Patent Publication Number: US-11651121-B2

Title: System and method for building cooling optimization using periodic building fuel consumption with the aid of a digital computer

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This patent application is a continuation of U.S. patent application Ser. No. 16/708,021, filed Dec. 9, 2019, pending, which is a continuation of U.S. Pat. No. 10,503,847, Issued Dec. 10, 2019, which is a continuation of U.S. Pat. No. 10,339,232, issued Jul. 2, 2019, the priority dates of which are claimed and the disclosures of which are incorporated by reference. 
    
    
     FIELD 
     This application relates in general to energy conservation and, in particular, to a system and method for building cooling optimization using periodic building fuel consumption with the aid of a digital computer. 
     BACKGROUND 
     Concern has been growing in recent days over energy consumption in the United States and abroad. The cost of energy has steadily risen as power utilities try to cope with continually growing demand, increasing fuel prices, and stricter regulatory mandates. Power utilities must also maintain existing infrastructure, while simultaneously finding ways to add more generation capacity to meet future needs, both of which add to the cost of energy. Moreover, burgeoning energy consumption continues to impact the environment and deplete natural resources. 
     A large proportion of the rising cost of energy is borne by consumers, who, despite the need, remain poorly-equipped to identify the most cost effective ways to lower their own energy consumption. Often, no-cost behavioral changes, such as adjusting thermostat settings and turning off unused appliances, and low-cost physical improvements, such as switching to energy-efficient light bulbs, are insufficient to offset utility bill increases. Rather, appreciable decreases in energy consumption are often only achievable by investing in upgrades to a building&#39;s heating or cooling envelope. Identifying and comparing the kinds of building shell improvements that will yield an acceptable return on investment in terms of costs versus likely energy savings, though, requires finding many building-specific parameters, especially the building&#39;s thermal conductivity (UA Total ). 
     The costs of energy for heating, ventilating, and air conditioning (HVAC) system operation are often significant contributors to utility bills for both homeowners and businesses, and HVAC energy costs are directly tied to a building&#39;s thermal efficiency. For instance, a poorly insulated house or a building with significant sealing problems will require more overall HVAC usage to maintain a desired interior temperature than would a comparably-sized but well-insulated and sealed structure. Lowering HVAC energy costs is not as simple as choosing a thermostat setting to cause an HVAC system to run for less time or less frequently. Rather, HVAC system efficiency, duration of heating or cooling seasons, differences between indoor and outdoor temperatures, and other factors, in addition to thermal efficiency, can weigh into overall energy consumption. 
     Conventionally, estimating periodic HVAC energy consumption and fuel costs begins with analytically determining a building&#39;s thermal conductivity UA Total  through an on-site energy audit. A typical energy audit involves measuring physical dimensions of walls, windows, doors, and other building parts; approximating R-values for thermal resistance; estimating infiltration using a blower door test; and detecting air leakage using a thermal camera, after which a numerical model is run to solve for thermal conductivity. The UA Total  result is combined with the duration of the heating or cooling season, as applicable, over the period of inquiry and adjusted for HVAC system efficiency, plus any solar savings fraction. The audit report is often presented in the form of a checklist of corrective measures that may be taken to improve the building&#39;s shell and HVAC system, and thereby lower overall energy consumption for HVAC. As an involved process, an energy audit can be costly, time-consuming, and invasive for building owners and occupants. Further, as a numerical result derived from a theoretical model, an energy audit carries an inherent potential for inaccuracy strongly influenced by mismeasurements, data assumptions, and so forth. As well, the degree of improvement and savings attributable to various possible improvements is not necessarily quantified due to the wide range of variables. 
     Therefore, a need remains for a practical model for determining actual and potential energy consumption for the heating and cooling of a building. 
     A further need remains for an approach to quantifying improvements in energy consumption and cost savings resulting from building shell upgrades. 
     SUMMARY 
     Fuel consumption for building heating and cooling can be calculated through two practical approaches that characterize a building&#39;s thermal efficiency through empirically-measured values and readily-obtainable energy consumption data, such as available in utility bills, thereby avoiding intrusive and time-consuming analysis with specialized testing equipment. While the discussion is herein centered on building heating requirements, the same principles can be applied to an analysis of building cooling requirements. The first approach can be used to calculate annual or periodic fuel requirements. The approach requires evaluating typical monthly utility billing data and approximations of heating (or cooling) losses and gains. 
     The second approach can be used to calculate hourly (or interval) fuel requirements. The approach includes empirically deriving three building-specific parameters: thermal mass, thermal conductivity, and effective window area. HVAC system power rating and conversion and delivery efficiency are also parametrized. The parameters are estimated using short duration tests that last at most several days. The parameters and estimated HVAC system efficiency are used to simulate a time series of indoor building temperature. In addition, the second hourly (or interval) approach can be used to verify or explain the results from the first annual (or periodic) approach. For instance, time series results can be calculated using the second approach over the span of an entire year and compared to results determined through the first approach. Other uses of the two approaches and forms of comparison are possible. 
     In one embodiment, a system and method for building cooling optimization using periodic building fuel consumption with the aid of a digital computer are provided. A thermal conductivity of a building is evaluated by a computer, the computer including a processor and a memory, based on a balance point up to which the building can be thermally sustained using only internal heating gains over a time period for a heating season. Fuel consumption for cooling of the building over a time period for a cooling season is determined using the balance point thermal conductivity. An effect of an optimization of cooling of the building on the determined fuel consumption is modeled by the computer, wherein the cooling is optimized based on the modeling. 
     In a further embodiment, a system and method for modeling building cooling fuel consumption with the aid of a digital computer are provided. A temperature difference between an average temperature outside and an average temperature inside the building is over a time period for a heating season is identified by a computer, the computer including a processor and a memory. Internal heating gains are identified by the computer within the building over the heating time period. A balance point thermal conductivity is found by the computer as a function of the internal heating gains over the temperature difference and duration of the heating time period. A fuel consumption is determined by the computer for cooling of the building over a cooling time period using the balance point thermal conductivity and a difference between an average temperature outside and an average temperature inside the building during the cooling time period. An effect of an optimization of cooling of the building on the determined fuel consumption is modeled by the computer, wherein the cooling is optimized based on the modeling. 
     The foregoing approaches, annual (or periodic) and hourly (or interval) improve upon and compliment the standard energy audit-style methodology of estimating heating (and cooling) fuel consumption in several ways. First, per the first approach, the equation to calculate annual fuel consumption and its derivatives is simplified over the fully-parameterized form of the equation used in energy audit analysis, yet without loss of accuracy. Second, both approaches require parameters that can be obtained empirically, rather than from a detailed energy audit that requires specialized testing equipment and prescribed test conditions. Third, per the second approach, a time series of indoor temperature and fuel consumption data can be accurately generated. The resulting fuel consumption data can then be used by economic analysis tools using prices that are allowed to vary over time to quantify economic impact. 
     Moreover, the economic value of heating (and cooling) energy savings associated with any building shell improvement in any building has been shown to be independent of building type, age, occupancy, efficiency level, usage type, amount of internal electric gains, or amount solar gains, provided that fuel has been consumed at some point for auxiliary heating. The only information required to calculate savings includes the number of hours that define the winter season; average indoor temperature; average outdoor temperature; the building&#39;s HVAC system efficiency (or coefficient of performance for heat pump systems); the area of the existing portion of the building to be upgraded; the R-value of the new and existing materials; and the average price of energy, that is, heating fuel. 
     Still other embodiments will become readily apparent to those skilled in the art from the following detailed description, wherein are described embodiments by way of illustrating the best mode contemplated. As will be realized, other and different embodiments are possible and the embodiments&#39; several details are capable of modifications in various obvious respects, all without departing from their spirit and the scope. Accordingly, the drawings and detailed description are to be regarded as illustrative in nature and not as restrictive. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG.  1    is a functional block diagram showing heating losses and gains relative to a structure. 
         FIG.  2    is a graph showing, by way of example, balance point thermal conductivity. 
         FIG.  3    is a flow diagram showing a computer-implemented method for modeling periodic building heating energy consumption in accordance with one embodiment. 
         FIG.  4    is a flow diagram showing a routine for determining heating gains for use in the method of  FIG.  3   . 
         FIG.  5    is a flow diagram showing a routine for balancing energy for use in the method of  FIG.  3   . 
         FIG.  6    is a process flow diagram showing, by way of example, consumer heating energy consumption-related decision points. 
         FIG.  7    is a table showing, by way of example, data used to calculate thermal conductivity. 
         FIG.  8    is a table showing, by way of example, thermal conductivity results for each season using the data in the table of  FIG.  7    as inputs into Equations (25) through (28). 
         FIG.  9    is a graph showing, by way of example, a plot of the thermal conductivity results in the table of  FIG.  8   . 
         FIG.  10    is a graph showing, by way of example, an auxiliary heating energy analysis and energy consumption investment options. 
         FIG.  11    is a functional block diagram showing heating losses and gains relative to a structure. 
         FIG.  12    is a flow diagram showing a computer-implemented method for modeling interval building heating energy consumption in accordance with a further embodiment. 
         FIG.  13    is a table showing the characteristics of empirical tests used to solve for the four unknown parameters in Equation (47). 
         FIG.  14    is a flow diagram showing a routine for empirically determining building- and equipment-specific parameters using short duration tests for use in the method of  FIG.  12   . 
         FIG.  15    is a graph showing, by way of example, a comparison of auxiliary heating energy requirements determined by the hourly approach versus the annual approach. 
         FIG.  16    is a graph showing, by way of example, a comparison of auxiliary heating energy requirements with the allowable indoor temperature limited to 2° F. above desired temperature of 68° F. 
         FIG.  17    is a graph showing, by way of example, a comparison of auxiliary heating energy requirements with the size of effective window area tripled from 2.5 m 2  to 7.5 m 2 . 
         FIG.  18    is a table showing, by way of example, test data. 
         FIG.  19    is a table showing, by way of example, the statistics performed on the data in the table of  FIG.  18    required to calculate the three test parameters. 
         FIG.  20    is a graph showing, by way of example, hourly indoor (measured and simulated) and outdoor (measured) temperatures. 
         FIG.  21    is a graph showing, by way of example, simulated versus measured hourly temperature delta (indoor minus outdoor). 
         FIG.  22    is a block diagram showing a computer-implemented system for modeling building heating energy consumption in accordance with one embodiment. 
     
    
    
     DETAILED DESCRIPTION 
     Conventional Energy Audit-Style Approach 
     Conventionally, estimating periodic HVAC energy consumption and therefore fuel costs includes analytically determining a building&#39;s thermal conductivity (UA Total ) based on results obtained through an on-site energy audit. For instance, J. Randolf and G. Masters,  Energy for Sustainability: Technology, Planning, Policy , pp. 247, 248, 279 (2008), present a typical approach to modeling heating energy consumption for a building, as summarized by Equations 6.23, 6.27, and 7.5. The combination of these equations states that annual heating fuel consumption Q Fuel  equals the product of UA Total , 24 hours per day, and the number of heating degree days (HDD) associated with a particular balance point temperature T Balance Point , as adjusted for the solar savings fraction (SSF) divided by HVAC system efficiency (η HVAC ): 
                     Q   Fuel     =       (     UA   Total     )     ⁢     (     24   *     HDD     T     Balance   ⁢         Point           )     ⁢     (     1   -   SSF     )     ⁢     (     1     η   HVAC       )               (   1   )               
such that:
 
                     T     Balance   ⁢         Point       =       T     Set   ⁢         Point       -       Internal   ⁢         Gains       UA   Total                 (   2   )               
and
 
η HVAC =η Furnace η Distribution   (3)
 
where T Set Point  represents the temperature setting of the thermostat, Internal Gains represents the heating gains experienced within the building as a function of heat generated by internal sources and auxiliary heating, as further discussed infra, η Furnace  represents the efficiency of the furnace or heat source proper, and η Distribution  represents the efficiency of the duct work and heat distribution system. For clarity, HDD T     Balance Point    will be abbreviated to HDD Balance Point Temp .
 
     A cursory inspection of Equation (1) implies that annual fuel consumption is linearly related to a building&#39;s thermal conductivity. This implication further suggests that calculating fuel savings associated with building envelope or shell improvements is straightforward. In practice, however, such calculations are not straightforward because Equation (1) was formulated with the goal of determining the fuel required to satisfy heating energy needs. As such, there are several additional factors that the equation must take into consideration. 
     First, Equation (1) needs to reflect the fuel that is required only when indoor temperature exceeds outdoor temperature. This need led to the heating degree day (HDD) approach (or could be applied on a shorter time interval basis of less than one day) of calculating the difference between the average daily (or hourly) indoor and outdoor temperatures and retaining only the positive values. This approach complicates Equation (1) because the results of a non-linear term must be summed, that is, the maximum of the difference between average indoor and outdoor temperatures and zero. Non-linear equations complicate integration, that is, the continuous version of summation. 
     Second, Equation (1) includes the term Balance Point temperature (T Balance Point ). The goal of including the term T Balance Point  was to recognize that the internal heating gains of the building effectively lowered the number of degrees of temperature that auxiliary heating needed to supply relative to the temperature setting of the thermostat T Set Point . A balance point temperature T Balance Point  of 65° F. was initially selected under the assumption that 65° F. approximately accounted for the internal gains. As buildings became more efficient, however, an adjustment to the balance point temperature T Balance Point  was needed based on the building&#39;s thermal conductivity (UA Total ) and internal gains. This further complicated Equation (1) because the equation became indirectly dependent on (and inversely related to) UA Total  through T Balance Point . 
     Third, Equation (1) addresses fuel consumption by auxiliary heating sources. As a result, Equation (1) must be adjusted to account for solar gains. This adjustment was accomplished using the Solar Savings Fraction (SSF). The SSF is based on the Load Collector Ratio (see Eq. 7.4 in Randolf and Masters, p. 278, cited supra, for information about the LCR). The LCR, however, is also a function of UA Total . As a result, the SSF is a function of UA Total  in a complicated, non-closed form solution manner. Thus, the SSF further complicates calculating the fuel savings associated with building shell improvements because the SSF is indirectly dependent on UA Total . 
     As a result, these direct and indirect dependencies significantly complicate calculating a change in annual fuel consumption based on a change in thermal conductivity. The difficulty is made evident by taking the derivative of Equation (1) with respect to a change in thermal conductivity. The chain and product rules from calculus need to be employed since HHD Balance Point Temp  and SSF are indirectly dependent on UA Total : 
                       dQ     F   ⁢   u   ⁢   e   ⁢   l         dUA   Total       =       {         (     UA   Total     )     [         (     HDD     Balance   ⁢         Point   ⁢         Temp       )     ⁢     (       -     dSSF   dLCR       ⁢     dLCR     dUA     T   ⁢   o   ⁢   t   ⁢   α   ⁢   l           )       +     (         dHDD     Balance   ⁢         Point   ⁢         Temp         d   ⁢     T     Balance   ⁢         Point           ⁢       dT     Balance   ⁢         Point         dUA     T   ⁢   o   ⁢   t   ⁢   α   ⁢   l         ⁢     (     1   -     S   ⁢   S   ⁢   F       )           ]     +       (     HDD     Balance   ⁢         Point   ⁢         Temp       )     ⁢     (     1   -   SSF     )         }     ⁢     (       2   ⁢   4       η   HVAC       )               (   4   )               
The result is Equation (4), which is an equation that is difficult to solve due to the number and variety of unknown inputs that are required.
 
     To add even further complexity to the problem of solving Equation (4), conventionally, UA Total  is determined analytically by performing a detailed energy audit of a building. An energy audit involves measuring physical dimensions of walls, windows, doors, and other building parts; approximating R-values for thermal resistance; estimating infiltration using a blower door test; and detecting air leakage. A numerical model is then run to perform the calculations necessary to estimate thermal conductivity. Such an energy audit can be costly, time consuming, and invasive for building owners and occupants. Moreover, as a calculated result, the value estimated for UA Total  carries the potential for inaccuracies, as the model is strongly influenced by physical mismeasurements or omissions, data assumptions, and so forth. 
     Empirically-Based Approaches to Modeling Heating Fuel Consumption 
     Building heating (and cooling) fuel consumption can be calculated through two approaches, annual (or periodic) and hourly (or interval) to thermally characterize a building without intrusive and time-consuming tests. The first approach, as further described infra beginning with reference to  FIG.  1   , requires typical monthly utility billing data and approximations of heating (or cooling) losses and gains. The second approach, as further described infra beginning with reference to  FIG.  11   , involves empirically deriving three building-specific parameters, thermal mass, thermal conductivity, and effective window area, plus HVAC system efficiency using short duration tests that last at most several days. The parameters are then used to simulate a time series of indoor building temperature and of fuel consumption. While the discussion herein is centered on building heating requirements, the same principles can be applied to an analysis of building cooling requirements. 
     First Approach: Annual (or Periodic) Fuel Consumption 
     Fundamentally, thermal conductivity is the property of a material, here, a structure, to conduct heat.  FIG.  1    is a functional block diagram  10  showing heating losses and gains relative to a structure  11 . Inefficiencies in the shell  12  (or envelope) of a structure  11  can result in losses in interior heating  14 , whereas gains  13  in heating generally originate either from sources within (or internal to) the structure  11 , including heating gains from occupants  15 , gains from operation of electric devices  16 , and solar gains  17 , or from auxiliary heating sources  18  that are specifically intended to provide heat to the structure&#39;s interior. 
     In this first approach, the concepts of balance point temperatures and solar savings fractions, per Equation (1), are eliminated. Instead, balance point temperatures and solar savings fractions are replaced with the single concept of balance point thermal conductivity. This substitution is made by separately allocating the total thermal conductivity of a building (UA Total ) to thermal conductivity for internal heating gains (UA Balance Point ), including occupancy, heat produced by operation of certain electric devices, and solar gains, and thermal conductivity for auxiliary heating (UA Auxiliary Heating ). The end result is Equation (31), further discussed in detail infra, which eliminates the indirect and non-linear parameter relationships in Equation (1) to UA Total . 
     The conceptual relationships embodied in Equation (31) can be described with the assistance of a diagram.  FIG.  2    is a graph  20  showing, by way of example, balance point thermal conductivity UA Balance Point , that is, the thermal conductivity for internal heating gains. The x-axis  21  represents total thermal conductivity, UA Total , of a building (in units of Btu/hr-° F.). The y-axis  22  represents total heating energy consumed to heat the building. Total thermal conductivity  21  (along the x-axis) is divided into “balance point” thermal conductivity (UA Balance Point )  23  and “heating system” (or auxiliary heating) thermal conductivity (UA Auxiliary Heating )  24 . “Balance point” thermal conductivity  23  characterizes heating losses, which can occur, for example, due to the escape of heat through the building envelope to the outside and by the infiltration of cold air through the building envelope into the building&#39;s interior that are compensated for by internal gains. “Heating system” thermal conductivity  24  characterizes heating gains, which reflects the heating delivered to the building&#39;s interior above the balance point temperature T Balance Point , generally as determined by the setting of the auxiliary heating source&#39;s thermostat or other control point. 
     In this approach, total heating energy  22  (along the y-axis) is divided into gains from internal heating  25  and gains from auxiliary heating energy  25 . Internal heating gains are broken down into heating gains from occupants  27 , gains from operation of electric devices  28  in the building, and solar gains  29 . Sources of auxiliary heating energy include, for instance, natural gas furnace  30  (here, with a 56% efficiency), electric resistance heating  31  (here, with a 100% efficiency), and electric heat pump  32  (here, with a 250% efficiency). Other sources of heating losses and gains are possible. 
     The first approach provides an estimate of fuel consumption over a year or other period of inquiry based on the separation of thermal conductivity into internal heating gains and auxiliary heating.  FIG.  3    is a flow diagram showing a computer-implemented method  40  for modeling periodic building heating energy consumption in accordance with one embodiment. Execution of the software can be performed with the assistance of a computer system, such as further described infra with reference to  FIG.  22   , as a series of process or method modules or steps. 
     In the first part of the approach (steps  41 - 43 ), heating losses and heating gains are separately analyzed. In the second part of the approach (steps  44 - 46 ), the portion of the heating gains that need to be provided by fuel, that is, through the consumption of energy for generating heating using auxiliary heating  18  (shown in  FIG.  1   ), is determined to yield a value for annual (or periodic) fuel consumption. Each of the steps will now be described in detail. 
     Specify Time Period 
     Heating requirements are concentrated during the winter months, so as an initial step, the time period of inquiry is specified (step  41 ). The heating degree day approach (HDD) in Equation (1) requires examining all of the days of the year and including only those days where outdoor temperatures are less than a certain balance point temperature. However, this approach specifies the time period of inquiry as the winter season and considers all of the days (or all of the hours, or other time units) during the winter season. Other periods of inquiry are also possible, such as a five- or ten-year time frame, as well as shorter time periods, such as one- or two-month intervals. 
     Separate Heating Losses from Heating Gains 
     Heating losses are considered separately from heating gains (step  42 ). The rationale for drawing this distinction will now be discussed. 
     Heating Losses 
     For the sake of discussion herein, those regions located mainly in the lower latitudes, where outdoor temperatures remain fairly moderate year round, will be ignored and focus placed instead on those regions that experience seasonal shifts of weather and climate. Under this assumption, a heating degree day (HDD) approach specifies that outdoor temperature must be less than indoor temperature. No such limitation is applied in this present approach. Heating losses are negative if outdoor temperature exceeds indoor temperature, which indicates that the building will gain heat during these times. Since the time period has been limited to only the winter season, there will likely to be a limited number of days when that situation could occur and, in those limited times, the building will benefit by positive heating gain. (Note that an adjustment would be required if the building took advantage of the benefit of higher outdoor temperatures by circulating outdoor air inside when this condition occurs. This adjustment could be made by treating the condition as an additional source of heating gain.) 
     As a result, fuel consumption for heating losses Q Losses  over the winter season equals the product of the building&#39;s total thermal conductivity UA Total  and the difference between the indoor T Indoor  and outdoor temperature T Outdoor , summed over all of the hours of the winter season: 
                     Q   Losses     =       ∑     t   Start       t   End           (     UA   Total     )     ⁢     (       T   t   Indoor     -     T   t   Outdoor       )                 (   5   )               
where Start and End respectively represent the first and last hours of the winter (heating) season.
 
     Equation (5) can be simplified by solving the summation. Thus, total heating losses Q Losses  equal the product of thermal conductivity UA Total  and the difference between average indoor temperature  T   Indoor  and average outdoor temperature  T   Outdoor  over the winter season and the number of hours H in the season over which the average is calculated:
 
 Q   Losses =( UA   Total )(   T     Indoor   − T     Outdoor )( H )  (6)
 
     Heating Gains 
     Heating gains are calculated for two broad categories (step  43 ) based on the source of heating, internal heating gains Q Gains-Internal  and auxiliary heating gains Q Gains-Auxiliary Heating , as further described infra with reference to  FIG.  4   . Internal heating gains can be subdivided into heating gained from occupants Q Gains-Occupants , heating gained from the operation of electric devices Q Gains-Electric , and heating gained from solar heating Q Gains-Solar . Other sources of internal heating gains are possible. The total amount of heating gained Q Gains  from these two categories of heating sources equals:
 
 Q   Gains   =Q   Gains-Internal   +Q   Gains-Auxiliary Heating   (7)
 
where
 
 Q   Gains-Internal   =Q   Gains-Occupants   +Q   Gains-Electric   +Q   Gains-Solar   (8)
 
     Calculate Heating Gains 
     Equation (8) states that internal heating gains Q Gains-Internal  include heating gains from Occupant, Electric, and Solar heating sources.  FIG.  4    is a flow diagram showing a routine  50  for determining heating gains for use in the method  40  of  FIG.  3    Each of these heating gain sources will now be discussed. 
     Occupant Heating Gains 
     People occupying a building generate heat. Occupant heating gains Q Gains-Occupants  (step  51 ) equal the product of the heat produced per person, the average number of people in a building over the time period, and the number of hours (II) (or other time units) in that time period. Let  P  represent the average number of people. If a person produces 250 Btu of heat per hour, heating gains from the occupants Q Gains-Occupants  equal:
 
 Q   Gains-Occupants =250(   P   )( H )  (9)
 
     Electric Heating Gains 
     The operation of electric devices that deliver all heat that is generated into the interior of the building, for instance, lights, refrigerators, and the like, contribute to internal heating gain. Electric heating gains Q Gains-Electric  (step  52 ) equal the amount of electricity used in the building that is converted to heat over the time period. 
     Care needs to be taken to ensure that the measured electricity consumption corresponds to the indoor usage. Two adjustments may be required. First, many electric utilities measure net electricity consumption. The energy produced by any photovoltaic (PV) system needs to be added back to net energy consumption (Net) to result in gross consumption if the building has a net-metered PV system. This amount can be estimated using time- and location-correlated solar resource data, as well as specific information about the orientation and other characteristics of the photovoltaic system, such as can be provided by the Solar Anywhere SystemCheck service (http://www.SolarAnywhere.com), a Web-based service operated by Clean Power Research, L.L.C., Napa, Calif., with the approach described, for instance, in commonly-assigned U.S. Patent application, entitled “Computer-Implemented System and Method for Estimating Gross Energy Load of a Building,” Ser. No. 14/531,940, filed Nov. 3, 2014, pending, the disclosure of which is incorporated by reference, or measured directly. 
     Second, some uses of electricity may not contribute heat to the interior of the building and need be factored out as external electric heating gains (External). These uses include electricity used for electric vehicle charging, electric dryers (assuming that most of the hot exhaust air is vented outside of the building, as typically required by building code), outdoor pool pumps, and electric water heating using either direct heating or heat pump technologies (assuming that most of the hot water goes down the drain and outside the building—a large body of standing hot water, such as a bathtub filled with hot water, can be considered transient and not likely to appreciably increase the temperature indoors over the long run). 
     Including the conversion factor from kWh to Btu (since Q Gains-Electric  is in units of Btu), internal electric gains Q Gains-Electric  equal: 
                     Q     Gains   -   Electric       =       (       Net   +   PV   -   External     _     )     ⁢     (   H   )     ⁢     (       3   ,   412   ⁢         Btu           kWh       )               (   10   )               
where Net represents net energy consumption, PV represents any energy produced by a PV system, External represents heating gains attributable to electric sources that do not contribute heat to the interior of a building. The average delivered electricity  Net+PV−External  equals the total over the time period divided by the number of hours (H) in that time period.
 
     
       
         
           
             
               
                 
                   
                     
                       Net 
                       + 
                       PV 
                       - 
                       External 
                     
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                       - 
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                     H 
                   
                 
               
               
                 
                   ( 
                   11 
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     Solar Heating Gains 
     Solar energy that enters through windows, doors, and other openings in a building as sunlight will heat the interior. Solar heating gains Q Gains-Solar  (step  53 ) equal the amount of heat delivered to a building from the sun. In the northern hemisphere, Q Gains-Solar  can be estimated based on the south-facing window area (m 2 ) times the solar heating gain coefficient (SHGC) times a shading factor; together, these terms are represented by the effective window area (W). Solar heating gains Q Gains-Solar  equal the product of W, the average direct vertical irradiance (DVI) available on a south-facing surface (Solar, as represented by DVI in kW/m 2 ), and the number of hours (H) in the time period. Including the conversion factor from kWh to Btu (since Q Gains-Solar  is in units of Btu while average solar is in kW/m 2 ), solar heating gains Q Gains-Solar  equal: 
     
       
         
           
             
               
                 
                   
                     Q 
                     
                       Gains 
                       - 
                       Solar 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         Solar 
                         _ 
                       
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                     ⁢ 
                     
                       ( 
                       W 
                       ) 
                     
                     ⁢ 
                     
                       ( 
                       H 
                       ) 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           3 
                           , 
                           412 
                           ⁢ 
                               
                           Btu 
                         
                         
                             
                           kWh 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   12 
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     Note that for reference purposes, the SHGC for one particular high quality window designed for solar gains, the Andersen High-Performance Low-E4 PassiveSun Glass window product, manufactured by Andersen Corporation, Bayport, Minn., is 0.54; many windows have SHGCs that are between 0.20 to 0.25. 
     Auxiliary Heating Gains 
     The internal sources of heating gain share the common characteristic of not being operated for the sole purpose of heating a building, yet nevertheless making some measureable contribution to the heat to the interior of a building. The fourth type of heating gain, auxiliary heating gains Q Gains-Auxiliary Heating , consumes fuel specifically to provide heat to the building&#39;s interior and, as a result, must include conversion efficiency. The gains from auxiliary heating gains Q Gains-Auxiliary Heating  (step  53 ) equal the product of the average hourly fuel consumed  Q   Fuel  times the hours (H) in the period and HVAC system efficiency η HVAC .
 
 Q   Gains-Auxiliary Heating =(   Q     Fuel )η HVAC   (13)
 
     Divide Thermal Conductivity into Parts 
     Referring back to  FIG.  3   , a building&#39;s thermal conductivity UA Total , rather than being treated as a single value, can be conceptually divided into two parts (step  44 ), with a portion of UA Total  allocated to “balance point thermal conductivity” (UA Balance Point ) and a portion to “auxiliary heating thermal conductivity” (UA Auxiliary Heating ), such as pictorially described supra with reference to  FIG.  2   . UA Balance Point  corresponds to the heating losses that a building can sustain using only internal heating gains Q Gains-Internal . This value is related to the concept that a building can sustain a specified balance point temperature in light of internal gains. However, instead of having a balance point temperature, some portion of the building UA Balance Point  is considered to be thermally sustainable given heating gains from internal heating sources (Q Gains-Internal ). As the rest of the heating losses must be made up by auxiliary heating gains, the remaining portion of the building UA Auxillary heating  is considered to be thermally sustainable given heating gains from auxiliary heating sources (Q Gains-Auxiliary Heating ). The amount of auxiliary heating gained is determined by the setting of the auxiliary heating source&#39;s thermostat or other control point. Thus, UA Total  can be expressed as:
 
 UA   Total   =UA   Balance Point   +UA   Auxiliary Heating   (14)
 
where
 
 UA   Balance Point   =UA   Occupants   +UA   Electric   +UA   Solar   (15)
 
such that UA Occupants , UA Electric , and UA Solar  respectively represent the thermal conductivity of internal heating sources, specifically, occupants, electric and solar.
 
     In Equation (14), total thermal conductivity UA Total  is fixed at a certain value for a building and is independent of weather conditions; UA Total  depends upon the building&#39;s efficiency. The component parts of Equation (14), balance point thermal conductivity UA Balance Point  and auxiliary heating thermal conductivity UA Auxiliary Heating , however, are allowed to vary with weather conditions. For example, when the weather is warm, there may be no auxiliary heating in use and all of the thermal conductivity will be allocated to the balance point thermal conductivity UA Balance Point  component. 
     Fuel consumption for heating losses Q Losses  can be determined by substituting Equation (14) into Equation (6):
 
 Q   Losses =( UA   Balance Point   +UA   Auxiliary Heating )(   T     Indoor   − T     Outdoor )( H )  (16)
 
     Balance Energy 
     Heating gains must equal heating losses for the system to balance (step  45 ), as further described infra with reference to  FIG.  5   . Heating energy balance is represented by setting Equation (7) equal to Equation (16):
 
 Q   gain-Internal   +Q   Gains-Auxiliary Heating =( UA   Balance Point   +UA   Auxiliary Heating )(   T     Indoor   − T     Outdoor )( H )  (17)
 
The result can then be divided by ( T   Indoor − T   Outdoor )(H), assuming that this term is non-zero:
 
     
       
         
           
             
               
                 
                   
                     
                       UA 
                       
                         Balance 
                         ⁢ 
                             
                         Point 
                       
                     
                     + 
                     
                       UA 
                       
                         Auxiliary 
                         ⁢ 
                             
                         Heating 
                       
                     
                   
                   = 
                   
                     
                       
                         Q 
                         
                           Gains 
                           - 
                           Internαl 
                         
                       
                       + 
                       
                         Q 
                         
                           Gains 
                           - 
                           Auxiliary 
                           ⁢ 
                               
                           Heating 
                         
                       
                     
                     
                       
                         ( 
                         
                           
                             
                               T 
                               ¯ 
                             
                             Indoor 
                           
                           - 
                           
                             
                               T 
                               ¯ 
                             
                             Outdoor 
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         ( 
                         H 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
     Equation (18) expresses energy balance as a combination of both UA Balance Point  and UA Auxiliary Heating .  FIG.  5    is a flow diagram showing a routine  60  for balancing energy for use in the method  40  of  FIG.  3   . Equation (18) can be further constrained by requiring that the corresponding terms on each side of the equation match, which will divide Equation (18) into a set of two equations: 
                     UA     Balance   ⁢         Point       =       Q     Gains   -   Internal           (         T   ¯     Indoor     -       T   ¯     Outdoor       )     ⁢     (   H   )                 (   19   )                             UA     Auxiliary   ⁢         Heating       =       Q     Gains   -   Auxiliary   ⁢         Heating           (         T   ¯     Indoor     -       T   ¯     Outdoor       )     ⁢     (   H   )                 (   20   )               
Components of UA Balance Point  
 
     For clarity, UA Balance Point  can be divided into three component values (step  61 ) by substituting Equation (8) into Equation (19): 
     
       
         
           
             
               
                 
                   
                     UA 
                     
                       Balance 
                       ⁢ 
                           
                       Point 
                     
                   
                   = 
                   
                     
                       
                         Q 
                         
                           Gains 
                           - 
                           Occupants 
                         
                       
                       + 
                       
                         Q 
                         
                           Gains 
                           - 
                           Electric 
                         
                       
                       + 
                       
                         Q 
                         
                           Gains 
                           - 
                           Solar 
                         
                       
                     
                     
                       
                         ( 
                         
                           
                             
                               T 
                               ¯ 
                             
                             Indoor 
                           
                           - 
                           
                             
                               T 
                               ¯ 
                             
                             Outdoor 
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         ( 
                         H 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     Since UA Balance Point  equals the sum of three component values (as specified in Equation (15)), Equation (21) can be mathematically limited by dividing Equation (21) into three equations: 
     
       
         
           
             
               
                 
                   
                     UA 
                     Occupants 
                   
                   = 
                   
                     
                       Q 
                       
                         Gains 
                         - 
                         Occupants 
                       
                     
                     
                       
                         ( 
                         
                           
                             
                               T 
                               ¯ 
                             
                             Indoor 
                           
                           - 
                           
                             
                               T 
                               ¯ 
                             
                             Outdoor 
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         ( 
                         H 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     UA 
                     Electric 
                   
                   = 
                   
                     
                       Q 
                       
                         Gains 
                         - 
                         Electric 
                       
                     
                     
                       
                         ( 
                         
                           
                             
                               T 
                               ¯ 
                             
                             Indoor 
                           
                           - 
                           
                             
                               T 
                               ¯ 
                             
                             Outdoor 
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         ( 
                         H 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     UA 
                     Solar 
                   
                   = 
                   
                     
                       Q 
                       
                         Gains 
                         - 
                         Solar 
                       
                     
                     
                       
                         ( 
                         
                           
                             
                               T 
                               ¯ 
                             
                             Indoor 
                           
                           - 
                           
                             
                               T 
                               ¯ 
                             
                             Outdoor 
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         ( 
                         H 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
     Solutions for Components of UA Balance Point  and UA Auxiliary Heating    
     The preceding equations can be combined to present a set of results with solutions provided for the four thermal conductivity components as follows. First, the portion of the balance point thermal conductivity associated with occupants UA Occupants  (step  62 ) is calculated by substituting Equation (9) into Equation (22). Next, the portion of the balance point thermal conductivity associated with internal electricity consumption UA Electric  (step  63 ) is calculated by substituting Equation (10) into Equation (23). The portion of the balance point thermal conductivity associated with solar gains UA Solar  (step  64 ) is then calculated by substituting Equation (12) into Equation (24). Finally, thermal conductivity associated with auxiliary heating UA Auxiliary Heating  (step  64 ) is calculated by substituting Equation (13) into Equation (20). 
     
       
         
           
             
               
                 
                   
                     UA 
                     Occupants 
                   
                   = 
                   
                     
                       2 
                       ⁢ 
                       5 
                       ⁢ 
                       0 
                       ⁢ 
                       
                         ( 
                         
                           P 
                           ¯ 
                         
                         ) 
                       
                     
                     
                       ( 
                       
                         
                           
                             T 
                             ¯ 
                           
                           Indoor 
                         
                         - 
                         
                           
                             T 
                             ¯ 
                           
                           Outdoor 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     UA 
                     Electric 
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             Net 
                             + 
                             PV 
                             - 
                             External 
                           
                           _ 
                         
                         ) 
                       
                       
                         ( 
                         
                           
                             
                               T 
                               ¯ 
                             
                             Indoor 
                           
                           - 
                           
                             
                               T 
                               ¯ 
                             
                             Outdoor 
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           3 
                           , 
                           412 
                           ⁢ 
                               
                           Btu 
                         
                         
                             
                           kWh 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     UA 
                     Solar 
                   
                   = 
                   
                     
                       
                         
                           ( 
                           
                             Solar 
                             _ 
                           
                           ) 
                         
                         ⁢ 
                         
                           ( 
                           W 
                           ) 
                         
                       
                       
                         ( 
                         
                           
                             
                               T 
                               ¯ 
                             
                             Indoor 
                           
                           - 
                           
                             
                               T 
                               ¯ 
                             
                             Outdoor 
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           3 
                           , 
                           412 
                           ⁢ 
                               
                           Btu 
                         
                         
                             
                           kWh 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     UA 
                     
                       Auxiliary 
                       ⁢ 
                           
                       Heating 
                     
                   
                   = 
                   
                     
                       
                         
                           Q 
                           ¯ 
                         
                         Fuel 
                       
                       ⁢ 
                       
                         η 
                         HVAC 
                       
                     
                     
                       ( 
                       
                         
                           
                             T 
                             ¯ 
                           
                           Indoor 
                         
                         - 
                         
                           
                             T 
                             ¯ 
                           
                           Outdoor 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
           
         
       
     
     Determine Fuel Consumption 
     Referring back to  FIG.  3   , Equation (28) can used to derive a solution to annual (or periodic) heating fuel consumption. First, Equation (14) is solved for UA Auxiliary Heating :
 
 UA   Auxiliary Heating   =UA   Total   −UA   Balance Point   (29)
 
Equation (29) is then substituted into Equation (28):
 
                       UA   Total     -     UA     Balance   ⁢         Point         =           Q   ¯     Fuel     ⁢     η   HVAC         (         T   ¯     Indoor     -       T   ¯     Outdoor       )               (   30   )               
Finally, solving Equation (30) for fuel and multiplying by the number of hours (H) in (or duration of) the time period yields:
 
                     Q   Fuel     =         (       UA   Total     -     UA     Balance   ⁢         Point         )     ⁢     (         T   ¯     Indoor     -       T   _     Outdoor       )     ⁢     (   H   )         η   HVAC               (   31   )               
Equation (31) is valid where UA Total ≥UA Balance Point . Otherwise, fuel consumption is 0.
 
     Using Equation (31), annual (or periodic) heating fuel consumption Q Fuel  can be determined (step  46 ). The building&#39;s thermal conductivity UA Total , if already available through, for instance, the results of an energy audit, is obtained. Otherwise, UA Total  can be determined by solving Equations (25) through (28) using historical fuel consumption data, such as shown, by way of example, in the table of  FIG.  7   , or by solving Equation (49), as further described infra. UA Total  can also be empirically determined with the approach described, for instance, in commonly-assigned U.S. Patent application, entitled “System and Method for Empirically Estimating Overall Thermal Performance of a Building,” Ser. No. 14/294,087, filed Jun. 2, 2014, pending, the disclosure of which is incorporated by reference. Other ways to determine UA Total  are possible. UA Balance Point  can be determined by solving Equation (21). The remaining values, average indoor temperature  T   Indoor  and average outdoor temperature  T   Outdoor , and HVAC system efficiency η HVAC  can respectively be obtained from historical weather data and manufacturer specifications. 
     Practical Considerations 
     Equation (31) is empowering. Annual heating fuel consumption Q Fuel  can be readily determined without encountering the complications of Equation (1), which is an equation that is difficult to solve due to the number and variety of unknown inputs that are required. The implications of Equation (31) in consumer decision-making, a general discussion, and sample applications of Equation (31) will now be covered. 
     Change in Fuel Requirements Associated with Decisions Available to Consumers 
     Consumers have four decisions available to them that affects their energy consumption for heating.  FIG.  6    is a process flow diagram showing, by way of example, consumer heating energy consumption-related decision points. These decisions  71  include:
         1. Change the thermal conductivity UA Total  by upgrading the building shell to be more thermally efficient (process  72 ).   2. Reduce or change the average indoor temperature by reducing the thermostat manually, programmatically, or through a “learning” thermostat (process  73 ).   3. Upgrade the HVAC system to increase efficiency (process  74 ).   4. Increase the solar gain by increasing the effective window area (process  75 ).
 
Other decisions are possible. Here, these four specific options can be evaluated supra by simply taking the derivative of Equation (31) with respect to a variable of interest. The result for each case is valid where UA Total ≥UA Balance Point . Otherwise, fuel consumption is 0.
       

     Changes associated with other internal gains, such as increasing occupancy, increasing internal electric gains, or increasing solar heating gains, could be calculated using a similar approach. 
     Change in Thermal Conductivity 
     A change in thermal conductivity UA Total  can affect a change in fuel requirements. The derivative of Equation (31) is taken with respect to thermal conductivity, which equals the average indoor minus outdoor temperatures times the number of hours divided by HVAC system efficiency. Note that initial thermal efficiency is irrelevant in the equation. The effect of a change in thermal conductivity UA Total  (process  72 ) can be evaluated by solving: 
     
       
         
           
             
               
                 
                   
                     
                       dQ 
                       Fuel 
                     
                     
                       dUA 
                       Total 
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             
                               T 
                               ¯ 
                             
                             Indoor 
                           
                           - 
                           
                             
                               T 
                               ¯ 
                             
                             Outdoor 
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         ( 
                         H 
                         ) 
                       
                     
                     
                       η 
                       HVAC 
                     
                   
                 
               
               
                 
                   ( 
                   32 
                   ) 
                 
               
             
           
         
       
     
     Change in Average Indoor Temperature 
     A change in average indoor temperature can also affect a change in fuel requirements. The derivative of Equation (31) is taken with respect to the average indoor temperature. Since UA Balance Point  is also a function of average indoor temperature, application of the product rule is required. After simplifying, the effect of a change in average indoor temperature (process  73 ) can be evaluated by solving: 
     
       
         
           
             
               
                 
                   
                     
                       dQ 
                       Fuel 
                     
                     
                       d 
                       ⁢ 
                       
                         
                           T 
                           Indoor 
                         
                         _ 
                       
                     
                   
                   = 
                   
                     
                       ( 
                       
                         UA 
                         Total 
                       
                       ) 
                     
                     ⁢ 
                     
                       ( 
                       
                         H 
                         
                           η 
                           HVAC 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   33 
                   ) 
                 
               
             
           
         
       
     
     Change in HVAC System Efficiency 
     As well, a change in HVAC system efficiency can affect a change in fuel requirements. The derivative of Equation (31) is taken with respect to HVAC system efficiency, which equals current fuel consumption divided by HVAC system efficiency. Note that this term is not linear with efficiency and thus is valid for small values of efficiency changes. The effect of a change in fuel requirements relative to the change in HVAC system efficiency (process  74 ) can be evaluated by solving: 
     
       
         
           
             
               
                 
                   
                     
                       dQ 
                       Fuel 
                     
                     
                       d 
                       ⁢ 
                       
                         η 
                         HVAC 
                       
                     
                   
                   = 
                   
                     - 
                     
                       
                         Q 
                         Fuel 
                       
                       ( 
                       
                         1 
                         
                           η 
                           HVAC 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   34 
                   ) 
                 
               
             
           
         
       
     
     Change in Solar Gains 
     An increase in solar gains can be accomplished by increasing the effective area of south-facing windows. Effective area can be increased by trimming trees blocking windows, removing screens, cleaning windows, replacing windows with ones that have higher SHGCs, installing additional windows, or taking similar actions. In this case, the variable of interest is the effective window area W. The total gain per square meter of additional effective window area equals the available resource (kWh/m 2 ) divided by HVAC system efficiency, converted to Btus. The derivative of Equation (31) is taken with respect to effective window area. The effect of an increase in solar gains (process  74 ) can be evaluated by solving: 
     
       
         
           
             
               
                 
                   
                     
                       dQ 
                       
                         F 
                         ⁢ 
                         u 
                         ⁢ 
                         e 
                         ⁢ 
                         l 
                       
                     
                     
                       d 
                       ⁢ 
                       
                         W 
                         ¯ 
                       
                     
                   
                   = 
                   
                     
                       - 
                       
                         [ 
                         
                           
                             
                               ( 
                               
                                 Solar 
                                 _ 
                               
                               ) 
                             
                             ⁢ 
                             
                               ( 
                               H 
                               ) 
                             
                           
                           
                             η 
                             HVAC 
                           
                         
                         ] 
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           3 
                           , 
                           412 
                           ⁢ 
                               
                           Btu 
                         
                         
                             
                           kWh 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   35 
                   ) 
                 
               
             
           
         
       
     
     Discussion 
     Both Equations (1) and (31) provide ways to calculate fuel consumption requirements. The two equations differ in several key ways:
         1. UA Total  only occurs in one place in Equation (31), whereas Equation (1) has multiple indirect and non-linear dependencies to UA Total .   2. UA Total  is divided into two parts in Equation (31), while there is only one occurrence of UA Total  in Equation (1).   3. The concept of balance point thermal conductivity in Equation (31) replaces the concept of balance point temperature in Equation (1).   4. Heat from occupants, electricity consumption, and solar gains are grouped together in Equation (31) as internal heating gains, while these values are treated separately in Equation (1).       

     Second, Equations (25) through (28) provide empirical methods to determine both the point at which a building has no auxiliary heating requirements and the current thermal conductivity. Equation (1) typically requires a full detailed energy audit to obtain the data required to derive thermal conductivity. In contrast, Equations (25) through (28), as applied through the first approach, can substantially reduce the scope of an energy audit. 
     Third, both Equation (4) and Equation (32) provide ways to calculate a change in fuel requirements relative to a change in thermal conductivity. However, these two equations differ in several key ways:
         1. Equation (4) is complex, while Equation (32) is simple.   2. Equation (4) depends upon current building thermal conductivity, balance point temperature, solar savings fraction, auxiliary heating efficiency, and a variety of other derivatives. Equation (32) only requires the auxiliary heating efficiency in terms of building-specific information.       

     Equation (32) implies that, as long as some fuel is required for auxiliary heating, a reasonable assumption, a change in fuel requirements will only depend upon average indoor temperature (as approximated by thermostat setting), average outdoor temperature, the number of hours (or other time units) in the (heating) season, and HVAC system efficiency. Consequently, any building shell (or envelope) investment can be treated as an independent investment. Importantly, Equation (32) does not require specific knowledge about building construction, age, occupancy, solar gains, internal electric gains, or the overall thermal conductivity of the building. Only the characteristics of the portion of the building that is being replaced, the efficiency of the HVAC system, the indoor temperature (as reflected by the thermostat setting), the outdoor temperature (based on location), and the length of the winter season are required; knowledge about the rest of the building is not required. This simplification is a powerful and useful result. 
     Fourth, Equation (33) provides an approach to assessing the impact of a change in indoor temperature, and thus the effect of making a change in thermostat setting. Note that Equation (31) only depends upon the overall efficiency of the building, that is, the building&#39;s total thermal conductivity UA Total , the length of the winter season (in number of hours or other time units), and the HVAC system efficiency; Equation (31) does not depend upon either the indoor or outdoor temperature. 
     Equation (31) is useful in assessing claims that are made by HVAC management devices, such as the Nest thermostat device, manufactured by Nest Labs, Inc., Palo Alto, Calif., or the Lyric thermostat device, manufactured by Honeywell Int&#39;l Inc., Morristown, N.J., or other so-called “smart” thermostat devices. The fundamental idea behind these types of HVAC management devices is to learn behavioral patterns, so that consumers can effectively lower (or raise) their average indoor temperatures in the winter (or summer) months without affecting their personal comfort. Here, Equation (31) could be used to estimate the value of heating and cooling savings, as well as to verify the consumer behaviors implied by the new temperature settings. 
     Balance Point Temperature 
     Before leaving this section, balance point temperature should briefly be discussed. The formulation in this first approach does not involve balance point temperature as an input. A balance point temperature, however, can be calculated to equal the point at which there is no fuel consumption, such that there are no gains associated with auxiliary heating (Q Gains-Auxiliary Heating  equals 0) and the auxiliary heating thermal conductivity (UA Auxiliary Heating  in Equation (28)) is zero. Inserting these assumptions into Equation (18) and labeling T Outdoor  as T Balance Point  yields:
 
 Q   Gains-Internal   =UA   Total (   T     Indoor   − T     Balance Point )( H )  (36)
 
     Equation (36) simplifies to: 
     
       
         
           
             
               
                 T 
                 ¯ 
               
               
                 Balance 
                 ⁢ 
                     
                 Point 
               
             
             = 
             
               
                 
                   T 
                   ¯ 
                 
                 Indoor 
               
               - 
               
                 
                   
                     Q 
                     ¯ 
                   
                   
                     Gains 
                     - 
                     Internal 
                   
                 
                 
                   UA 
                   Total 
                 
               
             
           
         
       
     
     where 
     
       
         
           
             
               
                 
                   
                     
                       Q 
                       ¯ 
                     
                     
                       Gains 
                       - 
                       Internal 
                     
                   
                   = 
                   
                     
                       Q 
                       
                         Gains 
                         - 
                         Internal 
                       
                     
                     H 
                   
                 
               
               
                 
                   ( 
                   37 
                   ) 
                 
               
             
           
         
       
     
     Equation (37) is identical to Equation (2), except that average values are used for indoor temperature  T   Indoor , balance point temperature  T   Balance Point , and fuel consumption for internal heating gains  Q   Gains-Internal , and that heating gains from occupancy (Q Gains-Occupants ), electric (Q Gains-Electric ), and solar (Q Gains-Solar ) are all included as part of internal heating gains (Q Gains-Internal ). 
     Application: Change in Thermal Conductivity Associated with One Investment 
     An approach to calculating a new value for total thermal conductivity    Total  after a series of M changes (or investments) are made to a building is described in commonly-assigned U.S. Patent application, entitled “System and Method for Interactively Evaluating Personal Energy-Related Investments,” Ser. No. 14/294,079, filed Jun. 2, 2014, pending, the disclosure of which is incorporated by reference. The approach is summarized therein in Equation (41), which provides: 
                       UA   ^     Total     =       UA   Total     +       ∑     j   =   1     M         (       U   j     -       U   ^     j       )     ⁢     A   j         +     ρ   ⁢     c   ⁡   (     n   -     n   ˆ       )     ⁢   V               (   38   )               
where a caret symbol ({circumflex over ( )}) denotes a new value, infiltration losses are based on the density of air (ρ), specific heat of air (c), number of air changes per hour (n), and volume of air per air change (V). In addition, U j  and Û j  respectively represent the existing and proposed U-values of surface j, and A j  represents the surface area of surface j. The volume of the building V can be approximated by multiplying building square footage by average ceiling height. The equation, with a slight restatement, equals:
 
   Total   =UA   Total   +ΔUA   Total   (39)
 
     
       
         
           
             
               
                 
                   
                     Δ 
                     ⁢ 
                     
                       UA 
                       Total 
                     
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           1 
                         
                         M 
                       
                       
                         
                           ( 
                           
                             
                               U 
                               j 
                             
                             - 
                               
                             
                               
                                 Û 
                                   
                               
                               j 
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           A 
                           j 
                         
                       
                     
                     + 
                     
                       ρ 
                       ⁢ 
                       
                         c 
                         ⁡ 
                         ( 
                         
                           n 
                           - 
                           
                             n 
                             ˆ 
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         V 
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   40 
                   ) 
                 
               
             
           
         
       
     
     If there is only one investment, the m superscripts can be dropped and the change in thermal conductivity UA Total  equals the area (A) times the difference of the inverse of the old and new R-values R and  R : 
     
       
         
           
             
               
                 
                   
                     Δ 
                     ⁢ 
                     
                       UA 
                       Total 
                     
                   
                   = 
                   
                     
                       A 
                       ⁡ 
                       ( 
                       
                         U 
                         - 
                         
                           U 
                           ^ 
                         
                       
                       ) 
                     
                     = 
                     
                       
                         A 
                         ⁡ 
                         ( 
                         
                           
                             1 
                             R 
                           
                           - 
                           
                             1 
                             
                               R 
                               ˆ 
                             
                           
                         
                         ) 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   41 
                   ) 
                 
               
             
           
         
       
     
     Fuel Savings 
     The fuel savings associated with a change in thermal conductivity UA Total  for a single investment equals Equation (41) times (32): 
                     Δ   ⁢     Q   Fuel       =       Δ   ⁢     UA   Total     ⁢       dQ   Fuel       dUA   Total         =       A   ⁡   (       1   R     -     1     R   ˆ         )     ⁢         (         T   ¯     Indoor     -       T   ¯     Outdoor       )     ⁢     (   H   )         η   HVAC                   (   42   )               
where ΔQ Fuel  signifies the change in fuel consumption.
 
     Economic Value 
     The economic value of the fuel savings (Annual Savings) equals the fuel savings times the average fuel price (Price) for the building in question: 
     
       
         
           
             
               Annual 
               ⁢ 
                   
               Savings 
             
             = 
             
               
                 A 
                 ⁡ 
                 ( 
                 
                   
                     1 
                     R 
                   
                   - 
                   
                     1 
                     
                       R 
                       ˆ 
                     
                   
                 
                 ) 
               
               ⁢ 
               
                 
                   
                     ( 
                     
                       
                         
                           T 
                           ¯ 
                         
                         Indoor 
                       
                       - 
                       
                         
                           T 
                           ¯ 
                         
                         Outdoor 
                       
                     
                     ) 
                   
                   ⁢ 
                   
                     ( 
                     H 
                     ) 
                   
                 
                 
                   η 
                   HVAC 
                 
               
               ⁢ 
               
                 ( 
                 Price 
                 ) 
               
             
           
         
       
     
     where 
                   Price   =     {               Price   NG       1   ⁢     0   5         ⁢         if   ⁢         price   ⁢         has   ⁢         units   ⁢         of   ⁢         $   ⁢         per   ⁢         therm                   Price   Electrity       3   ,   412       ⁢         if   ⁢         price   ⁢         has   ⁢         units   ⁢         of   ⁢         $   ⁢         per   ⁢       kWh                     (   43   )               
where Price NG  represents the price of natural gas and Price Electricity  represents the price of electricity.
 
     Example 
     Consider an example. A consumer in Napa, Calif. wants to calculate the annual savings associating with replacing a 20 ft 2  single-pane window that has an R-value of 1 with a high efficiency window that has an R-value of 4. The average temperature in Napa over the 183-day winter period (4,392 hours) from October 1 to March 31 is 50° F. The consumer sets his thermostat at 68° F., has a 60 percent efficient natural gas heating system, and pays $1 per therm for natural gas. How much money will the consumer save per year by making this change? 
     Putting this information into Equation (43) suggests that he will save $20 per year: 
     
       
         
           
             
               
                 
                   
                     Annual 
                     ⁢ 
                          
                     Savings 
                   
                   = 
                   
                     
                       20 
                       ⁢ 
                       
                         ( 
                         
                           
                             1 
                             1 
                           
                           - 
                           
                             1 
                             4 
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         
                           
                             ( 
                             
                               
                                 6 
                                 ⁢ 
                                 8 
                               
                               - 
                               
                                 5 
                                 ⁢ 
                                 0 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             ( 
                             
                               4 
                               , 
                               392 
                             
                             ) 
                           
                         
                         
                           0 
                           . 
                           6 
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           1 
                           
                             1 
                             ⁢ 
                             
                               0 
                               5 
                             
                           
                         
                         ) 
                       
                     
                     = 
                     
                       $ 
                       ⁢ 
                       2 
                       ⁢ 
                       0 
                     
                   
                 
               
               
                 
                   ( 
                   44 
                   ) 
                 
               
             
           
         
       
     
     Application: Validate Building Shell Improvements Savings 
     Many energy efficiency programs operated by power utilities grapple with the issue of measurement and evaluation (M&amp;E), particularly with respect to determining whether savings have occurred after building shell improvements were made. Equations (25) through (28) can be applied to help address this issue. These equations can be used to calculate a building&#39;s total thermal conductivity UA Total . This result provides an empirical approach to validating the benefits of building shell investments using measured data. 
     Equations (25) through (28) require the following inputs: 
     1) Weather:
         a) Average outdoor temperature (° F.).   b) Average indoor temperature (° F.).   c) Average direct solar resource on a vertical, south-facing surface.       

     2) Fuel and energy:
         a) Average gross indoor electricity consumption.   b) Average natural gas fuel consumption for space heating.   c) Average electric fuel consumption for space heating.       

     3) Other inputs:
         a) Average number of occupants.   b) Effective window area.   c) HVAC system efficiency.       

     Weather data can be determined as follows. Indoor temperature can be assumed based on the setting of the thermostat (assuming that the thermostat&#39;s setting remained constant throughout the time period), or measured and recorded using a device that takes hourly or periodic indoor temperature measurements, such as a Nest thermostat device or a Lyric thermostat device, cited supra, or other so-called “smart” thermostat devices. Outdoor temperature and solar resource data can be obtained from a service, such as Solar Anywhere SystemCheck, cited supra, or the National Weather Service. Other sources of weather data are possible. 
     Fuel and energy data can be determined as follows. Monthly utility billing records provide natural gas consumption and net electricity data. Gross indoor electricity consumption can be calculated by adding PV production, whether simulated using, for instance, the Solar Anywhere SystemCheck service, cited supra, or measured directly, and subtracting out external electricity consumption, that is, electricity consumption for electric devices that do not deliver all heat that is generated into the interior of the building. External electricity consumption includes electric vehicle (EV) charging and electric water heating. Other types of external electricity consumption are possible. Natural gas consumption for heating purposes can be estimated by subtracting non-space heating consumption, which can be estimated, for instance, by examining summer time consumption using an approach described in commonly-assigned U.S. Patent application, entitled “System and Method for Facilitating Implementation of Holistic Zero Net Energy Consumption,” Ser. No. 14/531,933, filed Nov. 3, 2014, pending, the disclosure of which is incorporated by reference. Other sources of fuel and energy data are possible. 
     Finally, the other inputs can be determined as follows. The average number of occupants can be estimated by the building owner or occupant. Effective window area can be estimated by multiplying actual south-facing window area times solar heat gain coefficient (estimated or based on empirical tests, as further described infra), and HVAC system efficiency can be estimated (by multiplying reported furnace rating times either estimated or actual duct system efficiency), or can be based on empirical tests, as further described infra. Other sources of data for the other inputs are possible. 
     Consider an example.  FIG.  7    is a table  80  showing, by way of example, data used to calculate thermal conductivity. The data inputs are for a sample house in Napa, Calif. based on the winter period of October 1 to March 31 for six winter seasons, plus results for a seventh winter season after many building shell investments were made. (Note the building improvements facilitated a substantial increase in the average indoor temperature by preventing a major drop in temperature during night-time and non-occupied hours.) South-facing windows had an effective area of 10 m 2  and the solar heat gain coefficient is estimated to be 0.25 for an effective window area of 2.5 m 2 . The measured HVAC system efficiency of 59 percent was based on a reported furnace efficiency of 80 percent and an energy audit-based duct efficiency of 74 percent. 
       FIG.  8    is a table  90  showing, by way of example, thermal conductivity results for each season using the data in the table  80  of  FIG.  7    as inputs into Equations (25) through (28). Thermal conductivity is in units of Btu/h-° F.  FIG.  9    is a graph  100  showing, by way of example, a plot of the thermal conductivity results in the table  90  of  FIG.  8   . The x-axis represents winter seasons for successive years, each winter season running from October 1 to March 31. The y-axis represents thermal conductivity. The results from a detailed energy audit, performed in early 2014, are superimposed on the graph. The energy audit determined that the house had a thermal conductivity of 773 Btu/h-° F. The average result estimated for the first six seasons was 791 Btu/h-° F. A major amount of building shell work was performed after the 2013-2014 winter season, and the results show a 50-percent reduction in heating energy consumption in the 2014-2015 winter season. 
     Application: Evaluate Investment Alternatives 
     The results of this work can be used to evaluate potential investment alternatives.  FIG.  10    is a graph  110  showing, by way of example, an auxiliary heating energy analysis and energy consumption investment options. The x-axis represents total thermal conductivity, UA Total  in units of Btu/hr-° F. The y-axis represents total heating energy. The graph presents the analysis of the Napa, Calif. building from the earlier example, supra, using the equations previously discussed. The three lowest horizontal bands correspond to the heat provided through internal gains  111 , including occupants, heat produced by operating electric devices, and solar heating. The solid circle  112  represents the initial situation with respect to heating energy consumption. The diagonal lines  113   a ,  113   b ,  113   c  represent three alternative heating system efficiencies versus thermal conductivity (shown in the graph as building losses). The horizontal dashed line  114  represents an option to improve the building shell and the vertical dashed line  115  represents an option to switch to electric resistance heating. The plain circle  116  represents the final situation with respect to heating energy consumption. 
     Other energy consumption investment options (not depicted) are possible. These options include switching to an electric heat pump, increasing solar gain through window replacement or tree trimming (this option would increase the height of the area in the graph labeled “Solar Gains”), or lowering the thermostat setting. These options can be compared using the approach described with reference to Equations (25) through (28) to compare the options in terms of their costs and savings, which will help the homeowner to make a wiser investment. 
     Second Approach: Time Series Fuel Consumption 
     The previous section presented an annual fuel consumption model. This section presents a detailed time series model. This section also compares results from the two methods and provides an example of how to apply the on-site empirical tests. 
     Building-Specific Parameters 
     The building temperature model used in this second approach requires three building parameters: (1) thermal mass; (2) thermal conductivity; and (3) effective window area.  FIG.  11    is a functional block diagram showing thermal mass, thermal conductivity, and effective window area relative to a structure  121 . By way of introduction, these parameters will now be discussed. 
     Thermal Mass (M) 
     The heat capacity of an object equals the ratio of the amount of heat energy transferred to the object and the resulting change in the object&#39;s temperature. Heat capacity is also known as “thermal capacitance” or “thermal mass” ( 122 ) when used in reference to a building. Thermal mass Q is a property of the mass of a building that enables the building to store heat, thereby providing “inertia” against temperature fluctuations. A building gains thermal mass through the use of building materials with high specific heat capacity and high density, such as concrete, brick, and stone. 
     The heat capacity is assumed to be constant when the temperature range is sufficiently small. Mathematically, this relationship can be expressed as:
 
 Q   Δt   =M ( T   t+Δt   Indoor   −T   t   Indoor )  (45)
 
where M equals the thermal mass of the building and temperature units T are in ° F. Q is typically expressed in Btu or Joules. In that case, M has units of Btu/° F. Q can also be divided by 1 kWh/3,412 Btu to convert to units of kWh/° F.
 
     Thermal Conductivity (UA Total ) 
     The building&#39;s thermal conductivity UA Total  ( 123 ) is the amount of heat that the building gains or losses as a result of conduction and infiltration. Thermal conductivity UA Total  was discussed supra with reference to the first approach for modeling annual heating fuel consumption. 
     Effective Window Area (W) 
     The effective window area (in units of m 2 ) ( 124 ), also discussed in detail supra, specifies how much of an available solar resource is absorbed by the building. Effective window area is the dominant means of solar gain in a typical building during the winter and includes the effect of physical shading, window orientation, and the window&#39;s solar heat gain coefficient. In the northern hemisphere, the effective window area is multiplied by the available average direct irradiance on a vertical, south-facing surface (kW/m 2 ), times the amount of time (II) to result in the kWh obtained from the windows. 
     Energy Gain or Loss 
     The amount of heat transferred to or extracted from a building (Q) over a time period of Δt is based on a number of factors, including:
         1) Loss (or gain if outdoor temperature exceeds indoor temperature) due to conduction and infiltration and the differential between the indoor and outdoor temperatures.   2) Gain associated with:
           a) Occupancy and heat given off by people.   b) Heat produced by consuming electricity inside the building.   c) Solar radiation.   d) Auxiliary heating.   
               

     Mathematically, Q can be expressed as: 
                     Q     Δ   ⁢   t       =       [             UA   Total     (         T   ¯     Indoor     -       T   ¯     Outdoor       )     ︷       Envelope   ⁢         Gain   ⁢         or   ⁢         Loss       +           (     2   ⁢   5   ⁢   0     )     ⁢     P   ¯       ︷       Occupancy   ⁢         Gain       +           Electric   _     (       3   ,   412   ⁢         Btu       1   ⁢         kWh       )     ︷       Internal   ⁢         Electric   ⁢         Gain       +         W   ⁢     Solar   _     ⁢           (       3   ,   412   ⁢         Btu       1   ⁢         kWh       )       ︷       Solar   ⁢         Gain       +           R   Furnace     ⁢     η   HVAC     ⁢           Status   _       ︷       Auxliary   ⁢         Heating   ⁢         Gain         ]     ⁢   Δ   ⁢   t             (   46   )               
where:
         Except as noted otherwise, the bars over the variable names represent the average value over Δt hours, that is, the duration of the applicable empirical test. For instance,  T   Outdoor  represents the average outdoor temperature between the time interval oft and t+Δt.   UA Total  is the thermal conductivity (in units of Btu/hour-° F.).   W is the effective window area (in units of m 2 ).   Occupancy Gain is based on the average number of people ( P ) in the building during the applicable empirical test (and the heat produced by those people). The average person is assumed to produce 250 Btu/hour.   Internal Electric Gain is based on heat produced by indoor electricity consumption ( Electric ), as averaged over the applicable empirical test, but excludes electricity for purposes that do not produce heat inside the building, for instance, electric hot water heating where the hot water is discarded down the drain, or where there is no heat produced inside the building, such as is the case with EV charging.   Solar Gain is based on the average available normalized solar irradiance ( Solar ) during the applicable empirical test (with units of kW/m 2 ). This value is the irradiance on a vertical surface to estimate solar received on windows; global horizontal irradiance (GHI) can be used as a proxy for this number when W is allowed to change on a monthly basis.   Auxiliary Heating Gain is based on the rating of the furnace (R in Btu), HVAC system efficiency (η HVAC , including both furnace and delivery system efficiency), and average furnace operation status ( Status ) during the empirical test, a time series value that is either off (0 percent) or on (100 percent).       

     Energy Balance 
     Equation (45) reflects the change in energy over a time period and equals the product of the temperature change and the building&#39;s thermal mass. Equation (46) reflects the net gain in energy over a time period associated with the various component sources. Equation (45) can be set to equal Equation (46), since the results of both equations equal the same quantity and have the same units (Btu). Thus, the total heat change of a building will equal the sum of the individual heat gain/loss components: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               M 
                               ⁡ 
                               ( 
                               
                                 
                                   T 
                                   
                                     t 
                                     + 
                                     
                                       Δ 
                                       ⁢ 
                                       t 
                                     
                                   
                                   Indoor 
                                 
                                 - 
                                 
                                   T 
                                   t 
                                   Indoor 
                                 
                               
                               ) 
                             
                             ︷ 
                           
                           
                             Total 
                             ⁢ 
                                 
                             Heat 
                             ⁢ 
                                 
                             Cha 
                             ⁢ 
                             nge 
                           
                         
                         = 
                           
                         
                           [ 
                           
                             
                               
                                 
                                   UA 
                                   
                                     Total 
                                     ⁡ 
                                     ( 
                                     
                                       
                                         
                                           T 
                                           _ 
                                         
                                         Indoor 
                                       
                                       - 
                                       
                                         
                                           T 
                                           ¯ 
                                         
                                         Outdoor 
                                       
                                     
                                     ) 
                                   
                                 
                                 ︷ 
                               
                               
                                 Envelop 
                                 ⁢ 
                                     
                                 Gain 
                                 ⁢ 
                                     
                                 or 
                                 ⁢ 
                                     
                                 Loss 
                               
                             
                             + 
                             
                               
                                 
                                   
                                     ( 
                                     250 
                                     ) 
                                   
                                   ⁢ 
                                   
                                     P 
                                     ¯ 
                                   
                                 
                                 ︷ 
                               
                               
                                 Occupancy 
                                 ⁢ 
                                     
                                 Gain 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           + 
                             
                           
                             
                               
                                 
                                   
                                     Electric 
                                     _ 
                                   
                                   ⁢ 
                                       
                                   
                                     ( 
                                     
                                       
                                         3 
                                         , 
                                         412 
                                         ⁢ 
                                             
                                         Btu 
                                       
                                       
                                         1 
                                         ⁢ 
                                             
                                         kWh 
                                       
                                     
                                     ) 
                                   
                                 
                                 ︷ 
                               
                               + 
                             
                             
                               Internal 
                               ⁢ 
                                   
                               Electric 
                               ⁢ 
                                   
                               Gain 
                             
                           
                         
                         ⁢ 
                         
                           
                             
                               W 
                               ⁢ 
                               
                                 Solar 
                                 _ 
                               
                               ⁢ 
                                   
                               
                                 ( 
                                 
                                   
                                     3 
                                     , 
                                     412 
                                     ⁢ 
                                         
                                     Btu 
                                   
                                   
                                     1 
                                     ⁢ 
                                         
                                     kWh 
                                   
                                 
                                 ) 
                               
                             
                             ︷ 
                           
                           
                             Solar 
                             ⁢ 
                                 
                             Gain 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             + 
                               
                             
                               
                                 
                                   
                                     R 
                                     Furnace 
                                   
                                   ⁢ 
                                   
                                     η 
                                     HVAC 
                                   
                                   ⁢ 
                                       
                                   
                                     Status 
                                     _ 
                                   
                                 
                                 ︷ 
                               
                               
                                 Auxliary 
                                 ⁢ 
                                     
                                 Heating 
                                 ⁢ 
                                     
                                 Gain 
                               
                             
                           
                           ] 
                         
                         ⁢ 
                         Δ 
                         ⁢ 
                         t 
                       
                     
                   
                 
               
               
                 
                   ( 
                   47 
                   ) 
                 
               
             
           
         
       
     
     Equation (47) can be used for several purposes.  FIG.  12    is a flow diagram showing a computer-implemented method  130  for modeling interval building heating energy consumption in accordance with a further embodiment. Execution of the software can be performed with the assistance of a computer system, such as further described infra with reference to  FIG.  22   , as a series of process or method modules or steps. 
     As a single equation, Equation (47) is potentially very useful, despite having five unknown parameters. In this second approach, the unknown parameters are solved by performing a series of short duration empirical tests (step  131 ), as further described infra with reference to  FIG.  14   . Once the values of the unknown parameters are found, a time series of indoor temperature data can be constructed (step  132 ), which will then allow annual fuel consumption to be calculated (step  133 ) and maximum indoor temperature to be found (step  134 ). The short duration tests will first be discussed. 
     Empirically Determine Building- and Equipment-Specific Parameters Using Short Duration Tests 
     A series of tests can be used to iteratively solve Equation (47) to obtain the values of the unknown parameters by ensuring that the portions of Equation (47) with the unknown parameters are equal to zero.  FIG.  13    is a table  140  showing the characteristics of empirical tests used to solve for the five unknown parameters in Equation (47). The empirical test characteristics are used in a series of sequentially-performed short duration tests; each test builds on the findings of earlier tests to replace unknown parameters with found values. 
       FIG.  14    is a flow diagram showing a routine  150  for empirically determining building- and equipment-specific parameters using short duration tests for use in the method  130  of  FIG.  12   . The approach is to run a serialized series of empirical tests. The first test solves for the building&#39;s total thermal conductivity (UA Total ) (step  151 ). The second test uses the empirically-derived value for UA Total  to solve for the building&#39;s thermal mass (M) (step  152 ). The third test uses both of these results, thermal conductivity and thermal mass, to find the building&#39;s effective window area (W) (step  153 ). Finally, the fourth test uses the previous three test results to determine the overall HVAC system efficiency (step  145 ). Consider how to perform each of these tests. 
     Test 1: Building Thermal Conductivity (UA Total ) 
     The first step is to find the building&#39;s total thermal conductivity (UA Total ) (step  151 ). Referring back to the table in  FIG.  13   , this short-duration test occurs at night (to avoid any solar gain) with the HVAC system off (to avoid any gain from the HVAC system), and by having the indoor temperature the same at the beginning and the ending of the test by operating an electric controllable interior heat source, such as portable electric space heaters that operate at 100% efficiency, so that there is no change in the building temperature&#39;s at the beginning and at the ending of the test. Thus, the interior heart source must have sufficient heating capacity to maintain the building&#39;s temperature state. Ideally, the indoor temperature would also remain constant to avoid any potential concerns with thermal time lags. 
     These assumptions are input into Equation (47): 
     
       
         
           
             
               
                 
                   
                     M 
                     ⁡ 
                     ( 
                     0 
                     ) 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             UA 
                             Total 
                           
                           ( 
                           
                             
                               
                                 T 
                                 ¯ 
                               
                               Indoor 
                             
                             - 
                             
                               
                                 T 
                                 ¯ 
                               
                               Outdoor 
                             
                           
                           ) 
                         
                         + 
                         
                           
                             ( 
                             
                               2 
                               ⁢ 
                               5 
                               ⁢ 
                               0 
                             
                             ) 
                           
                           ⁢ 
                           
                             P 
                             ¯ 
                           
                         
                         + 
                         
                           
                             Electric 
                             _ 
                           
                           ⁢ 
                               
                           
                             ( 
                             
                               
                                 3 
                                 , 
                                 412 
                                 ⁢ 
                                     
                                 Btu 
                               
                               
                                 1 
                                 ⁢ 
                                     
                                 kWh 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           
                             W 
                             ⁡ 
                             ( 
                             0 
                             ) 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 3 
                                 , 
                                 412 
                                 ⁢ 
                                     
                                 Btu 
                               
                               
                                 1 
                                 ⁢ 
                                     
                                 kWh 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           
                             R 
                             Furnace 
                           
                           ⁢ 
                           
                             
                               η 
                               HVAC 
                             
                             ( 
                             0 
                             ) 
                           
                         
                       
                       ] 
                     
                     ⁢ 
                     Δ 
                     ⁢ 
                     t 
                   
                 
               
               
                 
                   ( 
                   48 
                   ) 
                 
               
             
           
         
       
     
     The portions of Equation (48) that contain four of the five unknown parameters now reduce to zero. The result can be solved for UA Total : 
                     UA   Total     =       [         (     2   ⁢   5   ⁢   0     )     ⁢     P   ¯       +       Electric   _     ⁢           (       3   ,   412   ⁢         Btu       1   ⁢         kWh       )         ]       (         T   ¯     Indoor     -       T   ¯     Outdoor       )               (   49   )               
where  T   Indoor  represents the average indoor temperature during the empirical test,  T   Outdoor  represents the average outdoor temperature during the empirical test,  P  represents the average number of occupants during the empirical test, and  Electric  represents average indoor electricity consumption during the empirical test.
 
     Equation (49) implies that the building&#39;s thermal conductivity can be determined from this test based on average number of occupants, average power consumption, average indoor temperature, and average outdoor temperature. 
     Test 2: Building Thermal Mass (M) 
     The second step is to find the building&#39;s thermal mass (M) (step  152 ). This step is accomplished by constructing a test that guarantees M is specifically non-zero since UA Total  is known based on the results of the first test. This second test is also run at night, so that there is no solar gain, which also guarantees that the starting and the ending indoor temperatures are not the same, that is, T t+Δt   Indoor ≠T t   Indoor , respectively at the outset and conclusion of the test by not operating the HVAC system. These assumptions are input into Equation (47) and solving yields a solution for M: 
                   M   =       [           UA   Total     (         T   ¯     Indoor     -       T   ¯     Outdoor       )     +       (     2   ⁢   5   ⁢   0     )     ⁢     P   ¯       +       Electric   _     ⁢           (       3   ,   412   ⁢         Btu       1   ⁢         kWh       )           (       T     t   +     Δ   ⁢   t       Indoor     -     T   t   Indoor       )       ]     ⁢   Δ   ⁢   t             (   50   )               
where UA Total  represents the thermal conductivity, T Indoor  represents the average indoor temperature during the empirical test,  T   Outdoor  represents the average outdoor temperature during the empirical test,  P  represents the average number of occupants during the empirical test,  Electric  represents average indoor electricity consumption during the empirical test, t represents the time at the beginning of the empirical test, Δ t represents the duration of the empirical test, T t+Δt   Indoor  represents the ending indoor temperature, T t   Indoor  represents the starting indoor temperature, and T t+Δt   Indoor ≠/T t   Indoor .
 
     Test 3: Building Effective Window Area (W) 
     The third step to find the building&#39;s effective window area (W) (step  153 ) requires constructing a test that guarantees that solar gain is non-zero. This test is performed during the day with the HVAC system turned off. Solving for W yields: 
                   W   =       {       [       M   ⁡   (       T     t   +     Δ   ⁢   t       Indoor     -     T   t   Indoor       )       3   ,   412   ⁢   Δ   ⁢   t       ]     +         UA   Total     (         T   ¯     Indoor     -       T   ¯     Outdoor       )       3   ,   412       -         (     2   ⁢   5   ⁢   0     )     ⁢     P   ¯         3   ,   412       -     Electric   _       }     [     1     Solar   _       ]             (   51   )               
where M represents the thermal mass, t represents the time at the beginning of the empirical test, Δt represents the duration of the empirical test, T t+Δt   Indoor  represents the ending indoor temperature, and T t   Indoor  represents the starting indoor temperature, UA Total  represents the thermal conductivity,  T   Indoor  represents the average indoor temperature,  T   Outdoor  represents the average outdoor temperature,  P  represents the average number of occupants during the empirical test,  Electric  represents average electricity consumption during the empirical test, and  Solar  represents the average solar energy produced during the empirical test.
 
     Test 4: HVAC System Efficiency (η Furnace η Delivery ) 
     The fourth step determines the HVAC system efficiency (step  154 ). Total HVAC system efficiency is the product of the furnace efficiency and the efficiency of the delivery system, that is, the duct work and heat distribution system. While these two terms are often solved separately, the product of the two terms is most relevant to building temperature modeling. This test is best performed at night, so as to eliminate solar gain. Thus: 
                     η   HVAC     =       [         M   ⁡   (       T     t   +     Δ   ⁢   t       Indoor     -     T   t   Indoor       )       Δ   ⁢   t       -       UA   Total     (         T   ¯     Indoor     -       T   ¯     Outdoor       )     -       (     2   ⁢   5   ⁢   0     )     ⁢     P   ¯       -       Electric   _     (       3   ,   412   ⁢         Btu       1   ⁢         kWh       )       ]     [     1       R   Furnace     ⁢     Status   _         ]             (   52   )               
where M represents the thermal mass, t represents the time at the beginning of the empirical test, Δt represents the duration of the empirical test, T t+Δt   Indoor  represents the ending indoor temperature, and T t   Indoor  represents the starting indoor temperature, UA Total  represents the thermal conductivity,  T   Indoor  represents the average indoor temperature, {circumflex over (T)} Outdoor  represents the average outdoor temperature,  P  represents the average number of occupants during the empirical test,  Electric  represents average electricity consumption during the empirical test,  Status  represents the average furnace operation status, and R Furnace  represents the rating of the furnace.
 
     Note that HVAC duct efficiency can be determined without performing a duct leakage test if the generation efficiency of the furnace is known. This observation usefully provides an empirical method to measure duct efficiency without having to perform a duct leakage test. 
     Time Series Indoor Temperature Data 
     The previous subsection described how to perform a series of empirical short duration tests to determine the unknown parameters in Equation (47). Commonly-assigned U.S. patent application Ser. No. 14/531,933, cited supra, describes how a building&#39;s UA Total  can be combined with historical fuel consumption data to estimate the benefit of improvements to a building. While useful, estimating the benefit requires measured time series fuel consumption and HVAC system efficiency data. Equation (47), though, can be used to perform the same analysis without the need for historical fuel consumption data. 
     Referring back to  FIG.  12   , Equation (47) can be used to construct time series indoor temperature data (step  132 ) by making an approximation. Let the time period (Δt) be short (an hour or less), so that the average values are approximately equal to the value at the beginning of the time period, that is, assume  T Outdoor   ≈T t   Outdoor . The average values in Equation (47) can be replaced with time-specific subscripted values and solved to yield the final indoor temperature. 
                     T     t   +     Δ   ⁢   t       Indoor     =       T   t   Indoor     +         [     1   M     ]     [         UA   Total     (       T   t   Outdoor     -     T   t   Indoor       )     +       (     2   ⁢   5   ⁢   0     )     ⁢     P   t       +       Electric   t     ⁢     (       3   ,   412   ⁢         Btu       1   ⁢         kWh       )       +     W   ⁢            Solar   t     (       3   ,   412   ⁢         Btu       1   ⁢         kWh       )       +       R   Furnace     ⁢     η   HVAC     ⁢     Status   t         ]     ⁢   Δ   ⁢   t               (   53   )               
Once T t+Δt   Indoor  is known, Equation (53) can be used to solve for T t+2Δt   Indoor  and so on.
 
     Importantly, Equation (53) can be used to iteratively construct indoor building temperature time series data with no specific information about the building&#39;s construction, age, configuration, number of stories, and so forth. Equation (53) only requires general weather datasets (outdoor temperature and irradiance) and building-specific parameters. The control variable in Equation (53) is the fuel required to deliver the auxiliary heat at time t, as represented in the Status variable, that is, at each time increment, a decision is made whether to run the HVAC system. 
     Annual Fuel Consumption 
     Equation (47) can also be used to calculate annual fuel consumption (step  133 ) by letting Δt equal the number of hours in the entire time period (H) (and not the duration of the applicable empirical test), rather than making Δt very short (such as an hour, as used in an applicable empirical test). The indoor temperature at the start and the end of the season can be assumed to be the same or, alternatively, the total heat change term on the left side of the equation can be assumed to be very small and set equal to zero. Rearranging the equation provides: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             R 
                             Furnace 
                           
                           ⁢ 
                           
                             η 
                             HVAC 
                           
                           ⁢ 
                           
                             
                               Status 
                               _ 
                             
                             ( 
                             H 
                             ) 
                           
                         
                         = 
                           
                         
                           
                             
                               UA 
                               Total 
                             
                             ( 
                             
                               
                                 
                                   T 
                                   ¯ 
                                 
                                 Indoor 
                               
                               - 
                               
                                 
                                   T 
                                   ¯ 
                                 
                                 Outdoor 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             ( 
                             H 
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         - 
                           
                         
                           [ 
                           
                             
                               
                                 ( 
                                 250 
                                 ) 
                               
                               ⁢ 
                               
                                 P 
                                 ¯ 
                               
                             
                             + 
                             
                               
                                 Electric 
                                 _ 
                               
                               ( 
                               
                                 
                                   3 
                                   , 
                                   412 
                                   ⁢ 
                                       
                                   Btu 
                                 
                                 
                                   1 
                                   ⁢ 
                                       
                                   kWh 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             
                               + 
                                 
                               W 
                             
                             ⁢ 
                                
                             
                               
                                 Solar 
                                 _ 
                               
                               ( 
                               
                                 
                                   3 
                                   , 
                                   412 
                                   ⁢ 
                                       
                                   Btu 
                                 
                                 
                                   1 
                                   ⁢ 
                                       
                                   kWh 
                                 
                               
                               ) 
                             
                           
                           ] 
                         
                         ⁢ 
                         
                           ( 
                           H 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   54 
                   ) 
                 
               
             
           
         
       
     
     The left side of the equation, R Furnace η HVAC   Status (H), equals total fuel consumption. The second term in the right side of the equation equals the internal gains described in Equation (21). When Equation (21) is rearranged, UA Balance Point ( T   Indoor − T   Outdoor )(H)=Q Gains-Occupants +Q Gains-Electric +(Q Gains-Solar . After substituting into Equation (54), the total fuel consumption using the time series method produces a result that is identical to the one produced using the annual method described in Equation (31). 
     
       
         
           
             
               
                 
                   
                     Q 
                     Fuel 
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             UA 
                             Total 
                           
                           - 
                           
                             UA 
                             
                               Balance 
                               ⁢ 
                                   
                               Point 
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               T 
                               ¯ 
                             
                             Indoor 
                           
                           - 
                           
                             
                               T 
                               _ 
                             
                             Outdoor 
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         ( 
                         H 
                         ) 
                       
                     
                     
                       η 
                       HVAC 
                     
                   
                 
               
               
                 
                   ( 
                   55 
                   ) 
                 
               
             
           
         
       
     
     Maximum Indoor Temperature 
     Allowing consumers to limit the maximum indoor temperature to some value can be useful from a personal physical comfort perspective. The limit of maximum indoor temperature (step  134 ) can be obtained by taking the minimum of T t+Δt   Indoor  and T Indoor-Max , the maximum indoor temperature recorded for the building during the heating season. There can be some divergence between the annual and detailed time series methods when the thermal mass of the building is unable to absorb excess heat, which can then be used at a later time. Equation (53) becomes Equation (56) when the minimum is applied. 
                     T     t   +     Δ   ⁢   t       Indoor     =     Min   ⁢     {       T     Indoor   -   Max       ,       T   t   Indoor     +                   (   56   )                       [     1   M     ]     [         UA   Total     (       T   t   Outdoor     -     T   t   Indoor       )     +       (   250   )     ⁢     P   t       +                           Electric   t     ⁢     (       3   ,   412   ⁢         Btu       1   ⁢         kWh       )       +     W   ⁢            Solar   t     (       3   ,   412   ⁢         Btu       1   ⁢         kWh       )       +       R   Furnace     ⁢     η   HVAC     ⁢     Status   t         ]     ⁢   Δ   ⁢   t     }         
Comparison to Annual Method (First Approach)
 
     Two different approaches to calculating annual fuel consumption are described herein. The first approach, per Equation (31), is a single-line equation that requires six inputs. The second approach, per Equation (56), constructs a time series dataset of indoor temperature and HVAC system status. The second approach considers all of the parameters that are indirectly incorporated into the first approach. The second approach also includes the building&#39;s thermal mass and the specified maximum indoor temperature, and requires hourly time series data for the following variables: outdoor temperature, solar resource, non-HVAC electricity consumption, and occupancy. 
     Both approaches were applied to the exemplary case, discussed supra, for the sample house in Napa, Calif. Thermal mass was 13,648 Btu/° F. and the maximum temperature was set at 72° F. The auxiliary heating energy requirements predicted by the two approaches was then compared.  FIG.  15    is a graph  160  showing, by way of example, a comparison of auxiliary heating energy requirements determined by the hourly approach versus the annual approach. The x-axis represents total thermal conductivity, UA Total  in units of Btu/hr-° F. They-axis represents total heating energy.  FIG.  15    uses the same format as the graph in  FIG.  10    by applying a range of scenarios. The red line in the graph corresponds to the results of the hourly method. The dashed black line in the graph corresponds to the annual method. The graph suggests that results are essentially identical, except when the building losses are very low and some of the internal gains are lost due to house overheating, which is prevented in the hourly method, but not in the annual method. 
     The analysis was repeated using a range of scenarios with similar results.  FIG.  16    is a graph  170  showing, by way of example, a comparison of auxiliary heating energy requirements with the allowable indoor temperature limited to 2° F. above desired temperature of 68° F. Here, the only cases that found any meaningful divergence occurred when the maximum house temperature was very close to the desired indoor temperature.  FIG.  17    is a graph  180  showing, by way of example, a comparison of auxiliary heating energy requirements with the size of effective window area tripled from 2.5 m 2  to 7.5 m 2 . Here, internal gains were large by tripling solar gains and there was insufficient thermal mass to provide storage capacity to retain the gains. 
     The conclusion is that both approaches yield essentially identical results, except for cases when the house has inadequate thermal mass to retain internal gains (occupancy, electric, and solar). 
     Example 
     How to perform the tests described supra using measured data can be illustrated through an example. These tests were performed between 9 PM on Jan. 29, 2015 to 6 AM on Jan. 31, 2015 on a 35 year-old, 3,000 ft 2  house in Napa, Calif. This time period was selected to show that all of the tests could be performed in less than a day-and-a-half. In addition, the difference between indoor and outdoor temperatures was not extreme, making for a more challenging situation to accurately perform the tests. 
       FIG.  18    is a table  190  showing, by way of example, test data. The sub columns listed under “Data” present measured hourly indoor and outdoor temperatures, direct irradiance on a vertical south-facing surface (VDI), electricity consumption that resulted in indoor heat, and average occupancy. Electric space heaters were used to heat the house and the HVAC system was not operated. The first three short-duration tests, described supra, were applied to this data. The specific data used are highlighted in gray.  FIG.  19    is a table  200  showing, by way of example, the statistics performed on the data in the table  190  of  FIG.  18    required to calculate the three test parameters. UA Total  was calculated using the data in the table of  FIG.  10    and Equation (49). Thermal Mass (M) was calculated using UA Total , the data in the table of  FIG.  10   , and Equation (50). Effective Window Area (W) was calculated using UA Total , M, the data in the table of  FIG.  10   , and Equation (51). 
     These test parameters, plus a furnace rating of 100,000 Btu/hour and assumed efficiency of 56%, can be used to generate the end-of-period indoor temperature by substituting them into Equation (53) to yield: 
     
       
         
           
             
               
                 
                   
                     T 
                     
                       t 
                       + 
                       
                         Δ 
                         ⁢ 
                         t 
                       
                     
                     Indoor 
                   
                   = 
                   
                     
                       T 
                       t 
                       Indoor 
                     
                     + 
                     
                       
                         
                           [ 
                           
                             1 
                             
                               18 
                               , 
                               084 
                             
                           
                           ] 
                         
                         [ 
                         
                           
                             4 
                             ⁢ 
                             2 
                             ⁢ 
                             9 
                             ⁢ 
                             
                               ( 
                               
                                 
                                   T 
                                   t 
                                   Outdoor 
                                 
                                 - 
                                 
                                   T 
                                   t 
                                   Indoor 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               ( 
                               
                                 2 
                                 ⁢ 
                                 5 
                                 ⁢ 
                                 0 
                               
                               ) 
                             
                             ⁢ 
                             
                               P 
                               t 
                             
                           
                           + 
                           
                             3412 
                             ⁢ 
                                 
                             
                               Electric 
                               t 
                             
                           
                           + 
                           
                             11 
                             , 
                             600 
                             ⁢ 
                                 
                             
                               Solar 
                               t 
                             
                           
                           + 
                           
                             
                               ( 
                               
                                 100 
                                 , 
                                 000 
                               
                               ) 
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   0 
                                   . 
                                   5 
                                 
                                 ⁢ 
                                 6 
                               
                               ) 
                             
                             ⁢ 
                                 
                             
                               Status 
                               t 
                             
                           
                         
                         ] 
                       
                       ⁢ 
                       Δt 
                     
                   
                 
               
               
                 
                   ( 
                   57 
                   ) 
                 
               
             
           
         
       
     
     Indoor temperatures were simulated using Equation (57) and the required measured time series input datasets. Indoor temperature was measured from Dec. 15, 2014 to Jan. 31, 2015 for the test location in Napa, Calif. The temperatures were measured every minute on the first and second floors of the middle of the house and averaged.  FIG.  20    is a graph  210  showing, by way of example, hourly indoor (measured and simulated) and outdoor (measured) temperatures.  FIG.  21    is a graph  220  showing, by way of example, simulated versus measured hourly temperature delta (indoor minus outdoor).  FIG.  20    and  FIG.  21    suggest that the calibrated model is a good representation of actual temperatures. 
     Energy Consumption Modeling System 
     Modeling energy consumption for heating (or cooling) on an annual (or periodic) basis, as described supra with reference  FIG.  3   , and on an hourly (or interval) basis, as described supra beginning with reference to  FIG.  12   , can be performed with the assistance of a computer, or through the use of hardware tailored to the purpose.  FIG.  22    is a block diagram showing a computer-implemented system  230  for modeling building heating energy consumption in accordance with one embodiment. A computer system  231 , such as a personal, notebook, or tablet computer, as well as a smartphone or programmable mobile device, can be programmed to execute software programs  232  that operate autonomously or under user control, as provided through user interfacing means, such as a monitor, keyboard, and mouse. The computer system  231  includes hardware components conventionally found in a general purpose programmable computing device, such as a central processing unit, memory, input/output ports, network interface, and non-volatile storage, and execute the software programs  232 , as structured into routines, functions, and modules. In addition, other configurations of computational resources, whether provided as a dedicated system or arranged in client-server or peer-to-peer topologies, and including unitary or distributed processing, communications, storage, and user interfacing, are possible. 
     In one embodiment, to perform the first approach, the computer system  231  needs data on heating losses and heating gains, with the latter separated into internal heating gains (occupant, electric, and solar) and auxiliary heating gains. The computer system  231  may be remotely interfaced with a server  240  operated by a power utility or other utility service provider  241  over a wide area network  239 , such as the Internet, from which fuel purchase data  242  can be retrieved. Optionally, the computer system  231  may also monitor electricity  234  and other metered fuel consumption, where the meter is able to externally interface to a remote machine, as well as monitor on-site power generation, such as generated by a photovoltaic system  235 . The monitored fuel consumption and power generation data can be used to create the electricity and heating fuel consumption data and historical solar resource and weather data. The computer system  231  then executes a software program  232  to determine annual (or periodic) heating fuel consumption  244  based on the empirical approach described supra with reference to  FIG.  3   . 
     In a further embodiment, to assist with the empirical tests performed in the second approach, the computer system  231  can be remotely interfaced to a heating source  236  and a thermometer  237  inside a building  233  that is being analytically evaluated for thermal performance, thermal mass, effective window area, and HVAC system efficiency. In a further embodiment, the computer system  231  also remotely interfaces to a thermometer  238  outside the building  163 , or to a remote data source that can provide the outdoor temperature. The computer system  231  can control the heating source  236  and read temperature measurements from the thermometer  237  throughout the short-duration empirical tests. In a further embodiment, a cooling source (not shown) can be used in place of or in addition to the heating source  236 . The computer system  231  then executes a software program  232  to determine hourly (or interval) heating fuel consumption  244  based on the empirical approach described supra with reference to  FIG.  12   . 
     Applications 
     The two approaches to estimating energy consumption for heating (or cooling), hourly and annual, provide a powerful set of tools that can be used in various applications. A non-exhaustive list of potential applications will now be discussed. Still other potential applications are possible. 
     Application to Homeowners 
     Both of the approaches, annual (or periodic) and hourly (or interval), reformulate fundamental building heating (and cooling) analysis in a manner that can divide a building&#39;s thermal conductivity into two parts, one part associated with the balance point resulting from internal gains and one part associated with auxiliary heating requirements. These two parts provide that:
         Consumers can compare their house to their neighbors&#39; houses on both a total thermal conductivity UA Total  basis and on a balance point per square foot basis. These two numbers, total thermal conductivity UA Total  and balance point per square foot, can characterize how well their house is doing compared to their neighbors&#39; houses. The comparison could also be performed on a neighborhood- or city-wide basis, or between comparably built houses in a subdivision. Other types of comparisons are possible.   As strongly implied by the empirical analyses discussed supra, heater size can be significantly reduced as the interior temperature of a house approaches its balance point temperature. While useful from a capital cost perspective, a heater that was sized based on this implication may be slow to heat up the house and could require long lead times to anticipate heating needs. Temperature and solar forecasts can be used to operate the heater by application of the two approaches described supra, so as to optimize operation and minimize consumption. For example, if the building owner or occupant knew that the sun was going to start adding a lot of heat to the building in a few hours, he may choose to not have the heater turn on. Alternatively, if the consumer was using a heater with a low power rating, he would know when to turn the heater off to achieve desired preferences.       

     Application to Building Shell Investment Valuation 
     The economic value of heating (and cooling) energy savings associated with any building shell improvement in any building has been shown to be independent of building type, age, occupancy, efficiency level, usage type, amount of internal electric gains, or amount solar gains, provided that fuel has been consumed at some point for auxiliary heating. As indicated by Equation (43), the only information required to calculate savings includes the number of hours that define the winter season; average indoor temperature; average outdoor temperature; the building&#39;s HVAC system efficiency (or coefficient of performance for heat pump systems); the area of the existing portion of the building to be upgraded; the R-value of the new and existing materials; and the average price of energy, that is, heating fuel. This finding means, for example, that a high efficiency window replacing similar low efficiency windows in two different buildings in the same geographical location for two different customer types, for instance, a residential customer versus an industrial customer, has the same economic value, as long as the HVAC system efficiencies and fuel prices are the same for these two different customers. 
     This finding vastly simplifies the process of analyzing the value of building shell investments by fundamentally altering how the analysis needs to be performed. Rather than requiring a full energy audit-style analysis of the building to assess any the costs and benefits of a particular energy efficiency investment, only the investment of interest, the building&#39;s HVAC system efficiency, and the price and type of fuel being saved are required. 
     As a result, the analysis of a building shell investment becomes much more like that of an appliance purchase, where the energy savings, for example, equals the consumption of the old refrigerator minus the cost of the new refrigerator, thereby avoiding the costs of a whole house building analysis. Thus, a consumer can readily determine whether an acceptable return on investment will be realized in terms of costs versus likely energy savings. This result could be used in a variety of places:
         Direct display of economic impact in ecommerce sites. A Web service that estimates economic value can be made available to Web sites where consumers purchase building shell replacements. The consumer would select the product they are purchasing, for instance, a specific window, and would either specify the product that they are replacing or a typical value can be provided. This information would be submitted to the Web service, which would then return an estimate of savings using the input parameters described supra.   Tools for salespeople at retail and online establishments.   Tools for mobile or door-to-door sales people.   Tools to support energy auditors for immediate economic assessment of audit findings. For example, a picture of a specific portion of a house can be taken and the dollar value of addressing problems can be attached.   Have a document with virtual sticky tabs that show economics of exact value for each portion of the house. The document could be used by energy auditors and other interested parties.   Available to companies interacting with new building purchasers to interactively allow them to understand the effects of different building choices from an economic (and environmental) perspective using a computer program or Internet-based tool.   Enable real estate agents working with customers at the time of a new home purchase to quantify the value of upgrades to the building at the time of purchase.   Tools to simplify the optimization problem because most parts of the problem are separable and simply require a rank ordering of cost-benefit analysis of the various measures and do not require detailed computer models that applied to specific houses.   The time to fix the insulation and ventilation in a homeowner&#39;s attic is when during reroofing. This result could be integrated into the roofing quoting tools.   Incorporated into a holistic zero net energy analysis computer program or Web site to take an existing building to zero net consumption.   Integration into tools for architects, builders, designers for new construction or retrofit. Size building features or HVAC system. More windows or less windows will affect HVAC system size.       

     Application to Thermal Conductivity Analysis 
     A building&#39;s thermal conductivity can be characterized using only measured utility billing data (natural gas and electricity consumption) and assumptions about effective window area, HVAC system efficiency and average indoor building temperature. This test could be used as follows:
         Utilities lack direct methods to measure the energy savings associated with building shell improvements. Use this test to provide a method for electric utilities to validate energy efficiency investments for their energy efficiency programs without requiring an on-site visit or the typical detailed energy audit. This method would help to address the measurement and evaluation (M&amp;E) issues currently associated with energy efficiency programs.   HVAC companies could efficiently size HVAC systems based on empirical results, rather than performing Manual J calculations or using rules of thumb. This test could save customers money because Manual J calculations require a detailed energy audit. This test could also save customers capital costs since rules of thumb typically oversize HVAC systems, particularly for residential customers, by a significant margin.   A company could work with utilities (who have energy efficiency goals) and real estate agents (who interact with customers when the home is purchased) to identify and target inefficient homes that could be upgraded at the time between sale and occupancy. This approach greatly reduces the cost of the analysis, and the unoccupied home offers an ideal time to perform upgrades without any inconvenience to the homeowners.   Goals could be set for consumers to reduce a building&#39;s heating needs to the point where a new HVAC system is avoided altogether, thus saving the consumer a significant capital cost.       

     Application to Building Performance Studies 
     A building&#39;s performance can be fully characterized in terms of four parameters using a suite of short-duration (several day) tests. The four parameters include thermal conductivity, that is, heat losses, thermal mass, effective window area, and HVAC system efficiency. An assumption is made about average indoor building temperature. These (or the previous) characterizations could be used as follows:
         Utilities could identify potential targets for building shell investments using only utility billing data. Buildings could be identified in a two-step process. First, thermal conductivity can be calculated using only electric and natural gas billing data, making the required assumptions presented supra. Buildings that pass this screen could be the focus of a follow-up, on-site, short-duration test.   The results from this test suite can be used to generate detailed time series fuel consumption data (either natural gas or electricity). This data can be combined with an economic analysis tool, such as the PowerBill service (http://www.cleanpower.com/products/powerbill/), a software service offered by Clean Power Research, L.L.C., Napa, Calif., to calculate the economic impacts of the changes using detailed, time-of-use rate structures.       

     Application to “Smart” Thermostat Users 
     The results from the short-duration tests, as described supra with reference to  FIG.  4   , could be combined with measured indoor building temperature data collected using an Internet-accessible thermostat, such as a Nest thermostat device or a Lyric thermostat device, cited supra, or other so-called “smart” thermostat devices, thereby avoiding having to make assumptions about indoor building temperature. The building characterization parameters could then be combined with energy investment alternatives to educate consumers about the energy, economic, and environmental benefits associated with proposed purchases. 
     While the invention has been particularly shown and described as referenced to the embodiments thereof, those skilled in the art will understand that the foregoing and other changes in form and detail may be made therein without departing from the spirit and scope.