# Single-Variable Regression Analysis Models

These models constitute the main procedures adopted by the International Performance Measurement and Verification Protocol (IPMVP, 1997, 2002, 2007). A simple linear correlation is assumed to exist between building energy use and one independent variable. The ambient temperature is typically selected as the independent variable, especially to predict commercial/residential building heating and cooling energy use. Degree-days with properly selected balance temperature can be used as another option for the independent variable.

Ambient-temperature-based regression models have been shown to predict building energy use with an acceptable level of accuracy even for daily datasets (Kissock et al., 1992; Katipamula, Reddy, and Claridge, 1994; Kissock and Fels, 1995) and can be used to estimate energy savings (Claridge et al., 1991; Fels and Keating, 1993). Four basic functional forms of the single-variable regression models have been proposed for measuring energy savings in commercial and residential buildings. The selection of the function form depends on the application and the building characteristics. Figure 15.2 illustrates the four basic functional forms commonly used for ambient-temperature linear regression models. The regression models, also called change-point or segmented-linear models, combine both search methods and least-squares regression techniques to obtain the best-fit correlation coefficients. Each change-point regression model is characterized by the number of correlation coefficients. Therefore, the two-parameter model has two correlation coefficients (p_{0} and p,) and consists of a simple linear regression model between building energy use and ambient temperature. Table 15.1 summarizes the mathematical expressions of four change-point models and their applications. In general, the change-point regression models are more suitable for predicting heating rather than cooling energy use. Indeed, these regression models assume steady-state conditions and are insensitive to the building dynamic effects, solar effects, and nonlinear HVAC system controls such as on-off schedules.

Figure 15.3 provides an example of a three-parameter model for gas usage of a home in Colorado using an ambient-temperature linear regression method. Figure 15.4 illustrates, for the same home, a two-parameter model using a heating degree-day linear regression model.

FIGURE 15.2 Basic forms of single-variable regression models: (a) two-parameter model: (b) three-parameter model; (c) four-parameter model: (d) five-parameter model.

**TABLE 15.1**

Mathematical Expressions and Applications of Change Point Regression Models

Model Type |
Mathematical Expression |
Applications |

Two-parameter (2-P) |
£=P.+ PiT |
Buildings with constant-air-volume systems and simple controls |

Three-parameter (3-P) |
Heating: £ = p |
Buildings with envelope-driven heating or cooling loads (i.e., residential and small commercial buildings) |

Four-parameter (4-P) |
Heating: Cooling: |
Buildings with variable-air-volume systems or with high latent loads. Also, buildings with nonlinear control features (such as economizer cycles and hot deck reset schedules) |

Five-parameter (5-P) |
£ = P„ + Pi ■ (рз - Л* - p2 • (T PrP |
Buildings with systems that use the same energy source for both heating and cooling (i.e., heat pumps, electric heating and cooling systems) |

FIGURE 15.3 Three-parameter regression model based on outdoor ambient temperature for a home in Colorado.

FIGURE 15.4 Two-parameter regression model based on heating degree-day for the same home considered in Figure 15.3.

Other types of single-variable regression models have been applied to predict the energy use of HVAC equipment such as pumps, fans, and chillers. For instance. Phelan. Brandemuehl, and Krarti (1996) used linear and quadratic regression models to obtain a correlation between the electrical energy used by fans and pumps and the fluid mass flow rate.

# Multivariable Regression Analysis Models

These regression models use several independent variables to predict building energy use or energy savings due to retrofit projects. Several studies have indicated that the multivariable regression models provide better predictions of monthly, daily, and even hourly energy use of large commercial buildings than the single-variable models (Haberl and Claridge, 1987; Katipamula, Reddy, and Claridge, 1994, 1995, 1998). In addition to outdoor temperature, multivariable regression models use internal gain, solar radiation, and humidity ratio as independent variables. For instance, the cooling energy use for commercial buildings conditioned with VAV systems can be obtained using the following functional form of a multivariate regression model (Katipamula, Reddy, and Claridge, 1998):

where

p_{0} through p_{6} are regression coefficients;

/ is an indicator variable that is used to model the change in slope of the energy use variation as a function of the outdoor temperature;

*T _{a}* is the ambient outdoor dry-bulb temperature;

*Tj _{r}* is the outdoor dew-point temperature;

<7, is the internal sensible heat gain; and *q _{s}* is the total global horizontal solar radiation.

For commercial buildings conditioned with a CV HVAC system, the cooling energy use can be predicted using a simplified model of Eq. (15.3) as follows:

The models represented by Eqs. (15.3) and (15.4) have been applied to predict cooling energy consumption in several commercial buildings using various time-scale resolutions: monthly, daily, hourly, and hour-of-day (HOD). The HOD predictions, which require a significant modeling effort because the energy use data has to be regrouped in hourly bins corresponding to each hour of the day and 24 individual hourly models have to be obtained, are found to have better accuracy (Katipamula, Reddy, and Claridge, 1994). Table 15.2 indicates some of the advantages and disadvantages of the different multivariable regression modeling approaches. Generally, monthly utility bills can be used to develop monthly regression models but metering is required to establish daily, hourly, and HOD models.

It should be noted that the multivariable regression models discussed above are developed without retaining the time-series nature of the data. Other regression models can be considered to preserve the time variation of the building energy use. For instance, Fourier series models can be used to capture the daily and seasonal variations of commercial buildings energy use (Dhar, Reddy, and Claridge, 1998).

Multivariable regression models have been applied not only to estimate total building energy use but also to predict the behavior of individual pieces of HVAC

TABLE 15.2

Typical Advantages and Disadvantages of Multivariable Regression Models with Different Time Resolution

Advantages/ Disadvantages |
Monthly |
Daily |
Hourly |
HOD |

Modeling effort |
Minimum |
Minimum |
Moderate |
Difficult |

Metering needs |
None |
Required |
Required |
Required |

Data requirements |
At least 12 months |
At least three months |
At least three months |
At least three months |

Application to savings estimation |
In some cases |
In most cases |
All cases |
All cases |

Prediction accuracy |
Low |
High |
Moderate |
High |

*Source:* Katipamula, Reddy, and Claridge (1998).

equipment such as chillers, fans, and pumps. In particular, polynomial models have been widely used to model chiller energy use as a function of part-load ratio, evaporator leaving temperature, and condenser entering temperature (LBL, 1982). Other regression models for chillers have been obtained based on fundamental engineering principles (Gordon and Ng, 1994).