US20140216704A1 - Method for operating an hvac system - Google Patents

Abstract

A method for operating an HVAC system is provided. The method includes providing a model for an indoor temperature, y, of a building, providing predicted future outdoor temperatures, and calculating an activation time or an adjustment time interval for the HVAC system utilizing at least the model for y and the predicted future outdoor temperatures. Operation of the HVAC system can be improved with the activation time or the adjustment time interval.

Description

FIELD OF THE INVENTION
• The present subject matter relates generally to HVAC systems, such as residential or commercial HVAC systems, and methods for operating the same.
BACKGROUND OF THE INVENTION
• Commercial and residential buildings or structures are commonly equipped with systems for regulating the temperature of air within the building for purposes of e.g., comfort, protection of temperature sensitive contents, etc. Sometimes referred to as heating, ventilating, and air conditioning or HVAC systems, such systems typically include one or more components for changing the temperature of air (i.e. air treatment components as used herein) along with one or more components for causing movement of air (i.e. blowers as used herein). For example, a refrigerant based heat pump may be provided for heating or cooling air. Alternatively, or in addition thereto, electrically resistant heat strips and/or gas burners may be provided for heating air. One or more blowers or fans may be provided for causing the heated or cooled air to circulate within the building in an effort to treat all or some controlled portion of air in the building. Ducting and vents may be used to help distribute and return air from different rooms or zones within the building.
• During heating and/or cooling of air, HVAC systems consume energy. In particular, HVAC systems' energy consumption can account for more than fifty percent of a building's total energy consumption. Despite consuming large amounts of energy, HVAC systems are generally set to a specific operating temperature, and the HVAC systems operate to maintain an associated building at the specific operating temperature.
• Certain HVAC systems also include features for switching the specific operating temperature between a high set temperature and a low set temperature to conserve energy. In particular, such HVAC systems can be programmed to switch between the high set temperature and the low set temperature at specific times. However, switching between the high and low set temperatures can create certain problems. In particular, HVAC systems require a certain amount of time to heat and or cool the building. Thus, the associated building's temperature can lag behind the specific operating temperature of the HVAC system, and such temperature lag can be uncomfortable or unpleasant to occupants of the associated building.
• Accordingly, methods for operating HVAC systems that can account for temperature lags between various operating temperatures of the HVAC system would be useful. In particular, methods for operating HVAC systems that that can preheat and/or precool an associated building in order to account for temperature lags between various operating temperatures of the HVAC system would be useful.
BRIEF DESCRIPTION OF THE INVENTION
• The present subject matter provides a method for operating an HVAC system. The method includes providing a model for an indoor temperature, y, of a building, providing predicted future outdoor temperatures, and calculating an activation time or an adjustment time interval for the HVAC system utilizing at least the model for y and the predicted future outdoor temperatures. Operation of the HVAC system can be improved with the activation time or the adjustment time interval. Additional aspects and advantages of the invention will be set forth in part in the following description, or may be apparent from the description, or may be learned through practice of the invention.
• In a first exemplary embodiment, a method for operating an HVAC system is provided. The HVAC system is configured for cooling air within a building, heating air within the building, or both. The method includes providing a model for an indoor temperature, y, of the building, providing predicted future outdoor temperatures, and calculating an adjustment time interval for the HVAC system utilizing at least the model for y and the predicted future outdoor temperatures.
• In a second exemplary embodiment, a method for operating an HVAC system is provided. The HVAC system is configured for cooling air within a building, heating air within the building, or both. The method includes providing a model for an interior temperature, y, of the building, providing predicted future exterior temperatures of the building, calculating an activation time for the HVAC system utilizing at least the model for y and the predicted future outdoor temperatures.
• These and other features, aspects and advantages of the present invention will become better understood with reference to the following description and appended claims. The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate embodiments of the invention and, together with the description, serve to explain the principles of the invention.
BRIEF DESCRIPTION OF THE DRAWINGS
• A full and enabling disclosure of the present invention, including the best mode thereof, directed to one of ordinary skill in the art, is set forth in the specification, which makes reference to the appended figures.
• FIG. 1 provides a schematic representation of an exemplary building as may be used with the present subject matter.
• FIG. 2 illustrates a method for operating an HVAC system according to an exemplary embodiment of the present subject matter.
• FIG. 3 illustrates a method for operating an HVAC system according to an additional exemplary embodiment of the present subject matter.
DETAILED DESCRIPTION
• Reference now will be made in detail to embodiments of the invention, one or more examples of which are illustrated in the drawings. Each example is provided by way of explanation of the invention, not limitation of the invention. In fact, it will be apparent to those skilled in the art that various modifications and variations can be made in the present invention without departing from the scope or spirit of the invention. For instance, features illustrated or described as part of one embodiment can be used with another embodiment to yield a still further embodiment. Thus, it is intended that the present invention covers such modifications and variations as come within the scope of the appended claims and their equivalents.
• FIG. 1 provides a schematic representation of an exemplary building 100 as may be used with the present subject matter. Building 100 includes an HVAC system 110. HVAC system 110 is configured for providing heated air to building 100, providing cooled air to building 100, or both. In particular, building 100 defines an inside or interior 102. Interior 102 of building 100 is separated or segregated from an exterior or outside 104. HVAC system 110 can heat and/or cool interior 102 of building 100.
• As will be understood by those skilled in the art, HVAC system 110 can be any suitable mechanism for heating and/or cooling interior 102 of building 100. In the exemplary embodiment shown in FIG. 1, HVAC system 110 includes an air treatment component 118 for heating and/or cooling air and at least one blower 119 for directing heated and/or cooled air into interior 102 of building 100, e.g., via a duct system within building 100. As an example, air treatment component 118 can be a heat pump that provides for both heating and cooling of the air circulated by blower 119 of HVAC system 110. Alternatively, air treatment component 118 of HVAC system 110 can be a heater based on e.g., one or more gas burners or electric strips.
• HVAC system 110 also includes a thermostat 112 for controlling HVAC system 110 and measuring a temperature of interior 102. A user can set an operating temperature of HVAC system 110 with thermostat 112, and HVAC system 110 can operate to maintain interior 102 of building 100 at the operating temperature. Further, HVAC system 110 includes a temperature sensor 116, such as a thermocouple or thermistor, for measuring a temperature of exterior 104 of building 100.
• HVAC system 110 also includes a processing device or controller 114, e.g., positioned within thermostat 112. Various operational processes or methods for operating HVAC system 110 can be programmed into controller 114. As used herein, “controller” may include a memory and one or more microprocessors, CPUs or the like, such as general or special purpose microprocessors operable to execute programming instructions or micro-control code associated with operation of HVAC system 110. The memory may represent random access memory such as DRAM, or read only memory such as ROM or FLASH. In one embodiment, the processor executes programming instructions stored in memory. The memory may be a separate component from the processor or may be included onboard within the processor.
• It should be understood that the shape and configuration of building 100 shown in FIG. 1 is provided by way of example only. Buildings having different shapes, configurations, different numbers of rooms, hallways, etc.—both residential and commercial—may be used with the present subject matter. Further, the location of HVAC system 110 relative to building 100 is also provided by way of example only.
• As will be understood by those skilled in the art, HVAC system 110 can operate to maintain building 100 at a first operating temperature when building 100 is unoccupied. Conversely, HVAC system 110 can operate to maintain building 100 at a second operating temperature when building 100 is occupied. Controller 114 can adjust HVAC system 110 between the first and second operating temperatures, e.g., in order to conserve energy and/or reduce operating costs of HVAC system 110. However, the first operating temperature can be uncomfortable, e.g., too hot or too cold, to occupants of building 100 relative to the second operating temperature.
• HVAC system 110 requires time to heat and/or cool interior 102 of building 100 and adjust a temperature of interior 102. As discussed in greater detail below, the present subject matter provides methods for operating an HVAC system, such as HVAC system 110. Such methods can assist with improving performance of HVAC system 110, e.g., during heating and/or cooling of interior 102 of building 100 between the first and second operating temperatures.
• FIG. 2 illustrates a method 200 for operating an HVAC system according to an exemplary embodiment of the present subject matter. Method 200 can be used to operate any suitable HVAC system, such as HVAC system 110 (FIG. 1). As an example, controller 114 of HVAC system 110 can be programmed to implement method 200. Utilizing method 200, an adjustment time interval for HVAC can be calculated. The adjustment time interval can assist with improving operation of HVAC system 110 as discussed in greater detail below.
• At step 210, a model for an indoor temperature, y, of building 100 is provided. The model for y can be programmed into controller 114 such that controller 114 can calculate a predicted future indoor temperature of building 100, e.g., a predicted future temperature of interior 102. The model for y can utilize any suitable input to calculate y. For example, y can be calculated based at least in part upon a previous indoor temperature of building 100, a previous outdoor temperature of building 100, and/or a previous operational state of HVAC system 110, e.g., whether HVAC system 110 is on or off.
• The model for y can be any suitable model for simulating or modeling the heat dynamics of building 100. As an example, the model for y can be a second order linear model, e.g., such that the model for y is given as
• 
  y _k =a ₁ y _k−1 +a ₂ y _k−2 +b ₁ v _k−1 +b ₂ u _k−1
• where
• • y_k is an indoor temperature of building 100 at time k,
  • y_k−1 is an indoor temperature of building 100 at time k−1,
  • y_k−2 is an indoor temperature of building 100 at time k−2,
  • v_k−1 is an outdoor temperature at time k−1,
  • u_k−1 is an operating state of HVAC system 110 at time k−1, and
  • a₁, a₂, b₁, and b₂ are constants.
    As will be understood by those skilled in the art, the model for y provided above is a discrete-time auto-regressive model with exogenous inputs, and constants a₁, a₂, b₁, and b₂ can be determined utilizing recursive least-square techniques or any other suitable technique. As an example, controller 114 can receive indoor temperature measurements from thermostat 112, outdoor temperature measurements from temperature sensor 116, and operating states from HVAC system 110 over time and calculate constants a₁, a₂, b₁, and b₂ in order to identify the model for y.
• The model for y provided above can also be provided as a state space model. Thus, the model for y can be given as
• 
  X _k+1 =AX _k +BU _k
• where
• 
  X _k =[y _k y _k−1]^T,
• and
• 
  U _k =[v _k u _k]^T.
• As discussed above, the model for y can be any suitable model in alternative exemplary embodiments. Thus, the model provided above is not intended to limit the present subject matter in any aspect and is provided by way of example only.
• At step 220, predicted future outdoor temperatures are provided. As an example, controller 114 can receive the predicted future outdoor temperatures, e.g., predicted future temperatures of exterior 104 of building 100, at step 220. The predicted future outdoor temperatures can come from any suitable source. For example, the predicted future outdoor temperatures can be based on weather forecast data or historical weather data.
• As an example, weather forecast data generally includes a daily maximum temperature and a daily minimum temperature. Further, outdoor temperatures generally have a sinusoidal shape between the daily maximum temperature and the daily minimum temperature. Thus, the predicted future outdoor temperatures can be provided using the following:
• $f\left(t\right)=\{\begin{array}{cc}\begin{array}{c}\frac{T_{\max }\left(k\right)+T_{\min }\left(k\right)}{2}-\frac{T_{\max }\left(k\right)-T_{\min }\left(k\right)}{2}\\ \cos \left(\frac{\pi \left(t-t_{\min }\left(k\right)\right)}{t_{\max }\left(k\right)-t_{\min }\left(k\right)}\right)\end{array}& t\in \left[t_{\min }\left(k\right),t_{\max }\left(k\right)\right)\\ \begin{array}{c}\frac{T_{\max }\left(k\right)+T_{\min }\left(k+1\right)}{2}+\frac{T_{\max }\left(k\right)-T_{\min }\left(k+1\right)}{2}\\ \cos \left(\frac{\pi \left(t-t_{\max }\left(k\right)\right)}{t_{\min }\left(k+1\right)-t_{\max }\left(k\right)}\right)\end{array}& t\in \left[t_{\max }\left(k\right),t_{\min }\left(k+1\right)\right)\end{array}$
• where
• • T_max(k) is a maximum temperature on day k,
  • T_min(k) is a minimum temperature on day k,
  • t_max(k) is a time of day for T_max(k), and
  • t_min(k) is a time of day for T_min(k).
    Utilizing the above formula, the predicted future outdoor temperatures can be provided throughout the day despite only having the daily maximum temperature and the daily minimum temperature from the weather forecast data. As discussed above, the predicted future outdoor temperatures can be determined in any suitable manner. Thus, the formula provided above is provided by way of example only and is not intended to limit the present subject matter.
• At step 230, the adjustment time interval for HVAC system 110 is calculated. The adjustment time interval can correspond to a period of time required for HVAC system 110 to adjust an indoor temperature of building 100 from an initial temperature, T₀, to a final temperature, T_f, where T₀ and T_f are unequal. Thus, HVAC system 110 can heat and/or cool interior 102 of building 100 between T₀ and T_f within the adjustment time interval.
• As an example, controller 114 can calculate the adjustment time interval at step 230 utilizing at least the model for y of step 210 and the predicted future outdoor temperatures of step 220. In particular, controller 114 can calculate the adjustment time interval with the following:
• $y_f={\mathrm{CA}}^kx_{k_0}+\sum _{i=0}^{N-1}{\mathrm{CA}}^{N-i-1}{\mathrm{BU}}_i$ $\mathrm{where}$ $C=\left[\begin{array}{cc}1& 0\end{array}\right],A=\left[\begin{array}{cc}{a}_1& a_2\\ 1& 0\end{array}\right],B=\left[\begin{array}{cc}{b}_1& b_2\\ 1& 0\end{array}\right],x_{k_0}=\left[\begin{array}{c}{y}_{k_0}\\ y_{k_0-1}\end{array}\right],U_i=\left[\begin{array}{c}{v}_i\\ u_i\end{array}\right],$
• and
• N=k_f−k₀ where k₀ is an initial time at which the indoor temperature of building 100 is an initial temperature, y_{k 0} , and k_f is a final time at which the indoor temperature of building 100 is a final temperature, y_{k f} .
• Further, utilizing the above process, the adjustment time interval can be calculated in order to minimize energy consumption of HVAC system 110. For example, the adjustment time interval can be calculated with the following:
• $T\leftarrow {\mathrm{CA}}^Nx_{k_0}+\sum _{i=0}^{N-1}{\left({\mathrm{CA}}^{N-i-1}B\right)}_1v_i$ $k^{*}\leftarrow k_f-1$ $\mathrm{while}T<T_f\mathrm{do}$ $T\leftarrow T+{\left({\mathrm{CA}}^{k_f-k^{*}}B\right)}_2$ $k^{*}\leftarrow k^{*}-1$ $\mathrm{end}\mathrm{while}$
• where
• • T is the indoor temperature of building 100 and
  • k* is a time value.
    With the adjustment time interval calculated at step 230, operation of HVAC system 110 can be improved. For example, controller 114 can determine an initial time, k* or t₀, at which the indoor temperature of building 100 is T₀ and a final time, t_f, at which the indoor temperature of building 100 is T_f. The adjustment time interval can correspond to the difference between t₀ and t_f. Controller 114 can also activate HVAC system 110 at t₀ such the indoor temperature of building 100 is T_f at t_f. In such a manner, method 200 can assist with preheating and/or precooling interior 102 of building 100.
• As will be understood by those skilled in the art, controller 114 can be programmed to adjust the operating temperature of HVAC system 110 between T₀ and T_f. As an example, HVAC system 110 can operate to maintain building 100 at one of T₀ or T_f when building 100 is unoccupied. Conversely, HVAC system 110 can operate to maintain building 100 the other of T₀ and T_f when building 100 is occupied. Controller 114 can adjust HVAC system 110 between T₀ and T_f, e.g., in order to conserve energy and/or reduce operating costs of HVAC system 110.
• With the activation time interval calculated at step 230, controller 114 can operate HVAC system 110 to pre-heat and/or pre-cool interior 102 of building 100 between T₀ and T_f. Such pre-heating or pre-cooling can increase comfort of occupants within building 100 and improve satisfaction of such occupants with HVAC system 110. For example, before such occupants enter or return to building 100, controller 114 can operate to adjust interior 102 of building 100 from T₀ to T_f such the building 100 is pre-heated or pre-cooled and comfortable for such occupants.
• FIG. 3 illustrates a method 300 for operating an HVAC system according to an additional exemplary embodiment of the present subject matter. Method 300 can be used to operate any suitable HVAC system, such as HVAC system 110 (FIG. 1). As an example, controller 114 of HVAC system 110 can be programmed to implement method 300. Utilizing method 300, an activation time for HVAC system 110 can be calculated. The activation time can assist with improving operation of HVAC system 110 as discussed in greater detail below.
• Like in step 210 of method 200 (FIG. 2), a model for an indoor temperature, y, of building 100 is provided at step 310. The model for y provided in step 310 can be any suitable model for y, such as the model for y discussed above in method 200. The model for y can be programmed into controller 114 such that controller 114 can calculate a predicted future indoor temperature of building 100, e.g., a predicted future temperature of interior 102.
• Like in step 220 of method 200 (FIG. 2), predicted future outdoor temperatures are provided at step 320. The predicted future outdoor temperatures can come from any suitable source, such as described above in method 200. As an example, controller 114 can receive the predicted future outdoor temperatures, e.g., predicted future temperatures of exterior 104 of building 100, at step 320.
• At step 330, the activation time for HVAC system 110 is calculated. The activation time can correspond to a time at which HVAC system 110 is activated in order to adjust an indoor temperature of building 100 from an initial temperature, T₀, to a final temperature, T_f, where T₀ and T_f are unequal. Thus, HVAC system 110 can heat and/or cool interior 102 of building 100 between T₀ and T_f by activating HVAC system 110 at the activation time.
• As an example, controller 114 can calculate the activation time at step 330 utilizing at least the model for y of step 310 and the predicted future outdoor temperatures of step 320. In particular, controller 114 can calculate the activation time with the following:
• $y_f={\mathrm{CA}}^kx_{k_0}+\sum _{i=0}^{N-1}{\mathrm{CA}}^{N-i-1}{\mathrm{BU}}_i$ $\mathrm{where}$ $C=\left[\begin{array}{cc}1& 0\end{array}\right],A=\left[\begin{array}{cc}{a}_1& a_2\\ 1& 0\end{array}\right],B=\left[\begin{array}{cc}{b}_1& b_2\\ 1& 0\end{array}\right],x_{k_0}=\left[\begin{array}{c}{y}_{k_0}\\ y_{k_0-1}\end{array}\right],U_i=\left[\begin{array}{c}{v}_i\\ u_i\end{array}\right],$
• and
• N=k_f−k₀ where k₀ is the activation time at which the indoor temperature of building 100 is an initial temperature, y_{k 0} , and k_f is a final time at which the indoor temperature of building 100 is a final temperature, y_{k f} .
• Further, utilizing the above process, the activation time can be calculated in order to minimize energy consumption of HVAC system 110. For example, the activation time can be calculated with the following:
• $T\leftarrow {\mathrm{CA}}^Nx_{k_0}+\sum _{i=0}^{N-1}{\left({\mathrm{CA}}^{N-i-1}B\right)}_1v_i$ $k^{*}\leftarrow k_f-1$ $\mathrm{while}T<T_f\mathrm{do}$ $T\leftarrow T+{\left({\mathrm{CA}}^{k_f-k^{*}}B\right)}_2$ $k^{*}\leftarrow k^{*}-1$ $\mathrm{end}\mathrm{while}$
• where
• • T is the indoor temperature of building 100 and
  • k* is the activation time.
    With the activation time calculated at step 330, operation of HVAC system 110 can be improved. For example, controller 114 can activate or turn on HVAC system 110 at the activation time such the indoor temperature of building 100 is T_f at a final time, t_f. In such a manner, method 300 can assist with preheating and/or precooling interior 102 of building 100, e.g., in a similar manner as described above for method 200.
• This written description uses examples to disclose the invention, including the best mode, and also to enable any person skilled in the art to practice the invention, including making and using any devices or systems and performing any incorporated methods. The patentable scope of the invention is defined by the claims, and may include other examples that occur to those skilled in the art. Such other examples are intended to be within the scope of the claims if they include structural elements that do not differ from the literal language of the claims, or if they include equivalent structural elements with insubstantial differences from the literal languages of the claims.

Claims (19)

What is claimed is:

1. A method for operating an HVAC system, the HVAC system configured for cooling air within a building, heating air within the building, or both, the method comprising:

providing a model for an indoor temperature, y, of the building;

providing predicted future outdoor temperatures;

calculating an adjustment time interval for the HVAC system utilizing at least the model for y and the predicted future outdoor temperatures.

2. The method of claim 1, wherein the adjustment time interval corresponds to a period of time required for the HVAC system to adjust the indoor temperature of the building from an initial temperature, T₀, to a final temperature, T_f, wherein T₀ and T_f are unequal.

3. The method of claim 2, further comprising:

determining an initial time, t₀, at which the indoor temperature of the building is T₀ and a final time, t_f, at which the indoor temperature of the building is T_f, the adjustment time interval corresponding to the difference between t₀ and t_f; and

activating the HVAC system at t₀ such the indoor temperature of the building is T_f at t_f.

4. The method of claim 1, wherein the model for y comprises a second order linear model.

5. The method of claim 4, wherein the model for y comprises

y _k =a ₁ y _k−1 +a ₂ y _k−2 +b ₁ v _k−1 +b ₂ u _k−1

where

y_k is an indoor temperature of the building at time k,

y_k−1 is an indoor temperature of the building at time k−1,

y_k−2 is an indoor temperature of the building at time k−2,

v_k−1 is an outdoor temperature at time k−1,

u_k−1 is an operating state of the HVAC system at time k−1, and

a₁, a₂, b₁, and b₂ are constants.

6. The method of claim 1, wherein said step of calculating comprises calculating the adjustment time interval with the following:

$y_f={\mathrm{CA}}^kx_{k_0}+\sum _{i=0}^{N-1}{\mathrm{CA}}^{N-i-1}{\mathrm{BU}}_i$ $\mathrm{where}$ $C=\left[\begin{array}{cc}1& 0\end{array}\right],A=\left[\begin{array}{cc}{a}_1& a_2\\ 1& 0\end{array}\right],B=\left[\begin{array}{cc}{b}_1& b_2\\ 1& 0\end{array}\right],x_{k_0}=\left[\begin{array}{c}{y}_{k_0}\\ y_{k_0-1}\end{array}\right],U_i=\left[\begin{array}{c}{v}_i\\ u_i\end{array}\right],$

and

N=k_f−k₀ where k₀ is an initial time at which the indoor temperature of the building is an initial temperature, y_{k 0} , and k_f is a final time at which the indoor temperature of the building is a final temperature, y_{k f} .

7. The method of claim 1, wherein said step of calculating comprises calculating the adjustment time interval in order to minimize energy consumption of the HVAC system.

8. The method of claim 7, wherein said step of calculating comprises calculating the adjustment time interval with the following:

$T\leftarrow {\mathrm{CA}}^Nx_{k_0}+\sum _{i=0}^{N-1}{\left({\mathrm{CA}}^{N-i-1}B\right)}_1v_i$ $k^{*}\leftarrow k_f-1$ $\mathrm{while}T<T_f\mathrm{do}$ $T\leftarrow T+{\left({\mathrm{CA}}^{k_f-k^{*}}B\right)}_2$ $k^{*}\leftarrow k^{*}-1$ $\mathrm{end}\mathrm{while}$

where

T is the indoor temperature of the building and

k* is a time value.

9. The method of claim 1, wherein said step of providing predicted future outdoor temperatures comprises determining predicted future outdoor temperatures based upon weather forecast data.

10. The method of claim 9, wherein said step of providing predicted future outdoor temperatures comprises providing predicted future outdoor temperatures using the following:

$f\left(t\right)=\{\begin{array}{cc}\begin{array}{c}\frac{T_{\max }\left(k\right)+T_{\min }\left(k\right)}{2}-\frac{T_{\max }\left(k\right)-T_{\min }\left(k\right)}{2}\\ \cos \left(\frac{\pi \left(t-t_{\min }\left(k\right)\right)}{t_{\max }\left(k\right)-t_{\min }\left(k\right)}\right)\end{array}& t\in \left[t_{\min }\left(k\right),t_{\max }\left(k\right)\right)\\ \begin{array}{c}\frac{T_{\max }\left(k\right)+T_{\min }\left(k+1\right)}{2}+\frac{T_{\max }\left(k\right)-T_{\min }\left(k+1\right)}{2}\\ \cos \left(\frac{\pi \left(t-t_{\max }\left(k\right)\right)}{t_{\min }\left(k+1\right)-t_{\max }\left(k\right)}\right)\end{array}& t\in \left[t_{\max }\left(k\right),t_{\min }\left(k+1\right)\right)\end{array}$

where

T_max(k) is a maximum temperature on day k,

T_min(k) is a minimum temperature on day k,

t_max(k) is a time of day for T_max(k), and

t_min(k) is a time of day for T_min(k).

11. A method for operating an HVAC system, the HVAC system configured for cooling air within a building, heating air within the building, or both, the method comprising:

providing a model for an interior temperature, y, of the building;

providing predicted future exterior temperatures of the building;

calculating an activation time for the HVAC system utilizing at least the model for y and the predicted future outdoor temperatures.

12. The method of claim 1, further comprising:

turning on the HVAC system at the activation time; and

running the HVAC system in order to adjust the indoor temperature of the building from an initial temperature, T₀, to a final temperature, T_f, after said step of turning on.

13. The method of claim 1, wherein the model for y comprises a second order linear model.

14. The method of claim 13, wherein the model for y comprises

y _k =a ₁ y _k−1 +a ₂ y _k−2 +b ₁ v _k−1 +b ₂ u _k−1

where

y_k is an indoor temperature of the building at time k,

y_k−1 is an indoor temperature of the building at time k−1,

y_k−2 is an indoor temperature of the building at time k−2,

v_k−1 is an outdoor temperature at time k−1,

u_k−1 is an operating state of the HVAC system at time k−1, and

a₁, a₂, b₁, and b₂ are constants.

15. The method of claim 1, wherein said step of calculating comprises calculating the activation time with the following:

$y_f={\mathrm{CA}}^kx_{k_0}+\sum _{i=0}^{N-1}{\mathrm{CA}}^{N-i-1}{\mathrm{BU}}_i$ $\mathrm{where}$ $C=\left[\begin{array}{cc}1& 0\end{array}\right],A=\left[\begin{array}{cc}{a}_1& a_2\\ 1& 0\end{array}\right],B=\left[\begin{array}{cc}{b}_1& b_2\\ 1& 0\end{array}\right],x_{k_0}=\left[\begin{array}{c}{y}_{k_0}\\ y_{k_0-1}\end{array}\right],U_i=\left[\begin{array}{c}{v}_i\\ u_i\end{array}\right],$

and

N=k_f−k₀ where k₀ is the activation time at which the indoor temperature of the building is an initial temperature, y_{k 0} , and k_f is a final time at which the indoor temperature of the building is a final temperature, y_{k f} .

16. The method of claim 1, wherein said step of calculating comprises calculating the activation time in order to minimize energy consumption of the HVAC system.

17. The method of claim 16, wherein said step of calculating comprises calculating the activation time with the following:

$T\leftarrow {\mathrm{CA}}^Nx_{k_0}+\sum _{i=0}^{N-1}{\left({\mathrm{CA}}^{N-i-1}B\right)}_1v_i$ $k^{*}\leftarrow k_f-1$ $\mathrm{while}T<T_f\mathrm{do}$ $T\leftarrow T+{\left({\mathrm{CA}}^{k_f-k^{*}}B\right)}_2$ $k^{*}\leftarrow k^{*}-1$ $\mathrm{end}\mathrm{while}$

where

T is the indoor temperature of the building and

k* is the activation time.

18. The method of claim 1, wherein said step of providing predicted future outdoor temperatures comprises determining predicted future outdoor temperatures based upon weather forecast data.

19. The method of claim 18, wherein said step of providing predicted future outdoor temperatures comprises providing predicted future outdoor temperatures using the following:

$f\left(t\right)=\{\begin{array}{cc}\begin{array}{c}\frac{T_{\max }\left(k\right)+T_{\min }\left(k\right)}{2}-\frac{T_{\max }\left(k\right)-T_{\min }\left(k\right)}{2}\\ \cos \left(\frac{\pi \left(t-t_{\min }\left(k\right)\right)}{t_{\max }\left(k\right)-t_{\min }\left(k\right)}\right)\end{array}& t\in \left[t_{\min }\left(k\right),t_{\max }\left(k\right)\right)\\ \begin{array}{c}\frac{T_{\max }\left(k\right)+T_{\min }\left(k+1\right)}{2}+\frac{T_{\max }\left(k\right)-T_{\min }\left(k+1\right)}{2}\\ \cos \left(\frac{\pi \left(t-t_{\max }\left(k\right)\right)}{t_{\min }\left(k+1\right)-t_{\max }\left(k\right)}\right)\end{array}& t\in \left[t_{\max }\left(k\right),t_{\min }\left(k+1\right)\right)\end{array}$

where

T_max(k) is a maximum temperature on day k,

T_min(k) is a minimum temperature on day k,

t_max(k) is a time of day for T_max(k), and

t_min(k) is a time of day for T_min(k).

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