The objective is to predict the output variables of a specified model with known structure and known parameters when subject to specified input variables. To ensure accuracy, models have tended to become increasingly detailed. This approach presumes knowledge not only of the various natural phenomena affecting system behavior but also of the magnitude of various interactions (e.g., effective thermal mass, heat and mass transfer coefficients). The main advantage of this approach is that the system need not be physically built to predict its behavior. The forward-modeling approach is ideal in the preliminary design when design details are limited.

Forward modeling of building energy use begins with a physical description of the building system or component of interest. For example, building geometry, geographical location, physical characteristics (e.g., wall material and thickness), type of equipment and operating schedules, type of HVAC system, building operating schedules, plant equipment, etc., are specified. The peak and average energy use of such a building can then be predicted or simulated by the forward-simulation model. The primary benefits of this method are that it is based on sound engineering principles and has gained widespread acceptance by the design and professional community. Major simulation codes, such as DOE-2, EnergyPlus, ESP-r, and TRNSYS are based on forward-simulation models.

Although procedures for estimating energy requirements vary considerably in their degree of complexity, they all have three common elements: calculation of (1) space load, (2) secondary equipment load and energy requirements, and (3) primary equipment energy requirements. Here, secondary refers to equipment that distributes the heating, cooling, or ventilating medium to conditioned spaces, whereas primary refers to central plant equipment that converts fuel or electric energy to heating or cooling effect.

The space load is the amount of energy that must be added to or extracted from a space to maintain thermal comfort. The simplest procedures assume that the energy required to maintain comfort is only a function of the outdoor dry-bulb temperature. More detailed methods consider humidity, solar effects, internal gains, heat and moisture storage in walls and interiors, and effects of wind on both building envelope heat transfer and infiltration. The section on Thermal Loads Modeling addresses some of these factors. ASHRAE Standard 183 and Chapters 17 and 18 discuss load calculation in detail.

Although energy calculations are similar to the heating and cooling design load calculations used to size equipment, they are not the same. Energy calculations are based on average use and typical weather conditions rather than on maximum use and worst-case weather. Currently, most procedures are based on hourly profiles for climatic conditions and operational characteristics for a number of typical days of the year or on 8760 hours of operation per year.

The space load is converted to a load on the secondary equipment. This can be a simple estimate of duct or piping losses or gains, or a complex hour-by-hour simulation of an air system, such as variable-air-volume with outdoor-air cooling. This step must include calculation of all forms of energy required by the secondary system (e.g., electrical energy to operate fans and/or pumps, energy in heated or chilled water).

The secondary equipment load is converted to the fuel and energy required by the primary equipment and the peak demand on the utility system. It considers equipment efficiencies and part-load characteristics. It is often necessary to keep track of the different forms of energy, such as electrical, natural gas, and/or oil. In some cases, where calculations are required to ensure compliance with codes or standards, these energies must be converted to source energy or resource consumed, as opposed to energy delivered to the building boundary.

Previously, the steps were performed independently: each step was completed for the entire year and hourly results were passed to the next step. Current software usually performs all steps at each time interval, allowing effects such as insufficient plant capacity to be reflected in room conditions.

Often, energy calculations lead to an economic analysis to establish the cost effectiveness of efficiency measures (as in ASHRAE Standard 90.1). Thus, thorough energy analysis provides intermediate data, such as time of energy use and maximum demand, so utility charges can be accurately estimated. Although not part of the energy calculations, capital equipment costs should also be estimated to assess the life-cycle costs of alternative efficiency measures.

In this approach, input and output variables are known and measured, and the objective is to determine a mathematical description of the system and to estimate system parameters. In contrast to the forward approach, the data-driven approach is relevant only when the system has already been built and actual performance data are available for model development, calibration (see the section on Model Calibration), and/or identification. Two types of performance data can be used: nonintrusive and intrusive. Intrusive data are gathered under conditions of predetermined or planned experiments on the system to elicit system response under a wider range of system performance than would occur under normal system operation to allow more accurate model identification. When constraints on system operation do not allow such tests to be performed, the model must be identified from nonintrusive data obtained under normal operation.

The data-driven model has to meet requirements very different from the forward model. The data-driven model can only contain a relatively small number of parameters because of the limited and often repetitive information contained in the performance data. (For example, building operation from one day to the next is fairly repetitive.) It is thus a much simpler model that contains fewer terms representative of aggregated or macroscopic parameters (e.g., overall building heat loss coefficient and time constants). Because model parameters are deduced from actual building performance, it is much more likely to accurately capture as-built system performance, thus allowing more accurate prediction of future system behavior under certain specific circumstances. Performance data collection and model formulation need to be appropriately tailored for the specific circumstance, which often requires a higher level of user skill and expertise. In general, data-driven models are less flexible than forward models in evaluating energy implications of different design and operational alternatives, and so are not substitutes in this regard.

To better understand the uses of data-driven models, consider some of the questions that a building professional may ask about an existing building with known energy consumption (Rabl 1988):

All these questions are better addressed by the data-driven approach. The forward approach could also be used, for example, by going back to the blueprints of the building and of the HVAC system, and repeating the analysis performed at the design stage using actual building schedules and operating modes, but this is tedious and labor intensive, and materials and equipment often perform differently in reality than as specified. Tuning the forward-simulation model is often problematic, although it is an option (see the section on Model Calibration).

Energy models are useful for a range of applications, and the typical application of forward-modeling tools falls into three categories:

Forward-modeling programs are useful for evaluating many types of design elements, whereas they may be unnecessary or inappropriate for evaluating other elements. They are most useful for evaluating elements that affect heating and cooling loads. Many programs also provide the ability to compare the energy performance of different HVAC designs. The following are examples of design elements that are typically good candidates for evaluation with forward-modeling programs:

Some design elements do not necessarily require an energy model for evaluation, and a reasonably accurate savings estimate may be possible using other methods. Although it may still be useful to represent them in an energy model along with other measures, it may not be worth the effort for an analysis looking exclusively at these measures:

Limitations to the capabilities of most forward-modeling programs are important to understand. Some programs may not be appropriate candidates for performing an evaluation, depending on the feature to be analyzed:

Energy modeling has appropriate applications throughout the stages of design, construction, and operation of a building. At the earliest conceptual design phase, comparative studies help designers compare options for building form, system types, and other decisions that will be hard to change at later stages. During subsequent phases, energy modeling studies can be used to refine design decisions such as component specifications. At the conclusion of design, the models can be used to document energy code or rating-system compliance. Although use of energy models during construction is less common, their use during commissioning to compare expected performance to observed performance is growing. Similarly, building performance can be verified by comparing actual energy consumption to modeled performance, and any observed differences can help the building operator identify potential areas for improvement. The American Institute of Architects has published a guide to integrating energy modeling in the design process (AIA 2012). Proposed ASHRAE Standard 209P specifies mandatory and optional modeling tasks for new construction projects.

Creation of energy models has typically been done by specialists, and specialists continue today to perform a significant amount of energy modeling work. However, increasing user-friendliness of modeling software and the integration of building information models and energy modeling tools have made energy modeling more accessible to nonspecialists.

Anyone performing energy modeling should understand the way their program of choice works, understand assumptions being used by the program, and know how to interpret the results obtained from the program.

ASHRAE offers a Building Energy Modeling Professional (BEMP) qualification exam for modeler certification (www.ashrae.org/education--certification/certification/bemp-building-energy-modeling-professional-certification).

Selecting a building energy analysis program depends on its application, number of times it will be used, experience of the user, and hardware available to run it. The first criterion is the ability of the program to deal with the application. For example, if the effect of a shading device is to be analyzed on a building that is also shaded by other buildings part of the time, the ability to analyze detached shading is an absolute requirement, regardless of any other factors.

The cost of the computer facilities and the software itself are typically a small part of running a building energy analysis; the major costs are of learning to use the program and of using it. Major issues that influence the cost of learning a program include (1) complexity of input procedures, (2) quality of the user’s manual, and (3) of a good support system to answer questions. As the user becomes more experienced, the cost of learning becomes less important, but the need to obtain and enter a complex set of input data continues to consume the time of even an experienced user until data are readily available in electronic form compatible with simulation programs.

Complexity of input is largely influenced by the availability of default values for the input variables. Default values can be used as a simple set of input data when detail is not needed or when building design is very conventional, but additional detail can be supplied when needed. Secondary defaults, which can be supplied by the user, are also useful in the same way. Some programs allow the user to specify a level of detail. Then the program requests only the information appropriate to that level of detail, using default values for all others. However, whenever default values are used, even internally in the software, the user should take care to understand the default values and whether they apply to the situation being analyzed.

Quality of output is another factor to consider. Reports should be easy to read and uncluttered. Titles and headings should be unambiguous. Units should be stated explicitly. The user’s manual should explain the meanings of data presented. Graphic output can be very helpful. In most cases, simple summaries of overall results are the most useful, but very detailed output is needed for certain studies and also for debugging program input during the early stages of analysis.

Before a final decision is made, manuals for the most suitable programs should be obtained and reviewed, and, if possible, demonstration versions of the programs should be obtained and run, and support from the software supplier should be tested. The availability of training should be considered when choosing a more complex program.

Availability of weather data and a weather data processing subroutine or program are major features of a program. Some programs include subroutine or supplementary programs that allow the user to create a weather file for any site for which weather data are available. Programs that do not have this capability must have weather files for various sites created by the program supplier. In that case, the available weather data and the terms on which the supplier will create new weather data files must be checked. More information on weather data can be found in Chapter 14.

Auxiliary capabilities, such as economic analysis and design calculations, are a final concern in selecting a program. An economic analysis may include only the ability to calculate annual energy bills from utility rates, or it might extend to calculations or even to life-cycle cost optimization. An integrated program may save time because some input data have been entered already for other purposes.

Results of computer calculations should be accepted with caution, because software vendors do not accept responsibility for the correctness of calculation methods and have no control over program use. Manual calculation should be done to develop a good understanding of underlying physical processes and building behavior. In addition, the user should (1) review the computer program documentation to determine what calculation procedures are used, (2) compare results with manual calculations and measured data, and (3) conduct sample tests to confirm that the program delivers acceptable results.

The most accurate methods for calculating building energy consumption are the costliest because of their intense computational requirements and the expertise needed by the designer or analyst. Simulation programs that assemble component models into system models and then exercise those models with weather and occupancy data are preferred by experts for determining energy use in buildings.

Often, energy consumption at a system or whole-building level must be estimated quickly to study trends, compare systems, or study building effects such as envelope characteristics. For these purposes, simpler methods, such as degree-day and bin, may be used.

The International Building Performance Simulation Association (IBPSA-USA) maintains a website (www.buildingenergysoftwaretools.com) (IBPSA 2015) with information about hundreds of available software tools. Crawley et al. (2005) compare the capabilities of many building simulation tools.

Further described in Huang et al. (2001), the variable-base degree-day (VBDD) method accounts for balance point temperatures varying between buildings and even within a building, depending on the time of day. The 65°F base temperature is based on the assumption that heat gains from the sun and internal processes contribute on average 5°F of “free heat,” allowing a set point of 70°F with heating required only when the outdoor temperature drops below 65°F. Better-insulated, more airtight buildings can have lower balance point temperatures.The average balance point temperature for modern residential buildings is now 55 to 57°F. Commercial buildings’ low surface-to-volume ratios, high window-to-wall ratios, and high internal gains allow balance point temperatures of 50°F or lower.

Balance point temperatures differ markedly between daytime and nighttime hours. Figure 2 shows the variation of the balance point temperature for a typical house (Nisson and Dutt 1985). In daytime, the house receives heat gain from the sun and from occupant activity (e.g., equipment, lights). The balance point temperature may be around 15°F below the indoor temperature set point. However, at night, with no solar heat gain and reduced human activity, it may be only 3°F lower than set point. The variation can be even greater for commercial buildings because of higher internal heat gains during the day and very low heat gains at night.

The VBDD method accounts for these different building conditions by calculating (1) the balance point temperature of a building and (2) the heating and cooling degree-hours at that temperature. To divide the daytime and nighttime, degree-hours must be used instead of degree-days. Instead of using the average between the daily maximum and minimum temperatures as for degree-days, degree-hours are calculated from the temperature for each hour. The number of degree-hours divided by 24 can be slightly to significantly larger than the number of degree-days at the same base temperature. An example of why this can occur is a spring day where the average daily temperature may be below the base temperature, but several afternoon hours are above it. Such a day would have no cooling degree-days but a number of cooling degree-hours; these differences can be particularly significant for cooling. The calculations are repeated for daytime and nighttime conditions for each month of the year using degree-hour totals for each period with different base temperatures.

Figure 2. Variation of Balance Point Temperature and Internal Gains for Typical House (Nisson and Dutt 1985)

For cooling, the VBDD method can be adjusted to identify the number of cooling degree hours that are actually met by the HVAC system. The cooling balance point changes dramatically depending on whether the building is being vented. If the windows are open, heat gains are flushed from the building and do not affect its cooling load. However, if the windows are closed, solar and internal gains make the balance temperature significantly lower than the thermostat set point for cooling. The VBDD method accounts for these different conditions by calculating the cooling degree-hours using the balance temperature with the windows closed, but not counting degree-hours when the temperatures are below the thermostat set point, when the windows are assumed to be open. Figure 3 shows the one-to-one relationship between cooling degree-hours and the temperature difference between the balance point and outdoor air temperature. Those degree-hours occurring in the ventilation section when the outdoor temperatures are below the thermostat set point are not added to the running total of cooling degree-hours.

VBDD method calculations are, in most cases, too tedious for either hand calculations or implementation in a spreadsheet. In the early 1980s, several PC programs were written using the VBDD method, such as CIRA (Computerized Instrumented Residential Audit) (Sonderegger et al. 1982). Variable-base degree-day programs such as CIRA represent the apex of the VBDD method’s application; however, it remains a steady-state method that does not recognize a building's thermal history.

The most general and directly usable source for degree-day data is the ASHRAE Weather Data Viewer DVD (ASHRAE 2013). This application performs useful aggregations for more than 6000 locations. In addition to design conditions, the Weather DataViewer derives annual and monthly degree-days relative to any base temperature, and also provides bin data. Results are in spreadsheet form. Precalculated annual and monthly degree-days for several common bases are included in the climatic data that accompany Chapter 14, and additional online and printed collections of climatic data are listed in the section on Other Sources of Climatic Information in that chapter.

Annual and monthly degree-days relative to an arbitrary base can be estimated using the Weather Data Viewer or the procedures documented in the section on Estimation of Degree-Days in Chapter 14. Other degree-day estimation methods include the Erbs et al. (1983) model, which needs as input only the average t̅_o for each month of the year.

Figure 3. Uncounted Ventilation Degree-Hours versus Counted Cooling Degree-Hours

The heat balance method for calculating net space sensible loads is described in Chapter 18 and in the ASHRAE Toolkit for Building Load Calculations (Barnaby et al. 2005, 2009; Iu and Fisher 2004; Pedersen et al. 2001, 2003). Its development relies on the first law of thermodynamics (conservation of energy) at each surface. Because the heat balance method involves fewer assumptions than the weighting-factor method, it is more flexible and physically rigorous. However, the heat balance method requires more calculations at each point in the simulation process, using more computer time. This method has been validated in several studies (Chantrasrisalai et al. 2003a, 2003b; Eldridge et al. 2003; Iu et al. 2003).

The heat balance method allows the net instantaneous sensible heating and/or cooling load to be calculated on the space air mass. Generally, a heat balance equation is written for each enclosing surface, plus one equation for room air. Although not necessary, linearization is commonly used to simplify the radiative transfer formulation. This set of equations can then be solved for the unknown surface and air temperatures. Once these temperatures are known, they can be used to calculate the convective heat flow to or from the space air mass. The heat balance method is developed in Chapter 18 for use in design cooling load calculations.

The procedure described in Chapter 18 is aimed at obtaining the design cooling load for a fixed zone air temperature. For building energy analysis purposes, it is preferable to know the actual heat extraction rate. This may be determined by recasting Equation (27) of Chapter 18 so that the system heat transfer is determined simultaneously with the zone air temperature. The system heat transfer is the rate at which heat is transferred to the space by the system. Although this can be done by simultaneously modeling the zone and the system (Taylor et al. 1990, 1991), it is often convenient to make a simple, linearized representation of the system known as a control profile. This usually takes the form

(2)

where

System heat transfer q_sysj may be considered positive when heating is provided to the space and negative when cooling is provided. It is equal in magnitude but opposite in sign to the zone cooling load, as defined in Chapter 18, when zone air temperature is fixed.

Substituting Equation (2) into Equation (27) of Chapter 18 and solving for zone air temperature,

(3)

where

The zone air heat balance equation [Equation (3)] must be solved simultaneously with the interior and exterior surface heat balance equations [Equations (26) and (25) in Chapter 18]. Also, the correct temperature range must be found to use the proper set of a and b coefficients; this may be done iteratively. Once the zone air temperature is found, the actual system heat transfer rate may be found directly from Equation (2).

Beyond treatment of system heat transfer, other considerations that may be important in building energy analysis programs include treatment of radiant cooling and heating systems, treatment of interzone heat transfer, modeling convection heat transfer, and modeling radiation heat transfer.

The heat balance method in Chapter 18 assumes the use of a single design day. In a building energy analysis program, it is most commonly used with a year’s worth of design weather data. In this case, the first day of the year is usually simulated repeatedly until a steady-periodic response is obtained. Then, each day is simulated sequentially, and, where needed, historical data for surface temperatures and heat fluxes from the previous day are used.

When radiant cooling and heating systems are evaluated, the radiant source should be identified as a room surface. The calculation procedure considers the radiant source in the heat balance analysis. Therefore, the heat balance method is preferred over the weighting-factor method for evaluating radiant systems. Strand and Pedersen (1997) describe implementation of heat source conduction transfer functions that may be used for modeling radiant panels within a heat balance-based building simulation program.

In principle, this method extends directly to multiple spaces, with heat transfer between zones. In this case, some surface temperatures appear in the surface heat balance equations for two different zones. In practice, however, the size of the coefficient array required for solving the simultaneous equations becomes prohibitively large, and the solution time excessive. For this reason, some programs solve only one space at a time and assume that adjacent space temperatures are the same as the prior time step, or use iterative schemes to achieve a simultaneous solution. Other approaches may remove this limitation (Walton 1980).

Relatively simple exterior and interior convection models may be used for design cooling load calculation procedures. However, more sophisticated exterior convection models (Cooper and Tree 1973; Fracastoro et al. 1982; Melo and Hammond 1991; U.S. Department of Energy 1996-2016; Walton 1983; Yazdanian and Klems 1994) that incorporate the effects of wind speed, wind direction, surface orientation, etc., may be preferable. More detailed interior convection correlations for use in buildings are also available (Alamdari and Hammond 1982, 1983; Altmayer et al. 1983; Bauman et al. 1983; Bohn et al. 1984; Chandra and Kerestecioglu 1984; U.S. Department of Energy 1996-2016; Goldstein and Novoselac 2010; Khalifa and Marshall 1990; Peeters et al. 2011; Spitler et al. 1991; Walton 1983).

Also, more detailed models of exterior [e.g., Cole (1976); Walton (1983)] and interior [e.g., Carroll (1980); Davies (1988); Kamal and Novak (1991); Steinman et al. (1989); Walton (1980)] long-wave radiation transfer have been implemented in detailed building simulation programs.

The weighting-factor method of calculating instantaneous space sensible load is a compromise between simpler methods (e.g., steady-state calculation) that ignore the ability of building mass to store energy, and more complex methods (e.g., complete energy balance calculations). With this method, space heat gains at constant space temperature are determined from a physical description of the building, ambient weather conditions, and internal load profiles. Along with the characteristics and availability of heating and cooling systems for the building, space heat gains are used to calculate air temperatures and heat extraction rates. This discussion is in terms of heat gains, cooling loads, and heat extraction rates. Heat losses, heating loads, and heat addition rates are merely different terms for the same quantities, depending on the direction of the heat flow.

The weighting factors represent Z-transfer functions (Kerrisk et al. 1981; York and Cappiello 1982). The Z-transform is a method for solving differential equations with discrete data. Two groups of weighting factors are used: heat gain and air temperature.

Heat gain weighting factors represent transfer functions that relate space cooling load to instantaneous heat gains. A set of weighting factors is calculated for each group of heat sources that differ significantly in the (1) relative amounts of energy appearing as convection to the air versus radiation, and (2) distribution of radiant energy intensities on different surfaces.

Air temperature weighting factors represent a transfer function that relates room air temperature to the net energy load of the room. Weighting factors for a particular heat source are determined by introducing a unit pulse of energy from that source into the room’s network. The network is a set of equations that represents a heat balance for the room. At each time step, including the initial introduction, the energy flow to the room air represents the amount of the pulse that becomes a cooling load. Thus, a long sequence of cooling loads can be generated, from which weighting factors are calculated. Similarly, a unit pulse change in room air temperature can be used to produce a sequence of cooling loads.

A two-step process is used to determine the air temperature and heat extraction rate of a room or building zone for a given set of conditions. First, the room air temperature is assumed to be fixed at some reference value, usually the average air temperature expected for the room over the simulation period. Instantaneous heat gains are calculated based on this constant air temperature. Various types of heat gains are considered. Some, such as solar energy entering through windows or energy from lighting, people, or equipment, are independent of the reference temperature. Others, such as conduction through walls, depend directly on the reference temperature.

A space sensible cooling load for the room, defined as the rate at which energy must be removed from the room to maintain the reference value of the air temperature, is calculated for each type of instantaneous heat gain. The cooling load generally differs from the instantaneous heat gain because some energy from heat gain is absorbed by walls or furniture and stored for later release to the air. At time θ, the calculation uses present and past values of the instantaneous heat gain (q_θ, q_θ–1), past values of the cooling load (Q_θ–1, Q_θ–2, ...), and the heat gain weighting factors (v₀, v₁, v₂, ..., w₁, w₂, ...) for the type of heat gain under consideration. Thus, for each type of heat gain q_θ, cooling load Q_θ is calculated as

(4)

The heat gain weighting factors are a set of parameters that determine how much of the energy entering a room is stored and how rapidly stored energy is released later. Mathematically, the weighting factors are coefficients in a Z-transfer function relating the heat gain to the cooling load.

These weighting factors differ for different heat gain sources because the relative amounts of convective and radiative energy leaving various sources differ and because the distribution of radiative energy can differ. Heat gain weighting factors also differ for different rooms because room construction affects the amount of incoming energy stored by walls or furniture and the rate at which it is released. Sowell (1988) showed the effects of 14 zone design parameters on zone dynamic response. After the first step, cooling loads from various heat gains are added to give a total cooling load for the room.

In the second step, the total cooling load is used (with information on the room’s HVAC system and a set of air temperature weighting factors) to calculate the actual heat extraction rate and air temperature. The actual heat extraction rate differs from the cooling load (1) because, in practice, air temperature can vary from the reference value used to calculate the cooling load, or (2) because of HVAC system characteristics. Deviation of air temperature t_θ from the reference value at hour θ is calculated as

(5)

where ER_θ is the energy removal rate of the HVAC system at hour θ, and g₀, g₁, g₂, …, P₁, P₂, … are air temperature weighting factors, which incorporate information about the room, particularly thermal coupling between the air and the storage capacity of the building mass.

Example values of weighting factors for typical building rooms are presented in the following table. One of the three groups of weighting factors, for light, medium, and heavy construction rooms, can be used to approximate the behavior of any room. Some automated simulation techniques allow weighting factors to be calculated specifically for the building under consideration. This option improves the accuracy of the calculated results, particularly for a building with an unconventional design. McQuiston and Spitler (1992) provided electronic tables of weighting factors for a large number of parametrically defined zones.

Normalized Coefficients of Space Air Transfer Functions

Two assumptions are made in the weighting-factor method. First, the processes modeled are linear. This assumption is necessary because heat gains from various sources are calculated independently and summed to obtain the overall result (i.e., the superposition principle is used). Therefore, nonlinear processes such as radiation or natural convection must be approximated linearly. This assumption is not a significant limitation because these processes can be linearly approximated with sufficient accuracy for most calculations. The second assumption is that system properties influencing the weighting factors are constant (i.e., they are not functions of time). Often, a single set of weighting factors is used during the entire simulation period; however, multiple sets of weighting factors can be used to represent daily or seasonal variations. This assumption can limit the use of weighting factors in situations where important room properties vary more frequently during the calculation (e.g., the distribution of solar radiation incident on the interior walls of a room, which can vary over the day, and indoor surface heat transfer coefficients).

When the weighting-factor method is used, a combined radiative/convective heat transfer coefficient is used as the indoor surface heat transfer coefficient. This value is assumed constant even though, in a real room, (1) radiant heat transferred from a surface depends on the temperature of other room surfaces (not on room air temperature) and (2) the combined heat transfer coefficient is not constant. Under these circumstances, an average value of the property must be used to determine the weighting factors. Cumali et al. (1979) investigated extensions to the weighting-factor method to eliminate this limitation.

Weighting factors are derived by introducing a unit pulse in a heat balance analysis. These analyses are performed relatively quickly as a preprocess step. Once the weighting factors are established, the simulation time can be significantly reduced, compared with performing a heat balance at every time step. There is an implicit trade-off between the limitations of the weighting-factor method and its improved computation speed compared with that of the heat balance method.

The comprehensive room transfer function method (CRTF) is important as a reasonably accurate (comparable to weighting factor) model that is fast enough to be embedded in optimizations. One important application is model-predictive control. CRTF parameters may be estimated using forward (Seem et al. 1989) or inverse (Armstrong et al. 2006b; Gayeski et al. 2012) modeling. The interior terminating point of each wall is a common star node. Instead of separate surface flux and temperature for each wall, only the net star-node flux and temperature are evaluated. Exogenous radiant fluxes (solar and radiant shares of internal gains) are imposed on the star node as well. Radiant and convective exchange between walls also occurs through the star node. Convective heating and cooling by the system, on the other hand, as well as infiltration, ventilation, and the convective share of internal gains, enter the room model through its air node, which is coupled to the star node by a single, relatively small resistance. The resulting model has the topology of n conduction transfer functions terminating on a massless star node that is connected by a single resistance to an air capacitance node. However the c and d coefficients of all walls are combined, thus reducing the computational effort at each simulation time step. The CRTF thus has (up to the star node) the mathematical form of a multivariate autoregressive-moving-average with exogenous inputs (ARMAX) model. Methods of evaluating the common c and d coefficients and the star and air node resistances are described by Armstrong et al. (1992) and Seem et al. (1989).

Although implementations of the thermal-network method vary, they all have in common the discretization of the building into a network of nodes connected by heat transfer paths. In many respects, thermal-network models may be considered a variant of the heat balance method. Thermal-network models can include an arbitrary number of nodes as required for the problem under consideration. For example, heat balance models generally use simple methods for distributing radiation from lights; thermal-network models may model the lamp, ballast, and luminaire housing separately. Thermal-network models depend on a heat balance at each node to determine node temperature and energy flow between all connected nodes. Energy flows may include conduction, convection, and short- or long-wave radiation. Methods have been developed that reduce the number of node interconnections (e.g., by replacing the general delta radiant exchange network with a star network) (Carroll 1980, 1981).

For any mode of energy flow, a range of finite-difference or finite-volume techniques may be used to model the energy flow between nodes. Taking transient conduction heat transfer as an example, the simplest thermal-network model would be a one- or two-capacitor resistance/capacitance network (Hammarsten 1987; Sonderegger 1977; Sowell 1990). Others have used more refined network discretization (Clarke 2001; Lewis and Alexander 1990; Walton 1993).

Advanced thermal-network models generally use a set of algebraic and differential equations. In most implementations, the solution procedure is separated from the models so that, in theory, different solvers might be used to perform the simulation. In contrast, in most heat balance and weighting factor programs the solution technique takes advantage of a constrained model structure. Various solution techniques have been used in conjunction with thermal-network models. Examples include graph theory combined with Newton-Raphson and predictor/corrector ordinary differential equation integration (Buhl et al. 1990) and the use of Euler explicit integration combined with sparse matrix techniques (Walton 1993).

Of the four sensible-load zone models discussed, thermal-network models are the most flexible and have the greatest potential for high accuracy. However, they also require the most computation time, and, in current implementations, require more user effort to take advantage of the flexibility.

Heat transfer through above-grade, opaque envelope components (e.g., walls, roofs, ceilings) is often approximated using transient one-dimensional (1D) calculations. This approximation is common to most building energy simulation tools. Two- and three-dimensional effects, such as thermal bridging, are often accounted for by adjusting the properties of the materials in a 1D construction. Chapter 25 discusses the calculation of heat transfer through opaque building surfaces, including the overall effective thermal transmittance (or U-factor) of a flat building assembly.

Thermal properties of building materials and other solids can be found in Chapter 26 (Table 1) and Chapter 33 (Table 3), respectively.

Transient 1D heat transfer is calculated for multilayered constructions using one of three methods:

Iu and Fisher (2004) provide an overview and comparison of response factors and CTFs.

Thermal modeling of building foundations (Claridge et al. 1993), including guidelines for placement of insulation, is described in Chapter 27 of this volume and Chapter 44 of the 2019 ASHRAE Handbook—HVAC Applications. Chapter 18 of this volume provides information for calculating transmission heat losses through slab foundations and through basement walls and floors. These calculations are appropriate for design loads but are not intended for estimating annual energy usage. This section provides information about calculation methods suitable for energy estimates over time periods of arbitrary length.

The magnitude of foundation heat transfer relative to other loads in the building depends on several factors, including the insulation design of the foundation and the shape and size of the foundation relative to the overall size of the building. Foundation heat losses (or gains) are more significant in buildings with lower foundation area-to-perimeter ratios (Bahnfleth and Pedersen 1990; Kruis 2015). Low area-to-perimeter ratios often coincide with small buildings (e.g., detached residential construction) and building footprints with narrow cross sections. Where higher ground-coupled floor area-to-perimeter ratios may occur (e.g., in warehouses, shopping malls, other low-rise commercial buildings), thermal mass effects related to the core-area (nonperimeter) portion of the ground-coupled floor may also be important.

Ground-coupled foundation heat transfer involves three-dimensional (3D) thermal conduction, moisture transport, and the long-time-constant heat storage properties of the ground. During the early 1990s, only simplified models could be routinely used for calculating ground heat transfer. These models were based on 1D steady-state conduction or 1D dynamic thermal diffusion modeling. Because of continuing increases in computing power, the state of the art in ground heat transfer modeling has improved. Several calculation methods have been developed and applied to building energy simulation software (Bahnfleth and Pedersen 1990; Beausoleil-Morrison 1996; Beausoleil-Morrison and Mitalas 1997; Clements 2004; ISO 1998). These methods are further described and compared to more detailed 3D numerical methods (Ben-Nakhi 2007; Crowley 2007; Deru 2003; Thornton 2007) as part of the IEA BESTEST ground-coupled heat transfer modeling test cases (Neymark et al. 2008).

Other examples of ground heat transfer modeling techniques applied to floors and basements are described in Andolsun et al. (2010), Krarti (1994a, 1994b), and Krarti and Chuangchid (1999), and a simplification of that method in Chapter 19 of the 2009 ASHRAE Handbook—Fundamentals, and in Krarti et al. (1988a, 1988b) and Winkelmann (2002).

Methods of estimating foundation heat transfer can be highly constrained simplifications to ease calculation requirements, or very detailed and computationally intensive. Although the state of the art in ground heat transfer modeling is improving, many whole-building simulation tools still rely on a loosely defined set of ground temperatures that are applied as an exterior boundary condition for a simplified 1D approximation. These temperatures are often based on lagged, monthly average outdoor dry-bulb temperatures. In reality, the temperature in the ground varies both spatially and temporally, and is heavily influenced by the presence of a conditioned building. Finding a balance between computation speed and numerical accuracy is the focus of the dissertation work performed by Kruis (2015). Kruis found that two-dimensional (2D) calculations capture much of the accuracy of the more detailed 3D methods with computation times reasonable for whole-building energy simulation tools.

Fenestration systems (windows, skylights, and doors) affect building energy use through four basic mechanisms: (1) thermal heat transfer, (2) solar heat gain, (3) air leakage, and (4) daylighting. Details of each of these effects are covered in Chapter 15. Fenestration performance is characterized primarily by its overall coefficient of heat transfer (U-factor), solar heat gain coefficient (SHGC), and visible transmittance (VT). These metrics are typically used to rate fenestration products for a specific set of conditions that are not necessarily representative of the range of conditions encountered in a typical whole-building energy simulation.

The energy impact of fenestration systems depends on the thermal and spectral properties of each component, including

Fenestration system models span a wide range of complexity and input requirements. Some of the simpler models apply a constant U-factor and SHGC to the system, whereas more advanced models may require finer details of the fenestration system that are not often available to energy modelers, such as

Many of these inputs can be found or generated in the suite of fenestration-related software tools developed by Lawrence Berkeley National Lab (LBNL 2016):

In most energy modeling applications, the level of detail required to compose a fenestration system using these software tools is not readily available. Instead, energy modelers are at most provided with the product’s rated U-factor, SHGC, and VT, or specific values defined by compliance rules and incentive programs (e.g., ASHRAE Standard 90.1, ENERGY STAR^®). With this use case in mind, Arasteh et al. (2009) developed an approach in EnergyPlus to represent fenestration systems with angle-dependent optical properties using only the rated U-factor, SHGC, and VT. This approach offers a higher level of accuracy than the assumption of constant values, but does not explicitly simulate the opaque frame and dividers components or the fill gas in the fenestration system.

Infiltration is defined as the flow of outdoor air via (1) leakage through unintentional openings (e.g., cracks, porosities) in the building envelope and (2) natural ventilation through exterior windows, doors, vents, etc. It is typically treated as an additional load for the purposes of whole-building energy modeling. The flow is driven by pressure differences and buoyancy forces, and may be counteracted by pressurizing the building or by sealing openings with air barriers, weatherstripping, and similar measures. Outdoor air that has entered a conditioned space will at some point need to be heated or cooled to maintain the space at the desired temperature. There are a number of procedures available for calculating infiltration and associated loads. Empirical models, based on the results of blower door tests, are most readily available for residential buildings because of more readily available measured data, such as effective leakage area and flow exponent. Models that treat the building as a single zone are most directly applicable to smaller buildings (e.g., houses). Models that treat buildings as a set of zones (not necessarily corresponding to thermal zones) are also available, but these models require parameters that may be difficult to determine without building-specific data. More information on these models and on the general issue can be found in Chapter 16. Discussion of natural ventilation modeling is included in this chapter’s section on Low-Energy System Modeling. In the absence of the information necessary to use a more detailed approach, infiltration rates from appropriate standards or guideline documents (e.g., ASHRAE Standard 90.1; Gowri et al. 2009) can be used to determine the inputs for an energy model. For commercial buildings, which are typically designed to be pressurized, this approach may be more successful than for residential buildings, which may not be pressurized.

Environmental data provide boundary conditions for many energy estimating and modeling methods. Many energy estimating and modeling methods require climate data from specific sources or in specific formats. The most common application is full-year hourly simulation using a typical meteorological year (TMY) to represent typical building operation. Other applications may require historical climate data (for model calibration), real-time climate data (for model predictive controls), or projected climate data (for predicting the impact of climate change on building energy consumption). Climate data are available from a growing number of sources. Chapter 14 discusses the use of climatic design information in greater detail and includes data for more than 8000 locations worldwide. Hensen and Lamberts (2011) list several other sources of climate data and discuss the requirements of climate data for various simulation applications.

Accurate representation of internal heat gains in building simulation is important for several reasons. These heat gains often compose a significant portion of HVAC system loads. In addition, heat sources such as lighting and office equipment may be significant direct contributors to building energy consumption.

Typical sources of internal heat gain represented in a simulation include occupants, lighting, and plug loads. Some buildings also include significant heat from cooking, refrigeration, laboratory, or manufacturing equipment.

The following simulation inputs are generally necessary to characterize a source of internal heat gain:

Appropriate values for internal heat gain inputs may be different between energy calculations and HVAC system peak-load calculations. Although the basic form of input may be identical (e.g., both may require specifying office equipment power density), the objectives of the analyses are different. The typical objective of energy estimation is to determine likely energy consumption rather than worst-case energy consumption. Therefore, the appropriate values for internal heat gain inputs will typically be lower for energy estimation than for load calculations. ASHRAE research project RP-1093 developed different sets of diversity factors appropriate for energy calculations and for load calculations (Abushakra et al. 2000).

In many cases, information about actual internal heat gains is not available to the energy modeler, either because it is a new construction project and detailed equipment specifications are not available, or the project is an existing building and a detailed survey of heat sources is not practical. Therefore, it may be necessary to make assumptions. Chapter 18 provides valuable data on peak heat rates, radiant/convective splits, and fraction of heat to space for a range of equipment types. Other sources of internal gain assumptions include COMNET (2016), Building America simulation protocols (Wilson et al. 2014), and DOE reference buildings (Deru et al. 2011).

Technologies alone do not necessarily guarantee low energy use in buildings. Occupant behavior has a significant effect on building performance. Occupants’ expectations of satisfaction with their indoor environment drive various actions, such as adjusting thermostat settings, opening windows, turning on lights, closing window blinds, consuming domestic hot water, and moving around, to satisfy their physical and nonphysical needs. These actions affect the built environment and energy use. Clearly understanding and accurately modeling occupant behavior in buildings is crucial to reducing the gap between design and actual building energy performance (Gunay et al. 2013; Hoes et al. 2009; Turner and Frankel 2008; Yan et al. 2015), especially for low-energy buildings relying more on passive design features, occupancy-controlled technologies, and occupant engagement.

Occupant behavior is represented in predefined deterministic schedules or fixed settings to describe occupant movement and activities, the use of lighting, plug loads, and HVAC system operation. Assumptions of occupant input can be found in ASHRAE publications (e.g., Handbook, standards, guidelines), DOE reference buildings, simulation guides (COMNET 2016), building design documents, or measured data from existing buildings. Abushakra and Claridge (2008) developed a method for modeling occupancy in commercial buildings based on monitored lighting and receptacles energy use. The method provides typical occupancy load shapes that can be used in both forward building energy simulations and data-driven models. When these static schedules and settings are entered into building energy modeling (BEM) programs, the results are deterministic and homogeneous, ignoring the stochastic nature, dynamics, and diversity of occupant behavior. For example, occupants can open windows for various reasons: (1) feeling hot (thermal comfort driven), (2) feeling stuffy (indoor air quality driven), (3) arriving in a space (event driven), and (4) personal habit.

Alternative approaches used to integrate occupant behavior into models include embedded behavior modules adopted by BEM programs or agent-based modeling approaches that specify how agents (in this case, occupants) interact with one another and with their environment. Integration of occupant behavior with BEM programs ranges from built-in user functions to more flexible cosimulation. The user functions approach allows users, to a certain degree of flexibility, to write custom code to implement new or overwrite existing default building operation and supervisory controls. The cosimulation approach allows separate behavior tools to run simultaneously and exchange occupant information in a collaborative manner with BEM programs (Hong et al. 2016a; Wetter et al. 2011). Agent-based modeling specifies occupant attributes, behavioral rules, memory, resources, decision-making sophistication, and procedures for modifying current behavioral rules. Andrews et al. (2011), Langevin (2014), and Robinson (2011) developed agent-based modeling tools for cosimulation.

Despite advancements, there are significant and fundamental scientific problems remaining to be addressed in modeling occupant behavior, including (1) choosing the best modeling approach based on available data and for a particular application, (2) standardizing the representation of occupant behavior for interoperability (Hong et al. 2015), (3) selecting methods to evaluate occupant behavior models, (4) interpreting the simulation results from the use of stochastic models, and (5) considering social and behavioral influencing factors. IEA EBC (2013-2017) Annex 66 addresses some of these problems and provides a literature database on occupant behavior research.

Configuration of thermal zones for building energy simulation commonly follows the core and perimeter zone method (Hirsch 2003; LBNL 1984.). The core and perimeter approach divides a building into an approximate perimeter of thermal zones within 15 to 20 ft from the exterior walls and then further divides the building into separate perimeter zones for each orientation.

A commonly used thermal zoning guideline is provided in ASHRAE Standard 90.1, Appendix G, which describes the standard’s performance rating method. The standard allows combining two or more actual thermal zones into a single zone if they have the same space usage, their exterior glazed surfaces have the same orientation (within 45°), and they are served by the same type of HVAC system.

In certain cases, spaces that have similar thermal conditions may be aggregated into fewer zones within a simulation model with only a minor difference in simulation outcome, as stated in Lokmanhekim (1971), who proposed a method to define thermal zones by grouping spaces based on comparison of load profile plots.

Several recent studies have considered different approaches for thermal zoning:

Although these efforts provide promising steps toward improved thermal zoning for building energy simulation, a truly automated, one-size-fits-all approach remains to be developed.

Although some secondary components (e.g., heat exchangers, valves) are readily described by fundamental engineering principles, the complex nature of some equipment (e.g., chillers, boilers, fans coupled with air distribution systems) has discouraged the use of first-principles models for energy calculations in favor of regression-based component models based on manufacturers’ empirical performance data.

Energy consumption of complex equipment generally depends on equipment design, load conditions, environmental conditions, and equipment control strategies. For example, chiller performance depends on the basic equipment design features (e.g., heat exchange surfaces, compressor design), temperatures and flow through the condenser and evaporator, and methods for controlling the chiller at different loads and operating conditions (e.g., inlet guide vane control or variable-speed compressor control on centrifugal chillers to maintain leaving chilled-water temperature set point). In general, these variables vary constantly and require calculations on an hourly or subhourly basis.

Energy consumption characteristics of primary equipment have traditionally been modeled using equations developed by regression analysis of manufacturers’ published design data. Historically, published data were available only for full-load design conditions, and additional correction functions were used to correct the full-load data to part-load conditions. However, industry efforts to standardize reporting of performance data over a range of operating conditions (proposed ASHRAE Standard 205P) are making it possible to capture the performance of specific equipment via automated performance maps or custom regression fits. Use of such standard-format data is expected to become the preferred approach for transmitting equipment performance characteristics to energy modeling software.

The functional form of the regression equations and correction functions takes many forms, including exponentials, Fourier series, and, most commonly, second- or third-order polynomials. Selection of an appropriate functional form depends on the behavior of the equipment. In some cases, energy consumption is calculated using direct interpolation from data tables.

A typical approach to modeling primary equipment in energy simulation programs is to assume the following functional form for equipment power consumption:

(6)

(7)

where

The part-load ratio is the ratio of the load to the available equipment capacity at given off-design operating conditions. Like the power, the available, or full-load, capacity is a function of operating conditions.

The particular forms of off-design functions f₁ and f₃ depend on the specific type of equipment. For example, for fossil-fuel boilers, full-load capacity and power (or fuel use) can be affected by thermal losses to ambient temperature. However, these off-design functions are typically considered to be unity in most building simulation programs. For condensing boilers, efficiency is significantly affected by boiler entering water temperature, which is therefore an important input for the f₁ function (see also the section on Boilers). For chillers, both capacity and power are affected by condenser and evaporator temperatures, which are often characterized in terms of their secondary fluids. For direct-expansion air-cooled chillers, operating temperatures are typically the wet-bulb temperature of air entering the evaporator and the dry-bulb temperature of air entering the condenser. For liquid chillers, the temperatures are usually the leaving chilled-water temperature and the entering condenser water temperature (see also the section on Chillers). For fans and pumps, energy consumption is often represented as a polynomial function of airflow or fluid flow, as described in the section on Fans, Pumps, and Distribution Systems.

As an example, consider the cooling performance of a direct-expansion (DX) packaged single-zone rooftop unit. The nominal rated performance of these units is typically given for an outdoor air temperature of 95°F and evaporator entering coil conditions of 80°F db and 67°F wb. However, performance changes as outdoor temperature and entering coil conditions vary. To account for these effects, the DOE-2.1E simulation program expresses the off-design functions f₁ and f₃ with biquadratic functions of the outdoor dry-bulb temperature and the coil entering wet-bulb temperature. Many other programs use similar functions.

(8)

(9)

The constants in Equations (8) and (9) are given in Table 3.

Figure 5. Possible Part-Load Power Curves

The fraction full-load power function f₂ represents the change in equipment efficiency at part-load conditions and depends heavily on the control strategies used to match load and capacity. Figure 5 shows several possible shapes of these functional relationships. Curve 1 represents equipment with constant efficiency, independent of load. Curve 2 represents equipment that is most efficient in the middle of its operating range. Curve 3 represents equipment that is most efficient at full load. Note that these types of curves apply to both boilers and chillers.

Engineering first principles such as thermodynamics as well as heat, mass, and momentum transfer can be used to develop models of primary equipment as well as secondary components. Gordon and Ng (1994, 1995), Gordon et al. (1995), Jiang and Reddy (2003), Lebrun et al. (1999), and others have sought to develop such models in which unknown model parameters are extracted from measured or published manufacturers’ data.

The energy analyst often must choose the appropriate model for the job. For example, a complex boiler model is not appropriate if the boiler operates at virtually constant efficiency. Similarly, a regression-based model might be appropriate when the user has a full dataset of reliable in-situ measurements of the plant. A regression-based model may also be a good choice when manufacturer performance data is provided for a full range of anticipated operating conditions. However, first-principles physical models generally have several advantages over pure regression models:

Physical models of primary HVAC equipment can be found in many HVAC textbooks, but some of the models described here are specifically based on the work of Bourdouxhe et al. (1994a, 1994b, 1994c) in developing the ASHRAE HVAC 1 Toolkit (Lebrun et al. 1999). Each elementary component’s behavior is characterized by a limited number of physical parameters, such as heat exchanger heat transfer area or centrifugal compressor impeller blade angle. Values of these parameters are identified, or tuned, based on regression fits of overall performance compared to measured or published data.

Although physical models are based on physical characteristics, values obtained through a regression analysis of manufacturers’ data are not necessarily representative of the actual measured values. Strictly speaking, the parameter values are regression coefficients with estimated values, identified to minimize the error in overall system performance. In other words, errors in the fundamental models of equipment are offset by over- or underestimation of the parameter values.

Although many terminal units do not directly consume energy, they often have a significant effect on the energy consumed by primary and secondary equipment. For example, the minimum airflow set point for a variable-air-volume (VAV) terminal unit with a reheat coil affects supply fan energy, cooling energy consumption of the chiller or direct-expansion compressor, heating energy consumption of the electric reheat coil or central plant boiler, and pump energy for delivery of chilled and hot water. Therefore, the representation of terminal units in energy-estimating programs deserves special attention.

Several types of terminal units are commonly represented in energy-estimating programs: single-duct VAV boxes with and without reheat coils, single-duct VAV boxes with parallel or series induction fans, dual-duct VAV boxes, chilled-beam induction units, fan-coils, radiant panels, baseboard radiators, and others. Chapters 4, 5, and 20 of the 2020 ASHRAE Handbook—HVAC Systems and Equipment provide a description of many types of terminal units and their operation.

When selecting an energy-estimating program, it is important to note that not all programs explicitly represent some of these types of terminal units or they may not directly represent the desired control scheme.

The following considerations regarding accuracy and energy consumption are important when representing terminal units in energy-estimating programs:

Underfloor air distribution (UFAD) is sometimes considered as an energy-efficient HVAC strategy, and a method to estimate its energy performance is a valuable design tool. UFAD systems offer potential for energy savings in several ways. These systems typically operate at higher cooling supply air temperatures compared to overhead air delivery systems, and offer potential for additional economizer energy savings. Primary cooling equipment efficiency may improve when providing higher-temperature supply air. In addition, if the UFAD system is operated to maintain partially mixed air distribution, as described in Chapter 57 of the 2019 ASHRAE Handbook—HVAC Applications, then the zone air distribution effectiveness may be better than that of an overhead air delivery system and allow a reduction in outdoor air ventilation rate.

A limitation of many energy-estimating programs is the assumption that air within thermal zones is fully mixed, and these programs cannot directly represent the performance of a UFAD system that is operated to create some temperature stratification. Approximate methods have been used by energy modelers, such as raising the thermostat set point in the UFAD zones to represent a higher average air temperature or representing the actual zone with two stacked zones in the model. See Energy Design Resources (2012) for more information.

A method for explicit modeling of UFAD systems has been developed for the EnergyPlus program, with separate models for interior and perimeter zones (Webster et al. 2013).

Thermal displacement ventilation (TDV) is another air delivery strategy that offers energy efficiency benefits. A displacement system is typically designed to achieve fully stratified air distribution, as described in Chapter 57 of the 2019 ASHRAE Handbook—HVAC Applications. The potential efficiency benefits are similar to UFAD systems.

Modelers seeking to estimate the energy performance of a TDV system face the same basic limitations as with UFAD systems, and similar approximations have been used in practice. However, a method for modeling TDV represented by three nodes (floor temperature, occupied-zone temperature, and upper-zone temperature) is implemented in EnergyPlus (U.S. Department of Energy 1996-2016).

Heating and cooling a space using heated and cooled panels is fundamentally different from using conditioned air. In a radiant system, a significant fraction of the sensible heating and cooling occurs by radiant heat transfer between the radiant panel and the surfaces in the space. This process is described in Chapter 6 of the 2020 ASHRAE Handbook—HVAC Systems and Equipment.

As noted in the Heat Balance Method section, energy-estimating programs that use the heat balance method for space load calculations are generally preferred for representing the performance of radiant systems. Use of the heat balance method allows accounting for radiant heat transfer among surfaces in a space, and a heated or cooled panel may be represented as one of those surfaces.

The distribution system of an HVAC system affects energy consumption in two ways. First, fans and pumps consume electrical energy directly, based on the flow and pressures under which a device operates. Ducts and dampers, or pipes and valves, and the system control strategies affect flow and pressure at each fan or pump. Second, thermal energy is often transferred to (or from) the fluid by (1) heat transfer through pipes and ducts and (2) electrical input to fans and pumps. Analysis of system components should, therefore, account for both direct electrical energy consumption and thermal energy transfer.

Fan and pump performance are discussed in Chapters 21 and 44 of the 2020 ASHRAE Handbook—HVAC Systems and Equipment. In addition, Chapter 21 of this volume covers pressure loss calculations for airflow in ducts and duct fittings, and the effects of system air leakage. Chapter 22 presents a similar discussion for fluid flow in pipes. Although these chapters do not specifically focus on energy estimation, energy use is governed by the same performance characteristics and engineering relationships. Strictly speaking, performance calculations for a building’s fan and air distribution system require a detailed pressure balance on the entire network. For example, in an air distribution system, airflow through the fan depends on its physical characteristics, operating speed, and pressure differential across the fan. Pressure drop through the air distribution system depends on duct construction (i.e., friction and fitting losses), coil and filter characteristics, damper positions, air leakage, and airflow through each component. Interaction between the fan and air distribution system results in a set of coupled, nonlinear algebraic equations. Models and subroutines for performing these calculations are available in the ASHRAE HVAC2 Toolkit (Brandemuehl 1993) and CONTAM (NIST 2000-2016). A simplified, physics-based model of the system curve, which represents the system pressure drop as seen by the fan, is described by Sherman and Wray (2010).

Detailed analysis of a distribution system requires flow and pressure balancing among the components, but nearly all commercially available energy analysis methods approximate the effect of the interactions with part-load performance curves [empirical (regression-based) component models]. This approach eliminates the need to calculate pressure drop through the distribution system at off-design conditions. Two simplified methods are described in the following sections.

Polynomial Curve Fit. Part-load curves are often expressed in terms of a power input ratio as a function of the part-load ratio, defined as the ratio of part-load flow to design flow:

(10)

where

The exact shape of the part-load curve depends on the effect of flow control on the pressure and fan efficiency and may be calculated using a detailed analysis or measured field data. Figure 6 shows the relationship for three typical fan control strategies, as represented in a simulation program (York and Cappiello 1982). In the simulation program, the curves are represented by polynomial regression equations. Models and subroutines for performing these calculations are also available in the ASHRAE HVAC2 Toolkit (Brandemuehl 1993).

Figure 7 shows an example of a similar curve for the part-load operation of a fan system in a monitored building (Brandemuehl and Bradford 1999). In this case, the fan system represents 10 separate air handlers, each with supply and return fans, operating with variable-speed fan control to maintain a set duct static pressure.

Figure 6. Part-Load Curves for Typical Fan Operating Strategies (York and Cappiello 1982)

Figure 7. Fan Part-Load Curve Obtained from Measured Field Data under ASHRAE RP-823 (Brandemuehl and Bradford 1999)

Component Model. A second method can be used for fan system energy estimation in variable-airflow systems. It accounts separately for the substantial part-load performance variations that can occur for the fans, fan drive belts, fan motor, and variable-frequency drives (VFDs). For fans, dimensionless models are used to represent fan efficiency and speed variations with airflow. This method also uses a physics-based model (often called the “system curve”) for determining flow-dependent fan pressure rise (Sherman and Wray 2010). Note that system curves with duct static pressure controls or significant pressure drops through coils and filters do not follow the commonly assumed quadratic system curve. Fan power is calculated using airflow, pressure rise, and fan efficiency at each time step. Calculating fan power and speed allows determining fan shaft torque and the resulting loads on drive components (belts, motors, and VFDs). Note that fan laws do not apply to drive components. In the absence of physics-based models, regression models based on available data can be used to represent belt, motor, and VFD part-load efficiency variations. An implementation of this approach, developed by Wray in 2010, is included in EnergyPlus, and details are provided in the EnergyPlus Engineering Reference (U.S. Department of Energy 1996-2016).

A benefit of this approach is its ability to directly represent the impacts of low-pressure-drop distribution systems, air leakage reduction, and duct static-pressure reset control, which are important efficiency strategies for many systems. This approach can also represent the effects of fan system component oversizing and changes for components such as fans, belts, motors, and variable-frequency drives.

A disadvantage to the component model is that the additional capabilities require additional inputs. However, generic values for model parameters are provided in the EnergyPlus documentation.

Fan and Pump Heat. Heat dissipated to the airstream by fan operation increases airstream temperature. Fan shaft power is usually assumed to be dissipated fully in the airstream. Motor losses also contribute if the motor is mounted in the airstream. For pumps, these contributions are typically assumed to be zero because the motor is not mounted in the water stream, and the ratio of transport power to thermal capacitance rate is usually much less for water than for air.

The following equation provides a convenient and general model to calculate the heat transferred to the fluid by a motor:

(11)

where

The heat rejected by VFDs and the fraction of motor heat not transferred to the airstream should be accounted for as building loads.

Secondary HVAC systems comprise heat and mass transfer components (e.g., steam or hot-water air-heating coils, chilled-water cooling and dehumidifying coils, shell-and-tube liquid heat exchangers, air-to-air heat exchangers, evaporative coolers, steam injectors). Although these components do not consume energy directly, their thermal performance dictates interactions between building loads and energy-consuming primary components (e.g., chillers, boilers). In particular, secondary component performance determines the entering fluid conditions for primary components, which in turn determine energy efficiencies of primary equipment. In addition, heat and mass transfer components may affect air and fluid flow, which affect performance of energy-consuming secondary components such as fans and pumps. Accurate energy calculations cannot be performed without appropriate models of the system heat and mass transfer components.

For example, load on a chiller is typically described as the sum of zone sensible and latent loads, plus any heat gain from ducts, plenums, fans, pumps, and piping. However, the chiller’s energy consumption is determined not only by the load but also by the return chilled-water temperature and flow rate. The return water condition is determined by cooling coil performance and part-load operating strategy of the air and water distribution system. The cooling coil might typically be controlled to maintain a constant leaving air temperature by modulating water flow through the coil. In such a scenario, the cooling coil model must be able to calculate the leaving air humidity, water temperature, and water flow rate given the cooling coil design characteristics and entering air temperature and humidity, airflow, and water temperature.

Virtually all building energy simulation programs include, and require, models of heat and mass transfer components. These models are generally relatively simple. Whereas a coil designer might use a detailed tube-by-tube analysis of conduction and convection heat transfer and condensation on fin surfaces to develop an optimal combination of fin and tube geometry, an energy analyst is more interested in determining changes in leaving fluid states as operating conditions vary during the year. In addition, the energy analyst is likely to have limited design data on the equipment and, therefore, requires a model with very few parameters that depend on equipment geometry and detailed design characteristics.

A typical approach to modeling heat and mass transfer components for energy calculations is based on an effectiveness-NTU heat exchanger model (Kays and London 1984). The effectiveness-NTU (number of transfer units) model is described in most heat transfer textbooks and briefly discussed in Chapter 4. It is particularly appropriate for describing leaving fluid conditions when entering fluid conditions and equipment design characteristics are known. Also, this model requires only a single parameter to describe the characteristics of the exchanger: the overall transfer coefficient UA, which can be determined from limited design performance data.

Effectiveness methods are used to perform energy calculations for a variety of sensible heat exchangers in HVAC systems. For typical finned-tube air-heating coils, the cross-flow configuration with both fluid streams unmixed is most appropriate. Air-to-air heat exchangers may be cross- or counterflow. For liquid-to-liquid exchangers, tube-in-tube equipment can be modeled as parallel or counterflow, depending on flow directions; correlations for shell-and-tube effectiveness and NTU, which depend on the extent of baffling and the number of tube passes, are given in heat transfer texts (Mills 1999).

The energy analyst must determine the UA to describe the operations of a specific heat exchanger. There are three ways to determine this important parameter: direct calculation, measurement, and manufacturers’ data. Given detailed information about the materials, geometry, and construction of the heat exchanger, fundamental heat transfer principles can be applied to calculate the overall heat transfer coefficient. However, the method most appropriate for energy estimation is use of manufacturers’ performance data or measurement of installed performance. In reporting the design performance of a heat exchanger, a manufacturer typically gives the heat transfer rate under various operating conditions, with operating conditions described in terms of entering fluid flow rates and temperatures. The effectiveness and UA can be calculated from the given heat transfer rate and entering fluid conditions.

Analysis of air-cooling and dehumidifying coils requires coupled, nonlinear heat and mass transfer relationships. These relationships form the basis for all HVAC components with moisture transfer, including cooling coils, cooling towers, air washers, and evaporative coolers. Although the complex heat and mass transfer theory presented in many textbooks is often required for cooling coil design, simpler models based on effectiveness concepts are usually more appropriate for energy estimation. For example, the bypass factor is a form of effectiveness in the approach of the leaving air temperature to the apparatus dew-point, or coil surface, temperature.

The effectiveness-NTU method is typically developed and applied in analysis of sensible heat exchangers, but it can also be used to analyze other types of exchangers, such as cooling and dehumidifying coils, that couple heat and mass transfer. By redefining the state variables, capacity rates, and overall exchange coefficient of these enthalpy exchangers, the effectiveness concept may be used to calculate heat transfer rates and leaving fluid states. For sensible heat exchangers, the state variable is temperature, the capacity is the product of mass flow and fluid specific heat, and the overall transfer coefficient is the conventional overall heat transfer coefficient. For cooling and dehumidifying coils, the state variable becomes moist air enthalpy, the capacity has units of mass flow, and the overall heat transfer coefficient is modified to reflect enthalpy exchange. This approach is the basis for models by Brandemuehl (1993), Braun et al. (1989), and Threlkeld (1970). The same principles also underlie the coil model described in Chapter 23 of the 2012 ASHRAE Handbook—HVAC Systems and Equipment.

Figure 8. Psychrometric Schematic of Cooling Coil Processes

The effectiveness model is based on the observation that, for a given set of entering air and liquid conditions, the heat and mass transfer are bounded by thermodynamic maximum values. Figure 8 shows the limits for leaving air states on a psychrometric chart. Specifically, the leaving chilled-water temperature cannot be warmer than the entering air temperature, and the leaving air temperature and humidity cannot be lower than the conditions of saturated moist air at the temperature of the entering chilled water.

Figure 8 also shows that performance of a cooling coil requires evaluating two different effectivenesses to identify the leaving air temperature and humidity. An overall effectiveness can be used to describe the approach of the leaving air enthalpy to the minimum possible value. An air-side effectiveness, related to the coil bypass factor, describes the approach of the leaving air temperature to the effective wet-coil surface temperature.

Effectiveness analysis is accomplished for wet coils by establishing a common state variable for both the moist air and liquid streams. As implied by the lower limit of the entering chilled-water temperature, this common state variable is the moist air enthalpy. In other words, all liquid and coil temperatures are transformed to the enthalpy of saturated moist air at the liquid or coil temperature. Changes in liquid temperature can similarly be expressed in terms of changes in saturated moist air enthalpy through a saturation specific heat c_p,sat defined by the following:

(12)

Using the definition of Equation (12), the basic effectiveness relationships discussed in Chapter 4 can be written as

(13)

(14)

(15)

(16)

(17)

where

The cooling coil effectiveness of Equation (14) is defined, then, as the ratio of moist air enthalpies in Figure 8. As with sensible heat exchangers, effectiveness is also a function of the physical coil characteristics and can be obtained by modeling the coil as a counterflow heat exchanger. However, because heat transfer calculations are performed based on enthalpies, the overall transfer coefficient must be based on enthalpy potential rather than temperature potential. The enthalpy-based heat transfer coefficient UA_h is related to the conventional temperature-based coefficient by the specific heat:

(18)

A similar analysis can be performed to evaluate the air-side effectiveness, which identifies the leaving air temperature. Whereas the overall enthalpy-based effectiveness is based on an overall heat transfer coefficient between the chilled water and air, air-side effectiveness is based on a heat transfer coefficient between the coil surface and air.

As with sensible heat exchangers, the overall heat transfer coefficients UA can be determined either from direct calculation from coil properties or from manufacturers’ performance data. A sensible heat exchanger is modeled with a single effectiveness and can be described by a single parameter UA, but a wet cooling and dehumidifying coil requires two parameters to describe the two effectivenesses shown in Figure 8. These parameters are the internal and external UAs: one describes heat transfer between the chilled water and the air-side surface through the pipe wall, and the other between the surface and the moist air. UA values can be determined from the sensible and latent capacity of a cooling coil at a single rating condition. A significant advantage of the effectiveness-NTU method is that the component can be described with as little as one measured data point or one manufacturer’s design calculation.

The literature on boiler models is extensive, ranging from steady-state performance models (DeCicco 1990; Lebrun 1993) to detailed dynamic simulation models (Bonne and Jansen 1977; Lebrun et al. 1985), to a combination of these two schemes (Laret 1991; Malmström et al. 1985). However, the input information required for these models is not often readily available, and these models should be considered only in more complex situations (e.g., large boilers in large buildings, district heating systems, cogeneration systems), where a complete, detailed representation of heat distribution, emission, and operation and control under varying external conditions is warranted.

In most current energy-estimating programs, the performance of boilers is represented by regression equations and a full-load efficiency value as described in the section on Empirical (Regression-Based) Models. Those equations typically represent the boiler efficiency under varying load conditions and, in some cases, the equations also account for water temperature conditions. When using these regression-based models, two important issues to understand are efficiency rating conditions and the off-design efficiency calculation method used by the energy-estimating program.

Rating Conditions. The primary rating condition relevant for fuel-fired boiler efficiency is the entering water temperature, which is the return water temperature in most building heating applications. Efficiency improves as the return water temperature drops, because more heat can be extracted from exhaust gases. A condensing boiler can reach efficiencies up to 98% with entering water at 80°F. The minimum return water temperature for noncondensing boilers is typically about 140°F, because lower temperatures lead to condensation and damaging corrosion, and efficiency under that condition is usually in the range of 80% to 85%. The important point to consider when entering a value for fuel-fired boiler efficiency in an energy-estimating program is the assumption made by the program about rating conditions. For example, the DOE2.2 program expects noncondensing boiler efficiency values to correspond to 160°F entering water temperature and condensing water boiler efficiency values to correspond to 80°F entering water temperature. More information on boiler design and rating conditions is provided in Chapter 32 of the 2020 ASHRAE Handbook—HVAC Systems and Equipment.

Off-Design Conditions. Another important consideration in the use of energy-estimating programs is the type of model used to represent off-design conditions (i.e., how the efficiency is assumed to change at partial load or as the water temperature conditions change). Some models account only for changes in efficiency with changes in load. Other models consider both load and entering water temperature, and still others consider load and leaving water temperature. Some programs offer a choice of models. For best accuracy, models that account for load and entering water temperature are preferred.

Figures 9 and 10 show examples of how boiler efficiency is represented by different types of models. Figure 9 shows a model in which boiler efficiency is a function of part-load ratio alone. Different regression equation coefficients are used for atmospheric-draft and forced-draft boilers, and the forced-draft units are represented with better part-load efficiency. An implicit assumption of this model is that entering water temperature is in a range where it does not have a significant impact on efficiency, which may be reasonable for noncondensing boilers. Figure 10 shows a model for a condensing boiler in which efficiency is a function of both part-load ratio and entering water temperature. This model shows that efficiency increases as entering water temperature decreases and as load decreases.

Models representing energy consumption of chillers include both first-principles models and empirical models. Examples of first-principles models include detailed modeling techniques described in the ASHRAE HVAC 1 Toolkit (Lebrun et al. 1999) and the Gordon-Ng model (Gordon 2000). A discussion of the Gordon-Ng model is provided in the Data-Driven Modeling section, presented as an example of a physical model. From a practical point of view, the data required as inputs to these first-principles models will not always be available. Most current energy-estimating programs use empirical models to represent chiller performance. Hydeman et al. (2002) compares the prediction capabilities of several types of models.

Empirical chiller models typically comprise a set of regression equations that represent variations in cooling capacity and efficiency in operation. A common implementation consists of three polynomial equations. One represents variation in cooling capacity as a function of temperature conditions. A second represents the change in efficiency with changing temperature conditions. The third equation represents efficiency as a function of part-load ratio. The following are common inputs to empirical chiller models:

Figure 9. Example Boiler Model: Efficiency as Function of Part-Load Ratio

Figure 10. Example Boiler Model: Efficiency as Function of Part-Load Ratio and Entering Water Temperature

Different types of chillers have different performance characteristics, and their performance is typically represented by different sets of regression equation coefficients. Important chiller design differences include type of condenser (air- or water-cooled), type of compressor (reciprocating, screw, or centrifugal), and unloading mechanism (e.g., inlet vanes or variable-speed compressor control). Chapter 43 of the 2020 ASHRAE Handbook—Systems and Equipment provides descriptions of different types of chillers.

Coefficients for chiller models are often supplied as defaults by energy-estimating programs. However, when the intent of analysis is to compare the energy performance of specific chiller alternatives, the recommended practice is to obtain performance data for the specific chillers to create calibrated sets of coefficients for each chiller. Manufacturers can typically provide such data, and it is very important to obtain a set of data representing the full range of potential operating conditions. The process of calculating coefficients for a chiller model is described in Hydeman and Gillespie (2002). For another discussion of empirical chiller models and the process of determining model coefficients, see the HVAC Simulation Guidebook, vol. 1 (Energy Design Resources 2012).

A cooling tower is used in primary systems to reject heat from the chiller condenser. Controls typically manage tower fans and pumps to maintain a desired water temperature entering the condenser. Like cooling and dehumidifying coils in secondary systems, cooling tower performance has a strong influence on the chiller’s energy consumption. In addition, tower fans consume electrical energy directly.

Fundamentally, a cooling tower is a direct contact heat and mass exchanger. Equations describing the basic processes are given in Chapter 6 and in many HVAC textbooks. Chapter 40 of the 2020 ASHRAE Handbook—HVAC Systems and Equipment describes the specific performance of cooling towers. Performance subroutines are also available in Lebrun et al. (1999) and TRNSYS (2012).

For energy calculations, cooling tower performance is typically described in terms of the outdoor wet-bulb temperature, temperature drop of water flowing through the tower (range), and difference between leaving water and air wet-bulb temperatures (approach). Simple models assume constant range and approach, but more sophisticated models use rating performance data to relate leaving water temperature to the outdoor wet-bulb temperature, water flow, and airflow. Simple cooling tower models, such as those based on a single overall transfer coefficient that can be directly inferred from a single tower rating point, are often appropriate for energy calculations.

Although packaged equipment is most often modeled by empirical functions fit to manufacturers’ data, methods are also needed to model and evaluate advanced equipment such as variable-speed heat pumps. Zakula et al. (2011) detail a component-based heat pump model, including variable-speed fan and pump models; variable refrigerant-, air-, and water-side heat transfer coefficients; variable pressure drops; and staged compressors, one or more of which may operate over a wide speed range. Such models can produce performance maps for new equipment designs, including designs in which refrigerant and transport fluid flow rates and condenser subcooling are coordinated as optimized functions of conditions and imposed load (Zakula et al. 2012). A tricubic fit to such performance maps has been shown to represent performance accurately over a wide range of conditions and load.

As a special heat pump model, variable-refrigerant-flow (VRF) systems vary the refrigerant flow to meet the dynamic zone thermal loads. Hong et al. (2016) developed a new VRF heat pump model, which has physics-based component models. This VRF model, implemented in EnergyPlus, (1) enables advanced controls, including variable evaporating and condensing temperatures in the indoor and outdoor units, and variable fan speeds based on the temperature and zone load in the indoor units; (2) adds a detailed refrigerant pipe heat loss calculation using refrigerant flow rate, operational conditions, pipe length, and pipe insulation materials; (3) increases accuracy of simulation especially in partial load conditions; and (4) improves the usability of the model by significantly reducing the number of user input performance curves. A new heat recovery VRF model was also developed and is implemented in EnergyPlus, which enables more accurate modeling of heat recovery between zones with simultaneous cooling and heating loads. The new VRF models in EnergyPlus were validated with measured performance data from real buildings or laboratory experiments.

Ground-coupled systems use the ground as the source/sink for heating or cooling a building. These types of systems are described in detail in Chapter 34 of the 2019 ASHRAE Handbook—HVAC Applications. For the most part, the HVAC system components used in these systems are similar to those used in other HVAC systems and do not require any special techniques to estimate their performance. The exception to this is simulating energy exchange with the ground.

Energy is exchanged with the ground through direct use of water from the ground or through ground heat exchangers. With a direct-use system, the temperature entering the HVAC system from the ground is often considered to be constant or to vary only seasonally. This assumption can be valid when the extraction and injection wells are far enough apart to prevent mixing of the aquifer. A number of aquifer storage performance models have been developed for use with HVAC systems, but these have not yet reached mainstream simulation programs.

For simulating performance of ground heat exchangers, two different techniques have been developed: the duct ground storage (DST) model introduced by Hellstrom (1989) that calculates the transient thermal process of borehole fields, and a thermal response factor (g-function) approach based on work of Eskilson (1987). Both techniques give a relation between the heat extracted (or rejected) from the ground per unit borehole length and the borehole wall temperature. Both of these methods have been integrated into whole-building simulation programs.

Tools for modeling natural ventilation systems fall into three main categories, discussed in the following sections. Although these modeling tools can be applied individually, simulation and design are better served by using the capabilities of multiple types of tools to provide a more comprehensive analysis.

Simplified Models. Simplified models can be used at the initial design stages for solving algebraic (nondifferential) equations to determine climate suitability, ventilation flow rates, and zone-average temperatures. This can be valuable information, especially at the concept design stage, when architects require approximate information about the feasibility of implementing natural ventilation and determining size and position of ventilation openings. The equations typically used in these models can be found in Chapter 16 and in CIBSE (2005). However, there are many other examples of simplified techniques used to predict the behavior of specific aspects of natural ventilation. For example, Linden et al. (1990) developed analytical models for predicting interface heights in stratified regimes, and Chenvidyakarn and Woods (2005) used analytical methods to predict solution multiplicity in large-volume spaces.

Climate-suitability analysis methods and tools are available for the predesign phase. These relatively simple models provide a first-order estimate of the potential effectiveness of natural ventilation to provide direct ventilation cooling, nighttime cooling, and thermal comfort, based on climate data representative of the building location. Axley and Emmerich (2002) present a climate suitability design method based on a single-zone, steady-state energy balance, and thermal comfort criteria. Lomas et al. (2007) present an analysis of ambient temperature and moisture content with respect to thermal comfort criteria plotted on a psychrometric chart.

One common form of analytical model involves first solving equations to determine the driving pressures caused by wind and buoyancy forces. These driving pressures are calculated with reference to the neutral pressure level (CIBSE 2005) and are used at proposed opening locations as input to mathematical representations of the airflow through the openings (e.g., the orifice flow equation). Given the design airflow rate, these equations can then be used to calculate effective opening areas. Axley (2001) presents an opening sizing method that was subsequently formulated as a software tool (Emmerich et al. 2011; Dols et al. 2012). These methods and tools can also account for multiple openings and user-specified airflow patterns through a building (e.g., into the building via a trickle ventilator, through an air transfer grill, then out through a stack). As outlined in CIBSE (2005), driving pressures should be calculated for representative design scenarios when wind speeds and internal/external temperature differences are relatively low, because these will determine the maximum opening sizes required. Illustrations of how some of these simplified models can be used are given in Axley (2001), CIBSE (2005), and Emmerich et al. (2012).

Network Airflow Models. Network airflow models, also referred to as multizone models, are discussed in Chapter 13. They are more complex than the simplified models, because they solve the simultaneous (interdependent) airflows and pressure differences between multiple, interconnected building zones and the outdoors and are typically applied to the building in its entirety. In particular, using airflow network characteristics (e.g., airflow resistances) and climate data (along with assumed temperatures within the building), network airflow models calculate combined buoyancy- and wind-driven pressures and the rate and direction of airflows for all openings in the model. Some models can also be used to perform interzone contaminant mass transport calculations that can be useful in analyzing IAQ (Axley 2007) and simulating contaminant-based ventilation rate control schemes (e.g., CO₂-based demand controlled ventilation) (Dols et al. 2016a). Network airflow models can be coupled with dynamic thermal simulation programs to calculate the internal zone temperatures used in the network airflow model (Clarke 2001; Dols et al. 2016a, 2016b; U.S. Department of Energy 1996-2016).

A distinct challenge in the use of network airflow models is selecting mathematical representations and parameters that best capture the airflow characteristics of the ventilation components. For example, airflow resistances can be described using a discharge coefficient (or pressure loss factor) and the effective opening area. However, specifying the component characteristics requires a clear understanding of how the model interprets these values. Details of the terminology used can be found in Jones et al. (2016).

Computational Fluid Dynamics. Computational fluid dynamics (CFD) is also discussed in Chapter 13. CFD models represent the most complex class of airflow models and can be valuable for simulating natural ventilation, including the effects of wind on the building envelope pressures and internal airflow patterns that develop as a result of various boundary conditions and representative portions of the building interior. CFD is not typically applicable to simulating entire buildings over long-term, transient time frames. However, there are many strategies available for using CFD in the design and analysis of natural ventilation systems; for example, Malkawi et al. (2016) describes using both CFD and a combined network airflow and energy calculation method.

The main feature that sets CFD models apart from numerical and network airflow models is the use of a fine mesh comprising many thousands of cells throughout the computational domain. For each cell, nonlinear partial differential equations are solved to predict the air speed, air temperature, turbulence, and pressure in each cell. This provides a very detailed representation of the spatial distribution of the airflow and has the advantage of highlighting drafts, warm zones, and thermal stratification. Boundary conditions are required in CFD models to drive the flow. They include temperature or heat flux values at solid surfaces and conditions of air speed and turbulence at openings. Surface temperatures are often obtained from the results of a dynamic thermal network simulation model. This approach has the advantage of including the effects of radiation without solving a separate radiation model in the CFD simulation. Boundary conditions at natural ventilation openings must be specified with care, because neither the flow rate nor the internal temperature are known a priori. This can be overcome by using the orifice flow equation, which simply provides a relationship between the flow rate and pressure difference across openings [e.g., Durrani et al. (2015)]. Another important challenge to be aware of when using CFD is the need for an iterative approach to solve the governing, nonlinear equations. It is therefore important to appreciate that simulation run times can be long (i.e., several hours), and that techniques for controlling convergence are often needed, especially in buoyancy-driven natural ventilation flows where driving pressures are small. Often, this control can be achieved by using a transient simulation or false time-stepping [e.g., Kaye et al. (2009)].

When design tools have predicted that natural ventilation is unlikely to be sufficient for providing adequate IAQ and thermal comfort and a hybrid strategy has been proposed, it is useful to carry out some modeling of the system to provide confidence and identify likely energy implications. This can be done by enhancing the techniques described previously for modeling natural ventilation with network airflow and CFD models.

Ventilation and energy performance of buildings with natural or mixed-mode ventilation systems can be simulated using whole-building network airflow models, including those that may be combined with or directly incorporate a whole-building thermal analysis model (e.g., EnergyPlus) (Dols et al. 2015, 2016b). The energy module sends information about the building, ambient weather conditions, and zone temperatures to the network airflow module. This information is used to calculate infiltration, ventilation, and interzone airflow by the airflow network module; the results will be sent back to the energy module and used in the next time step’s heat balance. In addition to the airflow characteristics through openings such as windows and transfers grilles connecting spaces, models of hybrid systems must also include fan airflow models (e.g., design flow rates and fan performance curves).

One of the primary challenges in modeling hybrid ventilation systems is simulating the controls systems that are inherently associated with these systems. Hybrid ventilation schemes may require multiple modes of operation depending on variations in climate, including seasonal, daily, and short-term local variations. These modes of operation can vary in complexity because of the number and type of system components involved. In buildings with automatically controlled ventilation openings or fans, the control system may require parameters such as wind speed and direction, among others, to open, close, or adjust openings; turn fans on/off; or adjust flow rates. This range of flexibility varies among available models. However, it is likely that users will need to input sensors, actuators, and control logic per the input methods of the tool being used, and the model should predict system performance accordingly (e.g., variations in fan airflow, ventilation opening status). Some modeling tools or combinations of tools enable transient simulations to be performed with user-defined control algorithms (e.g., DOE 2015; Dols et al. 2016a; Wetter 2011), and these may be used for performing annual simulations or at least over periods of time that address the various modes of operation.

With an empirical approach, a simple or multivariate regression model is identified between measured energy use and the various influential parameters (e.g., climatic variables, building occupancy). The form of the regression models can be either purely statistical or loosely based on some basic engineering formulation of energy use in the building. In any case, the identified model coefficients are such that no (or very little) physical meaning can be assigned to them. This approach can be used with any time scale (monthly, daily, hourly or subhourly) if appropriate data are available. Single-variate, multivariate, change point, Fourier series, and artificial neural network (ANN) models fall under this category. Least-squares regression is the most common approach to model identification, although more sophisticated regression techniques such as maximum likelihood and two-stage regression schemes can also be used.

Statistical regression models are usually adequate for estimating savings in demand-side management (DSM) programs for conventional energy conservation measures (lighting retrofits, air handler retrofits such as CV to VAV retrofits) and for baseline model development in energy conservation measurement and verification (M&V) projects (Claridge 1998; Dhar 1995; Dhar et al. 1998, 1999a, 1999b; Fels 1986; Haberl and Culp 2012; Haberl et al. 1998; Katipamula et al. 1998; Kissock et al. 1998; Krarti et al. 1998; Kreider and Wang 1991; MacDonald and Wasserman 1989; Miller and Seem 1991; Reddy et al. 1997; Ruch and Claridge 1991).

Statistical models may also be appropriate for modeling equipment such as pumps and fans, and even more elaborate equipment such as chillers and boilers, if the necessary performance data are available (Braun 1992; Chen et al. 2005; Englander and Norford 1992; Lorenzetti and Norford 1993; Phelan et al. 1996). Although this approach allows detection or flagging of equipment or system faults, it is usually of limited value for diagnosis and online control.

The gray-box approach first formulates a physical model to represent the structure or physical configuration of the building or HVAC&R equipment or system, and then identifies important parameters representative of certain key and aggregated physical parameters and characteristics by statistical analysis (Rabl and Riahle 1992). This approach requires a high level of user expertise both in setting up the appropriate modeling equations and in estimating these parameters. Often an intrusive experimental protocol is necessary for proper parameter estimation. This approach has great potential, especially for fault detection and diagnosis (FDD) and online control, but its applicability to whole-building energy use is limited. Examples of parameter estimation studies applied to building energy use are Andersen and Brandemuehl (1992), Braun (1990), Gordon and Ng (1995), Guyon and Palomo (1999a), Hammarsten (1984), Rabl (1988), Reddy (1989), Reddy et al. (1999), Sonderegger (1977), and Subbarao (1988).

Several types of steady-state models are used for both building and equipment energy use: single-variate, multivariate, polynomial, and physical.

Single-Variate Models. Single-variate models (i.e., models with one regressor variable only) are perhaps the most widely used. They formulate energy use in a building as a function of one driving force that affects building energy use. An important aspect in identifying statistical models of baseline energy use is the choice of the functional form and the independent (or regressor) variables. Extensive studies (Fels 1986; Katipamula et al. 1994; Kissock et al. 1993; Reddy et al. 1997) have indicated that the outdoor dry-bulb temperature is the most important regressor variable for typical buildings, especially at monthly time scales but also at daily time scales.

The simplest steady-state data-driven model is one developed by regressing monthly utility consumption data against average billing-period temperatures. The model must identify the balance-point temperatures (or change points) at which energy use switches from weather-dependent to weather-independent behavior. In its simplest form, the 65°F degree-day model is a change-point model that has a fixed change point at 65°F. Other examples include three- and five-parameter Princeton scorekeeping methods (PRISM) based on the variable-base degree-day concept (Fels 1986). An allied modeling approach for commercial buildings is the four-parameter (4-P) model developed by Ruch and Claridge (1991), which is based on the monthly mean temperature (and not degree-days). Table 4 shows some commonly used model functional forms. The three parameters are a weather-independent base-level use, a change point, and a temperature-dependent energy use, characterized as a slope of a line that is determined by regression. The four parameters include a change point, a slope above the change point, a slope below the change point, and the energy use associated with the change point. A data-driven bin method has also been proposed to handle more than four change points (Thamilseran and Haberl 1995).

Figure 16 shows several types of steady-state, single-variate data-driven models. Figure 16A shows a simple one-parameter, or constant, model, and Table 4 gives the equivalent notation for calculating the constant energy use using this model. Figure 16B shows a steady-state two-parameter (2-P) model where b₀ is the y-axis intercept and b₁ is the slope of the regression line for positive values of x, where x represents the ambient air temperature. The 2-P model represents cases when either heating or cooling is always required.

Figure 16C shows a three-parameter change-point model, typical of natural gas energy use in a single-family residence that uses gas for space heating and domestic water heating. In the notation of Table 4 for the three-parameter model, b₀ represents the baseline energy use and b₁ is the slope of the regression line for values of ambient temperature less than the change point b₂. In this type of notation, the superscripted plus sign indicates that only positive values of the parenthetical expression are considered. Figure 16D shows a three-parameter model for cooling energy use, and Table 4 provides the appropriate analytic expression.

Figure 16. Steady-State, Single-Variate Models for Modeling Energy Use in Residential and Commercial Buildings

Figures 16E and 16F show four-parameter models for heating and cooling, respectively. The appropriate expressions for calculating the heating and cooling energy consumption are found in Table 4: b₀ represents the baseline energy exactly at the change point b₃, and b₁ and b₂ are the lower and upper region regression slopes for ambient air temperature below and above the change point b₃. Figure 16G shows a 5-P model (Fels 1986), which is useful for modeling buildings that are electrically heated and cooled. The 5-P model has two change points and a base level consumption value.

The advantage of these steady-state data-driven models is that their use can be easily automated and applied to large numbers of buildings where monthly utility billing data and average daily temperatures for the billing period are available. Steady-state single-variate data-driven models have also been applied with success to daily data (Kissock et al. 1998). In such a case, the variable-base degree-day method and monthly mean temperature models described previously for utility billing data analysis become identical in their functional form. Single-variate models can also be applied to daily data to compensate for differences such as weekday and weekend use by separating the data accordingly and identifying models for each period separately.

Disadvantages of steady-state single-variate data-driven models include insensitivity to dynamic effects (e.g., thermal mass) and to variables other than temperature (e.g., humidity and solar gain), and inappropriateness for some buildings (e.g., buildings with strong on/off schedule-dependent loads or buildings with multiple change points). Moreover, a single-variable, 3-P model such as the PRISM model (Fels 1986) has a physical basis only when energy use above a base level is linearly proportional to degree-days. This is a good approximation in the case of heating energy use in residential buildings where heating load never exceeds the heating system’s capacity. However, commercial buildings generally have higher internal heat generation with simultaneous heating and cooling energy use and are strongly influenced by HVAC system type and control strategy. This makes energy use in commercial buildings less strongly influenced by outdoor air temperature alone. Therefore, it is not surprising that blind use of single-variate models has had mixed success at modeling energy use in commercial buildings (MacDonald and Wasserman 1989).

Change-point regression models work best with heating data from buildings with systems that have few or no part-load nonlinearities (i.e., systems that become less efficient as they begin to cycle on/off with part loads). In general, single-variate change-point regression models do not predict cooling loads as well because outdoor humidity has a large influence on latent loads on the cooling coil. Other factors that decrease the accuracy of change-point models include solar effects, thermal lags, and on/off HVAC schedules. A change point model based on a linear combination of temperature and specific humidity, and in which direct and diffuse irradiance were included as separate explanatory variables, was found to work well for aggregates of buildings (Ali et al. 2011).

Four-parameter models are a better statistical fit than three-parameter models in buildings with continuous, year-round cooling or heating (e.g., grocery stores and office buildings with high internal loads). However, every model should be checked to ensure that the regression does not falsely indicate an unreasonable relationship. Paulus et al. (2015) provided an algorithm aiding in the selection of change-point linear regression models.

A major advantage of using a steady-state data-driven model to evaluate the effectiveness of energy conservation retrofits is its ability to factor out year-to-year weather variations by using a normalized annual consumption (NAC) (Fels 1986). Basically, annual energy conservation savings can be calculated by comparing the difference obtained by multiplying the pre- and post-retrofit parameters by the weather conditions for the average year. Typically, 10 to 20 years of average daily weather data from a nearby weather service site are used to calculate 365 days of average weather conditions, which are then used to calculate the average pre- and post-retrofit conditions.

Utilities and government agencies have found it advantageous to prescreen many buildings against test regression models. These data-driven models can be used to develop comparative figures of merit for buildings in a similar standard industrial code (SIC) classification. A minimum goodness of fit is usually established that determines whether the monthly utility billing data are well fitted by the one-, two-, three-, four-, or five-parameter model being tested. Comparative figures of merit can then be determined by dividing the parameters by the conditioned floor area to yield average daily energy use per unit area of conditioned space. For example, an area-normalized comparison of base-level parameters across residential buildings would be used to analyze weather-independent energy use. This information can be used by energy auditors to focus their efforts on those systems needing assistance (Haberl and Komor 1990a, 1990b).

Multivariate Models. Three types of steady-state, multivariate models have been reported:

These models are a logical extension of single-variate models, provided that the choice of variables to be included and their functional forms are based on the engineering principles under which HVAC and other building systems operate. The goal of modeling energy use by the multivariate approach is to characterize building energy use with a few readily available and reliable input variables. These input variables should be selected with care. The model should contain variables not affected by the retrofit and thus unlikely to change (e.g., climatic variables) from pre-retrofit to post-retrofit periods. Other less obvious variables, such as changes in operating hours, base load, and occupancy levels, should be included in the model if these are not energy conservation measures (ECMs) but variables that may change during the post-retrofit period.

Environmental variables that meet these criteria for modeling heating and cooling energy use include outdoor air dry-bulb temperature, solar radiation, and outdoor specific humidity. Some of these variables are difficult to estimate or measure in an actual building and hence are not good candidates for regressor variables. Further, some of the variables change little over time. Although the effect of these variables on energy use may be important, a data-driven model will implicitly lump their effect into the parameter that represents constant load. In commercial buildings, internally generated loads, such as the heat given off by people, lights, and electrical equipment, also affect heating and cooling energy use. These internal loads are difficult to measure in their entirety. However, monitored electricity used by internal lights and equipment may be a good surrogate for total internal sensible loads (Reddy et al. 1999). For example, when the building is fully occupied, it is also likely to be experiencing high internal electric loads, and vice versa.

The effect of environmental variables is important for buildings such as offices but may be less so for buildings with loads dominated by internal heat gain (e.g., data centers) and buildings with loads dominated by occupants (e.g. assembly buildings).

Differences in HVAC system behavior during occupied and unoccupied periods can be modeled by a dummy or indicator variable (Draper and Smith 1981). For some office buildings, there seems to be little need to include a dummy variable, but its inclusion in the general functional form adds flexibility.

Air-side heating and cooling use in various HVAC system types has been addressed by Reddy et al. (1995) and subsequently applied to monitored data in commercial buildings (Katipamula et al. 1994, 1998). Because quadratic and cross-product terms of engineering equations are not usually picked up by multivariate models, strictly linear energy use models are often the only option.

In addition to outdoor temperature T_o, internal electric equipment and lighting load E_int, solar loads q_sol, and latent effects via the outdoor dew-point temperature T_dp are candidate regressor variables. In commercial buildings, a major portion of the latent load derives from fresh air ventilation. However, this load appears only when the outdoor air dew-point temperature exceeds the cooling coil temperature. Hence, the term (T_dp – T_s)⁺ (where the + sign indicates that the term is to be set to zero if negative, and T_s is the mean surface temperature of the cooling coil, typically about 51 to 55°F) is a more realistic descriptor of the latent loads than is T_dp alone. Using (T_dp – T_s)⁺ as a regressor in the model is a simplification that seems to yield good accuracy.

Therefore, a multivariate linear regression model with an engineering basis has the following structure:

(19)

Based on the preceding discussion, b₄ = 0 because outdoor air does not introduce a latent cooling load when T_dp is less than the cooling coil surface temperature. Introducing indicator variable terminology (Draper and Smith 1981), Equation (19) becomes

(20)

where the indicator variable I is introduced to handle the change in slope of the energy use due to T_o. The variable I is set equal to 1 for T_o values to the right of the change point (i.e., for high T_o range) and set equal to 0 for low T_o values. As with the single-variate segmented models (i.e., 3-P and 4-P models), a search method is used to determine the change point that minimizes the total sum of squares of residuals (Fels 1986; Kissock et al. 1993).

Katipamula et al. (1994) found that Equation (20), appropriate for VAV systems, could be simplified for constant-volume HVAC systems:

(21)

Note that instead of using (T_dp – T_s)⁺, the absolute humidity potential (W₀ – W_s)⁺ could also be used, where W₀ is the outdoor absolute humidity, and W_s is the absolute humidity level at the dew point of the cooling coil (typically about 0.009 lb/lb). A final aspect to keep in mind is that the term should be omitted from the regressor variable set when regressing heating energy use, because there are no latent loads on a heating coil.

These multivariate models are very accurate for daily time scales and slightly less so for hourly time scales. The inaccuracy at hourly time scales is because changes in the way the building is operated during the day and the night lead to different relative effects of the various regressors on energy use, which cannot be accurately modeled by one single hourly model. Breaking up energy use data into hourly bins corresponding to each hour of the day and then identifying 24 individual hourly models leads to appreciably greater accuracy (Katipamula et al. 1994).

Another accurate method uses a piecewise linear regression with respect to temperature and a time-of-week indicator variable to account for changes in energy use throughout each day of the week. This model (Matthieu 2011) may be applied to daily, hourly, or subhourly data.

Gaussian process (GP) models are nonparametric regression models, meaning the relationship between input and output variables is not strictly defined at the outset of the regression. In GP modeling, the measured output data are samples of a multivariate Gaussian distribution with the distribution parameters being the input variables. The distribution is defined by matching the covariance of the distribution to the covariance between the measured input and output data. For any new set of input variables, a Gaussian distribution is defined and the mean of the distribution is interpreted to be the predicted output and the variance of the distribution the uncertainty in the prediction. Thus, GP modeling not only produces an output value given a set of multivariate inputs, it also produces an estimate of the uncertainty of the prediction. Heo and Zavala (2012) used GP models to predict the hourly energy use of a chilled-water system using ambient temperature, occupancy, relative humidity, and chiller supply temperature and showed much better performance than standard multivariate regression models.

Hybrid Inverse Change Point Model. ASHRAE RP-1404 (Abushakra et al. 2014) was conducted primarily to develop analysis methodologies by which less than a whole year of field monitoring can be used for whole-building energy use prediction while satisfying current accuracy guidelines. The methodologies were developed to benefit ongoing efforts to spur diffusion of high-performance buildings by actual monitoring and to benefit energy service companies (ESCOs) and energy professionals who need a more cost-effective and acceptable alternative to year-long energy monitoring. Applications for the hybrid inverse change point model include detailed audits, green-building performance verification, and verification of post-retrofit claims using pre/post monitored data.

The modeling method is called hybrid because two regressions are performed at different time scales. The weather-dependent portion uses a year of monthly day-normalized utility bills, and the weather-independent portion uses either daily or hourly data covering a shorter time span, possibly as short as two weeks. The research revealed that (1) two weeks of hourly monitored data can be adequate for producing accurate long-term whole-building energy predictions, if the monitoring is taken at a correct time, and (2) adding more data to the regression model (extending the length of the short-term monitoring period) can, counterintuitively, make the predictive model poorer.

The analysis was completed in detail at two different time scales: daily and hourly. The analysis with both hourly and daily data revealed that the proper monitoring period is highly dependent on the time at which the data are taken, with the best time to collect data being during the spring and fall, or periods where the average temperature is close to the annual average temperature and the temperature has a wide range.

The following is an example of the modeling procedure that results in an hourly energy regression model. An example of the method applied to daily data can be found in Singh et al. (2013). The first regression uses one of the steady-state change point models shown in Figure 16 and Table 4, possibly with an additional weather-related independent variable, such as humidity. For example, the form could be a 3-P cooling shape (using the nomenclature of Table 4) such as

(22)

where E_daily is day-normalized energy consumption in units of energy consumption per day.

For predicting cooling thermal loads, such as chilled-water use, investigate adding the outdoor humidity variable as a regressor in the model to account for latent loads. Equation (22) would become a multivariable linear regression of the form

(23)

The (ω_{daily average} – 0.009)⁺ term in Equation (23) is a surrogate for the latent load existing only when the absolute humidity ratio of air entering the cooling coil is above 0.009.

The dependent variable for the second regression is the residuals resulting from using the first equation to predict the temperature dependent energy portion at the hourly time scale, calculated by dividing the temperature dependent portion of the first regression by 24 hours/day. The second regression uses gathered hourly data consisting of weather and an internal driving variable(s), such as X_{internal,hourly}, often an occupancy indicator or submetered lighting and equipment load. The second regression is then performed to calculate the coefficients for the internal driving variables and constant term. Using Equation (22), the second regression has the form

(24)

The final equation for prediction at the hourly time scale is a rearrangement of Equation (24).

(25)

Polynomial Models. Historically, polynomial models have been widely used as pure statistical models to model the behavior of equipment such as pumps, fans, and chillers (Stoecker and Jones 1982), as discussed in the section on HVAC Component Modeling. The theoretical aspects of calculating pump performance are well understood and documented. Pump capacity and efficiency are calculated from measurements of pump head, flow rate, and pump electrical power input. Phelan et al. (1996) studied the predictive ability of linear and quadratic models for electricity consumed by pumps and water mass flow rate, and concluded that quadratic models are superior to linear models. For fans, Phelan et al. (1996) studied the predictive ability of linear and quadratic polynomial single-variate models of fan electricity consumption as a function of supply air mass flow rate, and concluded that, although quadratic models are superior in terms of predicting energy use, the linear model seems to be the better overall predictor of both energy use and demand (i.e., maximum monthly power consumed by the fan). This is a noteworthy conclusion given that a third-order polynomial is warranted analytically as well as from monitored field data presented by previous authors [e.g., Englander and Norford (1992), Lorenzetti and Norford (1993)].

Polynomial models have been used to correlate chiller (or heat pump) capacity Q_evap and the electrical power consumed by the chiller (or compressor) E_comp with the relevant number of influential physical parameters. For example, based on the functional form of the DOE-2 building simulation software (York and Cappiello 1982), models for part-load performance of energy equipment and plant, E_comp, can be modeled as the following triquadratic polynomial:

(26)

In this model, there are 11 model parameters to identify. However, because all of them are unlikely to be statistically significant, a step-wise regression to the sample data set yields the optimal set of parameters to retain in a given model. For example, Armstrong et al. (2006b) found the b, i, and k terms, using T_odb for T_cond and T_ewb for T_evap, to provide an adequate model of three different fixed-speed rooftop units. Other authors, such as Braun (1992) and Hydeman et al. (2002), used slightly different polynomial forms. Over a wide range of capacity and lift a full tricubic polynomial can be justified for VRF heat pump equipment (Gayeski et al. 2011).

Physical Models. In contrast to polynomial models, which have no physical basis, physical models are based on fundamental thermodynamic or heat transfer considerations. These types of models are usually associated with the parameter estimation approach. Often, physical models are preferred because they generally have fewer parameters, and their mathematical formulation can be traced to actual physical principles that govern the performance of the building or equipment. Hence, model coefficients tend to be more robust, leading to sounder model predictions. Only a few studies have used steady-state physical models for parameter estimation relating to commercial building energy use [e.g., Reddy et al. (1999)]. Unlike in single-family residences, it is difficult to perform elaborately planned experiments in large buildings and obtain representative values of indoor fluctuations.

For example, the generalized Gordon and Ng (GN) model (Gordon and Ng 2000) is a simple, analytical, universal model for chiller performance based on first principles of thermodynamics and linearized heat losses. The model predicts the dependent chiller coefficient of performance (COP) (the ratio of thermal cooling capacity Q_ch to electrical power E consumed by the chiller) with easily measurable parameters such as the fluid (water or air) temperature entering the condenser T_cdi, fluid temperature entering the evaporator T_chi, and the thermal cooling capacity of the evaporator. The GN model is a three-parameter model in the following form:

(27a)

where temperatures are in absolute units.

Substituting the following,

(27b)

the model given by Equation (27a) becomes

(27c)

which is a three-parameter linear model with no intercept term. The parameters of the model in Equation (27c) have the following physical meaning:

Gordon and Ng (2000) point out that Q_leak is typically an order of magnitude smaller than the other terms, but it is not negligible for accurate modeling, and should be retained in the model if the other two parameters identified are to be used for chiller diagnostics. The same linear model structure as Equation (27c) can be used if the fluid temperature leaving the evaporator T_cho is used instead of T_chi. However, the physical interpretation of the term a₃ is modified accordingly.

Reddy and Anderson (2002) and Sreedharan and Haves (2001) found that the GN and multivariate polynomial (MP) models were comparable in their predictive abilities. The GN model requires much less data if selected judiciously [even four well-chosen data points can yield accurate models, as demonstrated by Corcoran and Reddy (2003)]. Jiang and Reddy (2003) tested the GN model against more than 50 data sets covering various generic types and sizes of water-cooled chillers (single- and double-stage centrifugal chillers with inlet guide vanes and variable-speed drives, screw, scroll), and found excellent predictive ability (coefficient of variation of RMSE in the range of 2 to 5%).

In contrast to steady-state data-driven models, which are used with monthly and daily data containing one or more independent variables, dynamic data-driven models are typically used with hourly or subhourly data in cases where the building’s thermal mass is significant enough to delay heat gains or losses. Dynamic models traditionally require solving a set of differential equations. Disadvantages of dynamic data-driven models include their complexity and the need for more detailed measurements to tune the model. Unlike steady-state data-driven models, dynamic data-driven models usually require a high degree of user interaction and knowledge of the building or system being modeled.

Several residential energy studies have used dynamic data-driven models based on parameter estimation approaches, usually involving intrusive data gathering. Rabl (1988) classified the various types of dynamic data-driven models used for whole-building energy use, and identified the common underlying features of these models. There are essentially four different types of model formulations: thermal network, time series, differential equation, and modal, all of which qualify as parameter-estimation approaches.

A few studies (Hammarsten 1984; Rabl 1988; Reddy 1989) evaluated these different approaches with the same data set. Several papers reported results of applying different techniques, such as thermal network and autoregressive-moving-average (ARMA) models, to residential and commercial building energy use. Examples of dynamic data-driven models for commercial buildings are found in Andersen and Brandemuehl (1992), Braun (1990), and Rabl (1988), and for apartment buildings in Armstrong et al. (2006a). Successful use of ARMA models for model-predictive control is reported by Gayeski et al. (2012).

Dynamic data-driven models based on pure statistical approaches have also been reported. Two examples are machine learning (Miller and Seem 1991) and artificial neural networks (Kreider and Haberl 1994; Kreider and Wang 1991; Miller and Seem 1991).

The following example (taken from Sonderegger 1998) illustrates a utility bill analysis. Assume that values of utility bills over an entire year have been measured. To obtain the equation coefficients through regression, the utility bills must be normalized by the length of the time interval between utility bills. This is equivalent to expressing all utility bills, degree-days, and other independent variables by their daily averages.

Appropriate modeling software is used in which values are assumed for heating and cooling balance points; from these, the corresponding heating and cooling degree-days for each utility bill period are determined. Repeated regression is done till the regression equation represents the best fit to the meter data. The model coefficients are then assumed to be tuned. Some programs allow direct determination of these optimal model parameters (e.g., balance point temperature) without manual tuning of the parameters by the user.

Figure 17 shows how well a regression fit captures measured baseline energy use in a hospital building. Cooling degree-days are found to be a significant variable, with the best fit for a base temperature of 54°F.

In this analysis, some individual utility bills may be unsuitable to develop a baseline and should be excluded from the regression. For example, a bill may be atypically high because of a one-time equipment malfunction that was subsequently repaired. Given the justification to exclude some data, it is often tempting to look for more reasons to exclude bills that fall far from “the line” and not question those that are close to it. For example, bills for periods containing vacations or production shutdowns may look anomalously low, but excluding them from the regression would result in a chronic overestimate of the future baseline during the same period.

Figure 18 shows results for a single neural network typical of several hundred networks constructed for an academic engineering center located in central Texas. The cooling load is created by solar gains, internal gains, outdoor air sensible heat, and outdoor air humidity loads. The neural network is used to predict the pre-retrofit energy consumption for comparison with measured consumption of the retrofitted building. Six months of pre-retrofit data were available to train the network. Solid lines show the known building consumption data, and dashed lines show the neural network predictions. This figure shows that a neural network trained for one period (September 1989) can predict energy consumption well into the future (in this case, January 1990).

Figure 17. Variable-Base Degree-Day Model Identification Using Electricity Utility Bills at Hospital (Sonderegger 1998)

Figure 18. Neural Network Prediction of Whole-Building, Hourly Chilled-Water Consumption for Commercial Building

The network used for this prediction had two hidden layers. The input layer contained eight neurons that receive eight different types of input data as listed below. The output layer consisted of one neuron that gave the output datum (chilled-water consumption). Each training fact (i.e., training data set), therefore, contained eight input data (independent variables) and one pattern datum (dependent variable). The following eight hourly input data used in each hour’s data vector were selected on physical bases (Kreider and Rabl 1994):

These measured independent variables were able to predict chilled-water use to an RMS error of less than 4% (JCEM 1992).

Choosing an optimal network’s configuration for a given problem remains an art. The number of hidden neurons and layers must be sufficient to meet the requirement of the given application. However, if too many neurons and layers are used, the network tends to memorize data rather than learning (i.e., finding the underlying patterns in the data). Further, choosing an excessively large number of hidden layers significantly increases the required training time for certain learning algorithms. Anstett and Kreider (1993), Krarti et al. (1998), Kreider and Wang (1991), and Wang and Kreider (1992) report additional case studies for commercial buildings.

Establishing an absolute truth standard for evaluating a program’s ability to analyze physical behavior requires empirical validation, but this is only possible within the range of measurement uncertainty, including that related to instruments, spatial and temporal discretization, and the overall experimental design. Full-scale test cells and buildings are large, relatively complex experimental objects. The exact design details, material properties, and construction in the field cannot be perfectly known, so there is uncertainty about the simulation model inputs that accurately represent the experimental object. Meticulous care is required to describe the experimental apparatus as clearly as possible to modelers to minimize this uncertainty. This includes experimental determination of as many material properties and other simulation model inputs as possible, including overall building parameters such as overall steady-state heat transmission coefficient (UA_o), infiltration (in the form of an effective UA), and thermal capacitance (Judkoff et al. 2000; Neymark et al. 2005; Subbarao 1988). Measuring these overall parameters allows for a closure check on the individual parameters that comprise the overall parameters (e.g., building envelope material properties or individual steady-state envelope component conductances that sum up to the measured overall steady-state heat transmission coefficient). Also required are detailed on-site meteorological measurements. For example, many experiments measure global horizontal solar radiation, but very few experiments measure the splits between direct, diffuse, and ground-reflected radiation, all of which are inputs to many whole-building energy simulation programs. Small-scale test cells also have shortcomings because of similitude factors for air convection and the increased relative proportion of corner conditions, which increases the importance of 2D and 3D heat transfer. Side-by-side experimental design can help to dampen the effects of input uncertainties, if the effects of the intentional differences between test objects are robust compared to those of the unintentional differences.

The National Renewable Energy Laboratory (NREL) divides empirical validation into different levels, because many past validation studies produced inconclusive results. The levels of validation depend on the degree of control over possible sources of error in a simulation. The higher the degree of control over error sources, the better the potential for diagnosing the causes of measured to modeled output differences. The error sources consist of seven types, divided into two groups (Judkoff et al. 1983/2008):

The simplest level of empirical validation compares a building’s actual long-term energy use to that calculated by a computer program, with no attempt to eliminate sources of discrepancy. Because this is similar to how a simulation tool is used in practice, it is favored by many in the building industry. However, it is difficult to interpret the results because all possible error sources are acting simultaneously. Even if there is good agreement between measured and calculated performance, possible offsetting errors prevent a definitive conclusion about the model’s accuracy. More informative levels of validation involve controlling or eliminating various combinations of error types and increasing the information density of output-to-data comparisons (e.g., comparing temperature and energy results at time scales ranging from subhourly to annual). At the most detailed level, all known sources of error are controlled to identify and quantify unknown error sources and to reveal causal relationships associated with error sources, thereby facilitating diagnostics. Long-term (ideally annual) overall energy use needed to maintain comfort has extra meaning as a metric, because it is an indicator of how important a fault or simplification in a simulation program is in the context of one of the model’s most important uses. A $15/yr error out of a $1500/yr energy bill is tolerable, but a $500/yr error is not. Other measurements, such as fluxes through individual surfaces, or short-term temperatures, are important to establish cause and effect but are inherently less indicative by themselves of the overall importance of an error. For example, it is difficult to assess the importance of a 0.5°F absolute average hourly one-week error between simulated and measured zone temperature.

These principles also apply to intermodel comparative testing and analytical verification. The more realistic the test case, the more difficult it is to establish causality and diagnose problems; the simpler and more controlled the test case, the easier it is to pinpoint sources of error or inaccuracy. Methodically building up from simple, highly controlled cases to realistic cases one parameter at a time is useful for understanding the impact of assumptions and simplifications in building energy simulation programs.

Analytical verification compares outputs from a program, subroutine, algorithm, or software object to results from a known analytical solution or to results from a set of closely agreeing quasi-analytical solutions or verified numerical models. Here, the term analytical solution is the closed-form mathematical solution of a model that has an exact result for a given set of parameters and simplifying assumptions. The term quasi-analytical solution is the mathematical solution of a model for a given set of parameters and simplifying assumptions, which may require minor interpretation differences that cause minor results variations. Such a result may be computed by generally accepted numerical methods or other means, provided that such calculations occur outside the environment of a whole-building energy simulation program and can be scrutinized. The term verified numerical model is a numerical model with solution accuracy verified by close agreement with an analytical solution and/or other quasi-analytical or numerical solutions, according to a process that demonstrates convergence in the space and time domains. Such numerical models may be verified by applying an initial comparison with an analytical solution(s), followed by comparisons with other numerical models for incrementally more realistic cases where analytical solutions are not available.

Mathematical Truth Standards. An analytical solution provides an exact mathematical truth standard, limited to highly constrained cases for which exact analytical solutions can be derived. A secondary mathematical truth standard can be established based on the range of disagreement of a set of closely agreeing verified numerical models or other quasi-analytical solutions. Once verified against all available classical analytical solutions, and compared with each other for a number of other diagnostic test cases that do not have exact analytical solutions, the secondary mathematical truth standard can be used to test other models as implemented within whole-building simulation programs. Although a closed-form analytical solution provides the best possible mathematical truth standard, a secondary mathematical truth standard greatly enhances diagnostic capability for identifying software bugs and modeling errors as compared to the purely comparative method. This is because the range of disagreement among the results that comprise the secondary truth standard is typically much narrower than the range of disagreement among whole-building simulations. The secondary mathematical truth standard also allows more realistic (less constrained) boundary conditions to be used in the test cases, extending the analytical verification method beyond the constraints inherent for classical analytical solutions. This extends the usefulness of analytical verification methods by applying comparative techniques methodically, as outlined below.

Establishing Secondary Mathematical Truth Standards. The following methodology for verifying numerical models to develop a secondary mathematical truth standard facilitates extension of analytical verification techniques. The methodology applies to both development of test cases and implementation of the numerical models. The logic may be summarized as follows:

Example applications of establishing secondary mathematical truth standards are provided in Neymark et al. (2008, 2009).

Other Considerations. The following general guidelines are helpful when developing effective empirical validation, analytical verification, and comparative test cases:

A comparison between measured and calculated performance represents a small region in an immense N-dimensional parameter space. Investigators are constrained to exploring relatively few regions in this space, yet would like to ensure that the results are not coincidental (e.g., not a result of offsetting errors) and represent the validity of the simulation elsewhere in the parameter space. Analytical and comparative techniques minimize the uncertainty of extrapolations around the limited number of sampled empirical domains. Table 9 classifies these extrapolations.

Figure 19 shows one process to combine analytical, empirical, and comparative techniques. These three techniques may also be used together in other ways; for example, intermodel comparisons may be done before an empirical validation exercise, to better define the experiment and to help estimate experimental uncertainty by propagating all known error sources through one or more whole-building energy simulation programs (Hunn et al. 1982; Lomas et al. 1994).

Figure 19. Validation Method (Judkoff et al. 1983/2008)

For the path shown in Figure 19, the first step is running the code against analytical verification test cases to check its mathematical solutions. Discrepancies must be corrected before proceeding further. Second, the code is run against high-quality empirical validation data, and errors are corrected. Diagnosing error sources can be quite difficult. Comparative techniques can be used to create diagnostic procedures (Achermann and Zweifel 2003; Judkoff 1988; Judkoff and Neymark 1995a, 1995b; Judkoff et al. 1980, 1983/2008; Morck 1986; Neymark and Judkoff 2002, 2004; Purdy and Beausoleil-Morrison 2003; Spitler et al. 2001; Yuill and Haberl 2002) and better define the empirical experiments. The third step is to check agreement of several different programs with different thermal solution and modeling approaches (that have passed through steps 1 and 2) in a variety of representative cases. This uses the comparative technique as an extrapolation tool. Deviations in the program predictions indicate areas for further investigation.

When programs successfully complete these three stages, they are considered validated for cases where acceptable agreement was achieved (i.e., for the range of building, climate, and mechanical system types represented by the test cases). Once several detailed simulation programs have satisfactorily completed the procedure, other programs and simplified design tools can be tested against them. A validated code does not necessarily represent truth. It does represent a set of algorithms that have been shown, through a repeatable procedure, to perform according to the current state of the art. Similarly, the example results and associated computer programs in the nonnormative sections of Standard 140 do not represent truth, but rather an attempt to identify the range of uncertainty in the current state of the art via a well-defined and repeatable procedure. It is anticipated that, as building energy simulation programs improve, the informative example results in Standard 140 will be periodically updated, and the range of differences among test case results may be reduced.

The NREL methodology for validating building energy simulation programs has been generally accepted by the International Energy Agency (Irving 1988), ASHRAE Standard 140, ASHRAE Standard 90.1, and elsewhere, with refinements suggested by other researchers (Bland 1992; Bloomfield 1988, 1999; Guyon and Palomo 1999b; Irving 1988; Lomas 1991; Lomas and Bowman 1987; Lomas and Eppel 1992). Additionally, the Commission of European Communities has conducted considerable work under the PASSYS program (Jensen 1989; Jensen and van de Perre 1991).

Calibration is commonly used in conjunction with energy retrofit audit models (Judkoff et al. 2011a, 2011b; Reddy et al. 2006). This test method was initially developed by NREL for testing calibration procedures used with residential retrofit audit software; however, the fundamental concept could also be applied in a commercial building context. Other terms frequently used to describe model calibration include model tuning, model true-up, and model reconciliation.

Typically, residential and commercial model calibration has been implemented using monthly energy data collected from utility bills for an existing building that is about to receive an energy retrofit. Sometimes submetered, disaggregated, or higher-frequency data are also available. An audit gathers information about the building needed to construct an input file for a building energy simulation program. A calibration method reconciles model predictions with the data, and then the calibrated model is used to predict energy savings and energy cost savings from various combinations of retrofit measures. Many variations on this approach exist, including some where the savings predictions are subjected to calibration instead of, or along with, the model inputs.

Although it is logical to use the building’s actual performance data to tune the model, it is not certain that this results in a model that better predicts post-retrofit energy savings. When calibrating a large number of inputs to a limited number of outputs (in mathematics, an underdetermined or overparameterized problem), there are many combinations of input parameters that result in a close match to the utility bill data, so a close match is not in itself proof of good calibration (Reddy et al. 2006). The lower the frequency or informational content of the building performance data, the lower the probability that the calibration actually improves the model and associated energy savings predictions. Therefore, any method to test calibration techniques should use as many of the following three figures of merit as possible: (1) accuracy of the savings prediction, (2) how closely the calibrated input parameter values match the actual parameter values, and (3) the goodness of fit between the modeled and measured data. A limiting factor in attempting to empirically validate calibration techniques is the lack of high-quality annual monthly pre- and post-retrofit energy data, (higher frequency and submetered data are better), good pre- and post-retrofit building characteristics data, local pre- and post-retrofit weather data, and the dates of the retrofit installations. Until enough such empirical data are available to researchers, an alternative analytical method can be used in which a simulation program is used to generate its own synthetic pre and post-retrofit energy performance data. The synthetic data may be used as a surrogate for actual data. The method follows these general procedures:

The preceding method is a pure (isolated) test of the calibration technique: that is, the synthetic utility billing data are generated with the tested program, and the program accuracy related to building physics modeling is not tested. A pure calibration test requires (1) automated calibration where no human judgment is required that would be helped by knowing the explicit inputs, or (2) that the modeler running the calibration test does not know the explicit inputs used to develop the synthetic utility bills. This method facilitates self-testing of a calibration technique, and is useful in several ways, including (1) testing a single calibration method, (2) testing several calibration methods to determine under what test conditions each is best, and (3) investigating how much and what kind of informational content is needed in the synthetic calibration data to achieve good calibrations with different calibration methods. The pure calibration test, however, may not be practical for a certification test that must be administered by a third-party organization; for this case, a method developed by NREL (Judkoff et al. 2011a, 2011b) ensures that the person performing the test does not know the explicit inputs. The main feature of this test method is that several (preferably at least three) reference programs are used to generate the synthetic utility bills and create the reference energy savings data. The bills and savings are taken as the average of the reference program results. This method tests both the calibration technique, and how closely the physics models in the tested program match the physics models in the reference programs. Example acceptance criteria may be used to facilitate the comparison of energy savings predictions (Judkoff et al. 2011a). Figure 20 shows the overall conceptual approach to testing model calibration techniques.

Figure 20. Calibration Cases Conceptual Flow (Judkoff et al. 2011a)