Buildings as Dynamical Systems Part Two: Impact of Occupancy on a Building

Michael Georgescu, Ph.D.,  Director of Engineering and Research Ecorithm, Inc.

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 real-time data to provide instantaneous feedback to a device control loop but often over or under-estimate 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 day¹.  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 systems², weather, and ocean currents³ 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 systems-based 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 rules-based 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 sub-hourly 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 macro-level intuition of long term and recurrent occupancy patterns is ill-suited 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 four-week time span. The differences in magnitude of occupancy from day-to-day and week-to-week 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. 

[an error occurred while processing this directive]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 time³. 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. 

¹The concept behind Ecorithm’s True Analytics Software’s uses of weather were introduced by Ecorithm’s co-founder 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

²Susuki, Y. and Mezić, I., 2014. Nonlinear Koopman modes and power system stability assessment without models. IEEE Transactions on Power Systems, 29(2), pp.899-907.

³Mezić, I., Loire, S., Fonoberov, V.A. and Hogan, P., 2010. A new mixing diagnostic and Gulf oil spill movement. Science, 330(6003), pp.486-489.

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.