October 2016 
Innovations in Comfort, Efficiency, and Safety Solutions. 
Buildings as Dynamical Systems Part Two: Impact of Occupancy on a Building 
Michael Georgescu, Ph.D., Director of Engineering and Research Ecorithm, Inc. 
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In our September
blog post titled Buildings as Dynamical Systems: Part One,
we introduced the concept of buildings as dynamical systems and
discussed the impact that weather has on the operation of a building
and how a control system must compensate. This month we explain
the theory of dynamical systems further and the impact of occupancy on
the operation of a building.
Buildings are complex systems that are difficult to analyze in large
part due to the dynamic impact that occupancy and weather have on a
building’s operations. Current analysis techniques can identify
when a bad behavior results due to a specific operational condition;
however, they fall short when trying to determine a precise
relationship between environmental factors and a building’s
operations. Ecorithm applies dynamical systems theory to better
understand how a building performs over time with respect to both
occupancy and weather.
Statistical methods such as regression models have been used to
identify rough correlations between occupancy and HVAC data.
These methods can correlate long term average occupancy patterns to
HVAC use on a weekly or monthly scale but lack the ability to correlate
behavior on a shorter time scale such as an hour or a day. Other
methods may use realtime data to provide instantaneous feedback to a
device control loop but often over or underestimate actual occupancy
and are not able to identify occupancy patterns that emerge over
time.
Ecorithm combines dynamical systems theory with spectral analysis in a
unique automated fault detection and diagnostics (AFDD) solution to
provide building operators with the knowledge and actionable
intelligence to better understand their buildings and how best to
control them from day to day^{1}. This sophisticated
approach to analytics provides an AFDD system that is capable of
eliminating false alarms, applies machine learning and ultimately leads
to predictive analysis.
At a primary level, dynamical systems theory analyzes how physical
systems change over time. There are many examples of dynamical
systems that have been extensively studied, allowing analysis
techniques to be formed and evolve over time. Complex systems
such as the stock market, power systems^{2}, weather, and ocean currents^{3}
have been studied extensively under dynamical systems theory thus
analytic methods have been developed to understand these systems
mathematically. Surprisingly, buildings pose an even greater challenge,
which had previously prevented the development of dynamical
systemsbased analytical tools for building applications.
Ecorithm has found that a building too can be viewed as a dynamical
system since it is a collection of temperatures, pressures, flows, and
other physical quantities that interact with each other and change over
time with respect to primary factors such as weather and occupancy that
exhibit cyclical behavior. By applying a dynamical system
approach to fault detection and diagnostics, Ecorithm has the ability
to more explicitly identify faults and apply root cause analysis to
building management system (BMS) data compared to strictly statistical
based analysis or rulesbased fault detection systems. Thus,
applying dynamical system methodology to building systems can provide
better understanding of the operations of a building.
Everyone has an intuitive sense of occupancy in a building; offices
are typically occupied during a work day from 8am – 6pm and a
residential building will be occupied during nights and weekends.
On a macro level, intuition is derived from seemingly endless cyclical
patterns from day to day, week to week and even season to season.
However, on a micro level, the reliability of the occupancy of a
specific zone or of a specific occupant is not as predictable. For
example, a zone may contain any number of employees on a given day that
can be impacted by whether an occupant is sick, stuck in traffic, left
early for lunch, taking a vacation or working in a different
zone. There are a multitude of interactions and changes that take
place on a daily basis or even subhourly basis that make the process
difficult to forecast. As you begin to observe occupancy data over a
longer period of time using traditional statistical approaches, the
intricacies of these small variations start to fade as the system
regresses towards the average and resembles the aforementioned
intuitive cyclical pattern. Reliability increases when looking at
longer term averages but what can be done to also understand this
behavior on a shorter time scale that can have a greater influence how
a building system is controlled?
The main issue when looking at occupancy patterns on a shorter time
scale is not that occupancy is completely unpredictable but instead
that macrolevel intuition of long term and recurrent occupancy
patterns is illsuited when describing outcomes at a shorter time
scale. The dynamics of short term occupancy can be modeled using the
same mathematics as that of a coin flip which has a probabilistic
outcome. The model for these two systems can be similar in that
there is less of a determinant outcome for occupancy since there are a
range of possibilities each with some chance of
occurring. In a coin flip there is a 50% percent
chance that the outcome will be heads and a 50% chance that the outcome
will be tails. Using a dynamical system model, what happens if we can
determine and assign a probability that an occupant comes to a building
on any given day (hopefully more than 50%)?
The
figures above visualize observed occupancy patterns amongst a sample
building’s zones over a fourweek time span. The differences in
magnitude of occupancy from daytoday and weektoweek are readily
apparent. By simply examining the ratio of occupied to unoccupied zones
on a daily basis, these observations can be converted into a
probability estimating whether a given zone, chosen at random, is
likely to be occupied.
To test whether this mathematical representation improves the
predictability of a building’s behavior, a simple heating degree day /
cooling degree day energy usage regression model is created adding in
sensitivity multipliers based on magnitude of occupancy over this time
span. The figure shows the regression model output in energy use
prediction when comparing a prediction using the method above based on
the probability estimated of weekday occupancy magnitude versus an
estimate where the same level of occupancy is assumed during each
weekday.
Comparing
both models to the building’s actual energy consumption, one can see
that with both approaches, some days will over predict while others
will under predict. Over time in both cases, these errors cancel each
other out when creating an aggregated result (average). This is
consistent with the intuition that allows us to see reliable, cyclical
occupancy patterns when looking at magnitude of occupancy over a longer
period of time. The probabilistic model helps with giving a sense of
the building’s behavior under uncertain occupancy profiles. By taking
into consideration this previously unrepresented influence, a more
realistic assessment of the sensitivities of this input can be
calculated as it affects building performance.
In
practice, Ecorithm learns from the sensitivity of probabilistic models
to fine tune analysis and better understand what environmental impacts
most impact a building and what actionable insight can be generated for
a building operator to more accurately program a control system.
Applying sophisticated data science techniques such as probabilistic
models with spectral analysis gives Ecorithm the ability to understand
building operations at a granular level and provide valuable insight
with more certainty than other solutions. Ecorithm uses a unique
approach of examining spectral (frequency) content of building data to
understand the responses, patterns and sensitivities of different
building systems rather than utilizing only individual values in time^{3}.
This sophisticated analysis allows unseen cyclical behaviors of
building systems and equipment to be extracted and monitored.
Equipped with this unique knowledge of building cycles, Ecorithm’s
software can create and compare mathematical models to real world
dynamics in order to best understand how buildings operate and provide
superior Automated Fault Detection and Diagnostics (AFDD)
software.
^{1}The concept behind Ecorithm’s True Analytics Software’s
uses of weather were introduced by Ecorithm’s cofounder Dr. Igor Mezic
in the 2010 paper "Decomposing building system data for model
validation and analysis using the Koopman operator." Proceedings of the
National IBPSA USA Conference
^{2}Susuki, Y. and Mezić, I., 2014. Nonlinear Koopman modes and
power system stability assessment without models. IEEE Transactions on
Power Systems, 29(2), pp.899907.
^{3}Mezić,
I., Loire, S., Fonoberov, V.A. and Hogan, P., 2010. A new mixing
diagnostic and Gulf oil spill movement. Science, 330(6003), pp.486489.
About the Author
Dr.
Georgescu is the Director of Engineering & Research at Ecorithm.
Georgescu completed his PhD under Prof. Mezic with his thesis on the
Analysis of Systems in Building using Spectral Koopman Operator
Methods. Dr. Georgescu’s background is centered on studying the complex
behaviors exhibited in the building environment. Using spectral
methods, these complex behaviors can be understood by measuring the
oscillatory interactions that occur between various building systems.
Georgescu has established the use of spectral methods in various areas
of building analysis including design, fault detection, and modeling.
At Ecorithm,
these methods are utilized for automated analysis of the large datasets
which are produced by building sensor networks to characterize and
improve building operational efficiency.
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