nmecr provides a collection of 11 algorithms and 5 main utility functions for streamlining impact quantification for energy efficiency projects in buildings.

These are:

Additional influential independent variables, such as product quantity, number of occupants etc., can be added to all algorithms except the mean model to improve model accuracy over a dataset.

Mean Model

The mean model is simplest in the toolkit. The model form is:

and the model predictions are simply the average of the observed energy consumption. The mean model is a good benchmarking model and can be used for comparison against more complex models.

Change-point Models

In modeling building energy consumption, one of the most common regressor (or feature or independent variable) is the outdoor air temperature. Many of the building’s core energy systems, such as HVAC, are directly impacted by it. Change-point models are a class of models that capture the trends of energy consumption over various segments of the temperature profile. These models differ mainly in their derived temperature features. The figure below illustrates the difference between the various flavors of these models.

[Image Source](https://www.sciencedirect.com/science/article/abs/pii/S0378778814009645)

Note that all model forms below reference this figure and its naming convention.

Two-Parameter Model

A two-parameter model (or a simple linear regression), is a regression using one independent variable, and its model form is:

Three Parameter Cooling and Heating Models

Three Parameter models are appropriate for modeling building energy use that varies linearly with an independent variable over part of the range and remains constant over the other part (see the second panel in the image above).

The model forms are:

The + and - above indicate that the values of the terms in the parenthesis will be set to zero when they are negative and positive respectively.

When should use these models?

Imagine you are modeling a system that predominantly delivers cooling to a building. Up until a certain temperature, the building does not need cooling and this system stays dormant. However, above this temperature, known as a change-point in the industry, the system has to start cooling the building and use energy in the process. In a three-parameter cooling model, the energy use of the system is ‘modeled’ to vary linearly with an increase in outdoor air temperature above the identified change-point.

The three-parameter heating model is used in the opposite scenario, where the energy use starts to increase linearly with a decrease in outdoor air temperature.

Four Parameter Model

Four parameter models extend the three parameter models by replacing the initial flat-sloped segment with a positive or negative-sloped segment depending on the type of model: cooling vs heating. The third panel in the image above shows two examples of Four Parameter models. The model form is:

As before, the + and - above indicate that the values of the terms in the parenthesis will be set to zero when they are negative and positive respectively.

When should you use this model?

These models are appropriate for modeling heating and cooling energy use in variable-air-volume systems.

Five Parameter Model

The model form for a five parameter model is:

The 5P algorithm requires an initial set of change-points to determine the optimum changepoints. This can be set via the initial_breakpoints argument in nmecr::assign_model_inputs().

When should you use this model?

Five parameter models are appropriate for modeling whole-building electricity consumption from buildings with electric heat pumps or both electric chillers and electric resistance heating.

Degree Day Models

Degree day models are another type of regression model. They are similar to change point models, but instead of regressing energy consumption against outside air temperature, consumption is regressed against degree days.

Degree days, expressed as heating degree days (HDD) or cooling degree days (CDD) represent deviations of outside air temperature (OAT) from a baseline, and are calculated as:

where is the average temperature from the day, and is the base temperature of the building. The + indicates that only positive values of these differences are considered.

The heating and cooling degree day model forms are given by:

where the chosen model depends on the temperature regime being modeled (whether it is heating or cooling dominated), These models are analogous to three parameter models, and should be used in similar cases. As with the change point models, if HDD or CDD are negative, then a value of zero is used, and consumption is equal to a constant value (C).

To model a scenario that incorporates both heating and cooling, a combined model is used:

The base temperatures are not required to be equal. When = , the model is analogous to a four parameter model, and consists of two intersecting lines. When the base temperatures are different (with < , the result is the same as the five parameter model.

Time-of-Week and Temperature Model

The Time-of-Week and Temperature Model in nmecr is based on the original paper from Lawrence Berkeley National Laboratory (LBNL) and their implementation of the algorithm.

The two main components of the model are:

a time-of-week indicator variable, and
a piece-wise linear and continuous outdoor air temperature dependence

Original Algorithm

The original algorithm has the following key features:

A week is divided into 15-minute intervals, i.e. 672 time-of-week variables
Outdoor air temperature is divided into six equally-sized temperature intervals. For example, if a facility experienced a temperature profile ranging from 50F to 110F, the temperature segments would be 50-60F, 60-70F, 70-80F, 70-90F, 90-100F, and 100-110F. The paper recommends using at least twice the expected number of change-points.
The temperature parameters are only used when a facility is operating in occupied mode
The facility’s response to temperature is not expected to change at night. Essentially, this translates to six temperature segments in the occupied mode and 1 temperature segment (that includes all data) in the unoccupied mode.
The start and end of the occupied mode are manually determined by visualizing average load profiles on non-DR days.

Changes in the implementation (RMV2.0)

The following are the main clarifications/additions in LBNL’s implementation of the algorithm:

Occupied and unoccupied modes are based on a regression of load on outside air temperature. Times of the week that the regression usually underpredicts the load are occupied and the rest are called unoccupied. More specifically, the times of the week that are underpredicted 65% of the time are assumed to be in the occupied mode.
The algorithm checks to ensure there are at least 20 data points above the highest temperature change-point and 20 data points below the lowest change-point. The algorithm requires at least 1 change-point to run.
The temperature segment width is determined by the user-requested number of segments. Each temperature segment is equal in width.
The same temperature segments are applied to both occupied and unoccupied modes
In case of a lack of temperature data, the algorithm can be run as a Time-of-Week (see below)
Both TOWT and TOW can be run on 15-min, hourly, and daily intervals

Further changes implemented in nmecr

The occupied and unoccupied mode determination is based on a user input, accessed through the formal argument occupancy_threshold within nmecr::assign_model_inputs(). The default value is set to 0.65 to match the RMV2.0 implementation. In certain scenarios, the split into occupied and unoccupied may not be required or desired. In these cases, the occupancy_threshold can be set to 1 to create one overall model. A practical application of this is the case when occupancy information is available as an additional variable. The TOWT model can then be run by setting occupancy_threshold = 1 and the additional variable as a proxy for occupancy.
Temperature change-points can be manually specified when needed. This behavior can be accessed through has_temp_knots_defined and temp_knots_value within nmecr::assign_model_inputs(). If has_temp_knots_defines is set to False (F), the algorithm determines the change-points internally. On the other hand, if this argument is set to True (T), it looks to temp_knots_value for the specified change-points. Note that temperature change-points are referred to as temperature knots in the code.
The temperature segments can either be of equal width or have equal number of data points in them. This behavior is accessed through equal_temp_segment_points within nmecr::assign_model_inputs(). If set to True (T), the algorithm determines the temperature segments such that each has equal number of data points in it. If set to False (F), the segments are determined such that they are equal in width.

The model forms for TOWT are:

Note that the actual model forms depend on a number of factors, and the actual model form may be different than the ones shown above.

Notice that there is no (is_Fri) term or (is_Sun) term in the above eqauations. This is because the average energy use on these days is captured by the intercept the respective equations.

The is_* term implies that the value can be either a 0 or a 1. For example, is_Mon will be set to 1 when the timestamp is a Monday to 0 for all other days of the week.

Time-of-Week Model

The time-of-week model is a simplification of the time-of-week and temperature model. It is a linear model with indicator variables for each ‘time-of-week’. In a model based on daily interval data, there are 7 indicator variables, for each day of the week. Similarly, in a model based on hourly interval data, there are 168 indicator variables, for each hour of the week. The model form, therefore, is:

Daily Interval

Notice that there is no (is_Sun) term in the above equation. The average energy use for Sundays is captured by the intercept term (C) in the above equation.

Hourly Interval

Similarly, there is no (is_tow168) term in the above equation.