typically, a set of initial in the state-space of a system of ODEs are evolved in time into trajectories, and with some additional conditions on the system being examined, the behavior of trajectories are analyzed in relation to the stability of fixed points and their associated manifolds in the system. Because the equations describing a building model can be of a high dimension and are often not accessible analytically, methods are required which are measurement, or time-series based, to study such systems. In this context, the Koopman operator can be applied to building models for the visualization and analysis of these systems. By projecting the time-evolution of outputs from a building simulation onto eigenfunctions of the operator, spatial features of the system being studied can be extracted. To introduce the Koopman operator, consider the evolution of a nonlinear dynamical system given by
x(t + 1) = F (x(t))
(1)
where x ∈ M are the state space variables belonging to a finite, but multi-dimensional space M , and F : M → M 2
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maps the variables at time t to time t + 1. The Koopman operator U is a linear operator that acts on M in the following manner: for g : M → R, where g is a function describing observations of the state space variables, U maps g to a new function U g given by
U g(x) = g(F (x(t))) = g(x(t + 1)).
the relationship between Fourier analysis and the Koopman operator is first identified in [25]. There are several methods available for calculating Koopman modes such as using the Arnoldi algorithm [20], or by taking Fourier averages over a spatial field [13]. When observables are periodic, the decomposition can be computed using discrete Fourier transformation [19]. For more information about model decomposition using the Koopman operator, refer to the references above and the review paper [26]. In this paper, the Koopman operator is utilized to create zoning approximations by analyzing spatial features of a subset of Koopman modes that reflect the primary features of a building model’s temperature behavior. The calculated modes are observed at frequencies corresponding to physically significant time-scales of the building’s temperature behavior. By creating zoning approximations that preserve spatial characteristics of these modes, simplified models are created that aim to best preserve the thermal behavior of the original model for the simulated conditions.
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3. The Case Study Building
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The Student Resources Building (SRB) was constructed in 2007 at the University of California, Santa Barbara. The building is a hub for campus organizations and primarily serves student needs. The 68,000 sqft. floor plan contains three above ground floors and is separated into northern and southern halves by a 4 story 5500 sqft. atrium acting as a divider between these halves. The building includes several areas of different space utilization including offices, classrooms, child daycare, and auditorium / theater, as well as common areas were student groups can assemble. Energy saving features of the building design include extensive use of natural ventilation throughout the building floor space. The atrium and offices in the perimeter of the building are naturally ventilated with baseboard heaters while rooms in the core of the buildings are conditioned by one of six air handling units. Mechanically ventilated rooms are conditioned by either variable air volume units with terminal reheat or constant air volume units. Fan coil units condition unoccupied rooms that contain electrical or telecommunications equipment which require continuous space cooling. HVAC chilled water is provided by district cooling from a campus chilled water loop while hot water for heating is generated by a gas fired boiler located in the plant of the building. Because of the coastal setting, cool offshore winds flow perpendicular to the length of the atrium. Using motors along the perimeter of the atrium, vents can be opened allowing hot air to advect from the atrium ceiling while cool outdoor air is drawn through vents located on the ground floor of the building.
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Although the dynamical system may be nonlinear and evolve on a finite-dimensional space, the Koopman operator is linear, but infinite-dimensional. In this work, the function g is a time-series of space temperature measurements. The function g can be any time-series measurable quantity. There are no requirements on the smoothness of the function g, but in numerical implementation, it is convenient that measurements exist at equal time intervals. The Koopman operator describes the evolution of an observable one step in time, and iterative application of the operator describes the trajectory of the observable. Because the Koopman operator is linear, its eigenfunctions and eigenvalues are defined as follows: for eigenfunctions ψk : M → C and constant eigenvalues λk ∈ C
n X
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Vector-valued observables, G : M → Rm , that are in the span of eigenfunctions, ψk , k = 1, ..., n, where n may be infinity, can be expressed as projections onto the eigenfunctions of the Koopman operator by
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In Eq.4, {vk }nk=1 are a set of vectors called Koopman modes, and are coefficients of the projections of observables onto the eigenfunctions of the operator. Koopman modes describe the dynamics of observables at different eigenvalue frequencies and growth rates (proportional to θk and rk where λk = rk eiθk ), and will be the basis for creating zoning approximations described later in this paper. For dynamics on an attractor, Koopman modes do not grow (i.e. rk = 1), and Koopman eigenvalues are on the unit circle. When observables are bounded, G(x) then can be decomposed by calculating Fourier averages, G∗ω (x) ∈ Cm , of the form: n−1 1 X i2πjω e G(x(j)). n→∞ n j=0
G∗ω (x) = lim
(5)
If G∗ω is nonzero, for a given ω, the observable, G(x), has a nonzero component at the Koopman mode of frequency ω. The notion of Koopman modes was introduced in [13], and 3
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Figure 1: (Color Online) Exterior views of the university building studied. Northern entrance of the SRB (left), and southern entrance (right).
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Figure 2: Exterior views of the EnergyPlus model of the university building studied. Northern exterior (top), and southern exterior (bottom).
3.1. The EnergyPlus Model
building walkthroughs. Usage profiles were modeled after the operating hours of the building and observations of occupant activity. The HVAC system of the building model is defined based on design documents.
The whole building energy model of the SRB was created using DesignBuilder [27] and simulated using EnergyPlus v6.0.0.023. Weather input for the simulation was gathered from TMY3 weather data for the local area. In the model studied, each room is represented as its own thermal zone. A rendering of the building model is shown in Figure 2. The model contains 215 zones. With a timestep duration of 30 minutes, the model takes 3 hours to simulate one year of operation on a computer with 2.53 GHz CPU. Architectural data for the model was obtained from CAD drawings, and thermal properties of materials used in construction were estimated using information from literature. Internal heat gains include occupancy, lighting, and equipment operation. In the case of lighting, heat generation was calculated from specifications available in building drawings. Heat from equipment operation and occupancy were estimated by an audit conducted during
4. Koopman Mode Analysis of the EnergyPlus Model Applying properties of the Koopman operator, and the concept of Koopman modes, the thermal behavior of the detailed EnergyPlus model is studied. Analysis of Koopman modes allows identification of spatial structures of a system’s observables which exhibit similar temporal dynamics. For the task of creating zoning approximations, the observable of interest is zone temperature. Using the EnergyPlus model of the case study building, a yearlong simulation is performed, and from the simulation, a set of Koopman modes are calculated from model 4
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predictions of zone air temperature. The Koopman spectrum calculated from the EnergyPlus model is shown in Figure 3.
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Figure 4: Magnitude of Koopman modes ordered by normalized magnitude. Zoning approximations are based off of spatial structures of highlighted modes.
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Figure 3: (Color Online) Koopman spectrum for the zones of the detailed EnergyPlus model. Color reflects amplitude of temperature response. Period (Hours) of largest modes indicated: A(8760Hr), B(4380Hr), C(168Hr), D(24Hr), E(12Hr).
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Horizontal bands in the spectrum reveal that the temperature behavior of a particular zone has spectral content across multiple Koopman modes, while vertical bands indicate that a large group of zones is influential to a particular Koopman mode which indicates spatial coherency. Within the spectrum, the five modes that are largest in magnitude correspond to the time-scale periods of 8760 hours (year), 24 hours (day), 4380 hours (half-year), 168 hours (week), and 12 hours (half-day). The periods reflect time-scales that are physically significant and thus, in many cases, only several modes are required to describe important characteristics of the temperature response of the building. Figure 4 compares the magnitude of modes illustrating how their magnitude quickly decreases beyond the first few. Because Koopman modes are defined on the complex set of numbers, each mode has an associated amplitude and phase. The temperature response of building zones is related to the amplitude and phase of the calculated Koopman modes in the following manner:
Tm (t) =
∞ X
kVk,m ksin(ωk t + ∠Vk,m ),
Figure 5: (Color Online) Amplitude and phase of zones for the five modes considered. Adjacent zones with similar amplitudes and phases are grouped together when creating zoning approximations.
the yearlong and daily Koopman mode indicating that the largest oscillations in zone temperature occur at these two frequencies. Additionally, the daily mode has 2 clusters which are concentrated at a phase of roughly 0 and π radians. Considering Eq.6 for the 24 hour Koopman mode, mechanically ventilated rooms have a phase near 0 radians because their temperature generally rises between 00:00 - 12:00 (HVAC mainly off) and lowers between 12:00 24:00 (HVAC mainly on). Naturally ventilated rooms on the other hand have a phase near π radians because their temperature lowers between 00:00 - 12:00 (morning heat loss) and rises between 12:00 - 24:00 (afternoon heat gain). In Figures 6 - 8, the amplitudes and phases of zones from the five considered Koopman modes are shown spatially
(6)
k=1
where Vk = (vk,1 , ..., vk,m )T and ωk is the frequency corresponding to the k-th Koopman mode. In Figure 5, the distribution of amplitudes and phases of the 5 largest modes are shown. Each point in Figure 5 corresponds to a particular zone for a particular mode. Clusters in the distribution are informative of primary attributes of the temperature response of building zones. The clusters with the largest amplitude belong to 5
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against a plan view of the building to illustrate these observations. Koopman modes calculated from the temperature response of a building simulation are a result of oscillatory heat loads the building model is subject to. The modes are influenced by model parameters, simulated weather, internal heat loads, and HVAC heating and cooling for zones with mechanical conditioning. The modes capture the dynamics of a particular simulation, and changes to the simulation, such as the use of alternate weather conditions or different internal loads, will produce a different temperature response resulting in a different set of Koopman modes. Analysis of the modes requires some understanding of the heat flows the building is subject to, and some conclusions can be drawn from prior knowledge of the model inputs. In a physical context, the magnitude and phase of each mode is influenced by the inputs (heat loads) that the building is subject to from internal and external sources. In the model simulation performed, sources of heat input consist of weather, HVAC heating and cooling, occupancy, lighting, and equipment. Of these inputs, weather primarily contributes to building thermal behavior at the 8760 hour and 24 hour time-scale due to changes in building heat transfer between night and day as well as the different seasons. Internal heat generation (from occupants, HVAC heating and cooling, lighting, equipment) influences thermal behavior at the 168 hour and 24 hour time-scale due to defined set-points and operation schedules. The 4380 hour and 12 hour modes are sub-harmonics of the 8760 hour and 24 hour Koopman modes respectively and adds further delineation between the behavior of building zones. In Figure 6, the yearlong mode possesses a nearly uniform phase (between 2-3 radians), but has large changes in amplitude. The phase of zones are similar in magnitude since heat transfer at this time scale occurs at a slower rate compared to the time constant of the thermal mass of the building. The amplitudes of zones in this mode however vary greatly due to differences in construction throughout the building plan; the zones corresponding to more heavily insulated areas within the building core (not including the central atrium), have a lower amplitude of temperature oscillation compared to more heavily glazed, naturally ventilated, offices along the building perimeter. An exception to this is shown in a handful of zones which are representative of server rooms of the building modeled. These zones contain continuously operating HVAC equipment which condition the zone to a constant temperature. Therefore, these zones contain no oscillatory components in the modes considered. Modes at shorter time scales (168 hours and lower) contain large variations in phase. Compared to the yearlong mode, which is primarily influenced by weather, the modes at shorter time-scales are additionally affected by internal building loads. Different configurations of weather, defined internal loads, and spatial location for modeled zones
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can be seen in the Koopman mode phase at these scales. For example, the phase of zones within the 24 hour mode, shown in Figure 7, can be used to identify zones which are mechanically ventilated (phase ≈ 0 radians) and naturally ventilated (phase ≈ π radians). This is due to naturally ventilated zones experiencing maximum temperature during the day (due to solar radiation and higher outdoor air temperature), and mechanically ventilated zones experiencing minimum temperature during the day (due to HVAC cooling). Patterns between zones among the amplitude and phase of Koopman modes reveal building areas which are similarly influenced by simulated weather and internal loads. By identifying spatial structures of adjacent zones occurring across multiple modes, a zoning approximation can be made that best preserves the thermal behavior of the original model. 4.1. Creating Zoning Approximations
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From the calculated Koopman modes of the case study building model, adjacent zones are grouped together if their amplitudes and phases within the Koopman modes considered are within some tolerance. Under this criteria, the temperature behavior of the adjacent zones are coherent. From the discussion of Koopman modes presented, combining zones using this approximation physically corresponds to combining zones which behave similarly due to internal heat generation and environmental heat transfer. The following definition of coherency is used for comparing the amplitudes and phases of zones and creating zoning approximations: Definition 1. Coherency of Koopman Modes: consider a vector-valued observable G : M → Rm , where G = (g1 , ..., gm )T and gj : M → R for j = 1, ..., m. Let V1 (G), ..., Vk (G), k ≤ n be a collection of Koopman modes of interest. Note that Vk (G) = (vk,1 , ..., vk,m )T ∈ Rm . Fix k,1 , k,2 ≥ 0 and consider j1 , j2 ∈ 1, ..., m. Then, gj1 and gj2 are (k,1 , k,2 )-coherent (with respect to the k-th chosen Koopman mode) if |kvk,j1 k − kvk,j2 k| < k,1
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and |∠(vk,j1 ) − ∠(vk,j2 )| < k,2
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for all j1 , j2 = 1, ..., m where m is the number of zones. Note that in Eq.8, ∠(•) is the angle of a complex number and the equation is the shortest distance between two points on a circle. When creating zoning approximations, adjacent zones are combined if they satisfy this definition of coherency given the tolerances k,1 and k,2 . Because the amplitude and phase of zones can be distributed differently across the Koopman modes considered, the values of k,1 and k,2 can be chosen independently for each mode. The values of for each mode determine the aggressiveness of the algorithm 6
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ip t cr us an M ed pt Ac ce Figure 6: (Color Online) 8760 hour and 4380 hour Koopman mode amplitudes and phases plotted against a plan view of the building. (Note differences in magnitude of color scale between images)
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ip t cr us an M ed pt Ac ce Figure 7: (Color Online) 168 hour and 24 hour Koopman mode amplitudes and phases plotted against a plan view of the building. (Note differences in magnitude of color scale between images)
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Figure 8: (Color Online) 12 hour Koopman mode amplitude and phase plotted against a plan view of the building. (Note differences in magnitude of color scale between images)
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at combining adjacent zones. If = 0, only zones with identical temperature responses will be combined, but if a larger value for is chosen, a greater number of zones will be considered to be similarly behaving causing a coarser zoning approximation to be calculated. To summarize, the procedure of creating zoning approximations from an EnergyPlus model using Koopman modes is: (i) Simulate full-order EnergyPlus mode outputting observables of interest (in this case zone temperature). (ii) Calculate Koopman modes by projecting observables onto eigenfunctions of the Koopman operator. (iii) Combine adjacent zones which have coherent temperature responses. Using this procedure zoning approximations,with varying degrees of coarseness, are created by changing the values of k,1 and k,2 based on the number of desired unique zones. When zones are combined, shared walls become converted into internal zone objects acting as thermal mass for the combined zones, and internal loads from equipment, lighting, and occupancy are combined for the aggregated zone. The performance of these reduced-order models will now be discussed.
4.2. Comparison of Zoning Approximations Measuring the difference between zoning approximations can be difficult due to a lack of homogeneity present in building plans. It would make little sense to compare the temperature behavior of rooms supplied by an HVAC system since their temperature can be controlled to a preset value. Nor would it be relevant to measure the effect on HVAC energy consumption that naturally ventilated rooms have since they are not directly heated or cooled. Because there are no constraints in Eqs. 7 and 8 on the combining of zones, naturally conditioned and mechanically conditioned spaces can be merged. This is especially the case as simplified model realizations become coarse and the metric of Eqs. 7 and 8 become less stringent in comparing the temperature responses between adjacent zones. To measure the impact of the generated zoning approximations on model accuracy under a uniform criterion, the methodology for comparing a baseline model to a proposed energy model from ASHRAE Standard 90.1 Appendix G [24] is adopted. For the comparison between model realizations to be analogous, all zones of the evaluated models are simulated with heating and cooling. This methodology calculates the performance of a baseline, and compares that performance to those of proposed models (i.e. simplified models) under identical operating conditions. The approach eliminates errors caused by differences in conditioned floor area between model realizations when condi-
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Figure 9: Simplified model accuracy (relative to initial detailed model) as the values of 1,2 are increased.
tioned spaces of different types (e.g HVAC vs. naturally ventilated) are merged. The accuracy of a simplified model is measured by comparing the total HVAC energy needed to control the building to a uniform temperature. The accuracy of the simplified models with varying level of zoning refinement are shown in Figure 9 and Table 1. The baseline (non-simplified) model contains 215 zones. As different values of k,1 and k,2 are used to calculate coherency of Koopman modes, more adjacent zones are determined to be sufficiently similar to each other, reducing the total number of zones. As the number of zones reduces, the calculated HVAC energy decreases as shown in Table 1. This occurs when zones are combined due to increased air mixing within the building leading to a reduction of inter-zone heat transfer [16]. As the building model becomes increasing simplified and more zones are combined, the building is better able to resist changes in temperature causing a decrease in the cooling and heating load of the simplified building model. In the reduced models, zoning approximations using Koopman modes preserve features of the building floor plan as shown in Figures 6-8. For intuition on how this approach simplifies the building plan, the floor plans of the 215 zone, 103 zone, and 57 zone models are illustrated in Figure 10. For this particular building, when the number of zones is reduced by half, created models reach a point of oversimplification where accuracy greatly diminishes. Essentially, as the number of zones is reduced, ”clumps” are created in the representation of the building plan. Tracking the progress of plan layout with respect to the amount of model simplification, zones with similar loads, orientation, and spatial location (core vs. perimeter) are the first to combine. Error greatly increases when these guidelines begin to be violated. It should be noted that the reduction of zones is with respect to the metric shown in Eqs. 7 and 8 which is based on temperature behavior. With the metric used, some small zones remain in the reduced floorplans which correspond to areas such as vacant spaces and server rooms. Although these spaces have unique temper-
Table 1: Change in model predictive capability of heating and cooling loads as the number of zones is reduced.
1,2 0 0.01 0.06 0.1 0.15 0.20 0.25 0.30 0.35 0.45 0.70 0.92
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# of Zones 215 212 195 173 137 103 79 69 57 40 16 7
Htg. (% Error) 0 0.1 0.1 1 1.6 9.9 18.3 23.5 31.2 36.2 56.4 66.8
Clg. (% Error) 0 0.1 0.2 0.8 0.5 2.2 3.1 4.3 6.2 9.1 13.6 15.6
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ature responses compared to adjacent areas, they are small enough in volume that they could potentially be merged earlier in the reduction process with minimal impact to total HVAC energy consumption. Depending on the intent of the modeling practitioner, other metrics can be created which preserve additional aspects of building performance besides temperature during the reduction process. 5. Discussion and Conclusions In this paper, a detailed building energy model is analyzed using the Koopman operator in order to identify, and systematically develop, zoning approximations based off of observations of zone temperature. The goal of these approximations is to reduce model complexity while minimally impacting model accuracy. Through calculating Koopman modes, influences of the simulated heat loads on zones of the building model can be identified at different physically significant time-scales. By identifying spatial structures across these time-scales, models containing 10
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Figure 10: Boundaries of two zoning approximations compared to the original model zone layout (plan view of the building shown). Left plan is original model, middle plan is 103 zone simplified model (1,2 = 0.20), and right plan is 57 zone simplified model (1,2 = 0.35).
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a reduced number of zones are created. While reducing the number of zones of the building model studied, several observations were made on types of simplifications that had little impact on accuracy. The following simplifications can potentially be applied if a simplified model is initially being created, and this would be an approach to take if one is unable to first create, and simulate, a detailed building model. Some guidelines to observe which can help maintain model accuracy coming from the presented approach are: (i) The space use classification is the same throughout the thermal zone (ii) All rooms in a thermal zone that are adjacent to glazed exterior walls face the same orientation or their orientations vary by less than 45 degrees. (iii) Separate zones should be assumed for interior and perimeter rooms.
The guidelines mentioned are not definitive for every model, but instead serve as a rule of thumb and oversimplification should always be a concern to the practitioner. In the building model studied, there exists a balance point at which accuracy greatly diminishes due to over consolidation of zones from the original detailed building model. In this model, the balance point occurs when there are
roughly half as many zones as there are rooms. This balance point though depends on the layout of the building. Future work may be to investigate approaches of determining a bound for this balance point based off of characteristics of a building layout. To apply the method proposed in this paper, a detailed building model must first be created so that a simplified model can be produced. Although creating a detailed model can be time-consuming for a practitioner, software packages to create such models are evolving and allow buildings with complex geometries and constructions to be created in shorter amounts of time [28]. As software becomes more developed (e.g. integration of building information models (BIM) into the building energy modeling workflow), a potential burden to the practitioner may soon become the amount of data that becomes available for analysis rather than the time necessary to create a model. By creating zoning approximations for a detailed model, less parametric information is needed to describe the building plan. This approach can be of aid to building energy modeling in the future when such difficulties with model complexity are realized. 6. Acknowledgments The authors would like to acknowledge Bryan Eisenhower, Valerie Eacret, and Erika Eskenazi for their con-
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tributions, as well as Kazimir Gasljevic and UC Santa Barbara Facilities Management. This work was partially funded by Army Research Office Grant W911NF-11-1-0511, with Program Manager Dr. Sam Stanton.
k C References
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T ψk λk vk Vk
nonlinear function state space variables i-th state space variable high dimension manifold set of real numbers Koopman operator observation function vector-valued observation function (G = (g1 , ..., gm )T ) building temperature response k-th Koopman eigenfunction k-th Koopman eigenvalue k-th Koopman mode of observable k-th Koopman mode of vectorvalued observable (Vk (G) = (vk,1 , ..., vk,m )T ) eigenvalue/eigenfunction index set of complex number coherency tolerance
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7. Nomenclature
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