Online Model Estimation for Predictive Thermal Control of Buildings
Peter Radecki, Member, IEEE, and Brandon Hencey, Member, IEEE



Ideally, a Building Automation System (BAS) would
automatically modify set-points and load shedding based on
weather, occupancy, and utility pricing predictions [8].
Every building has unique and time-varying thermal
dynamics, occupancy, and heat loads which must be
characterized accurately if a BAS is to apply model
predictive controllers (MPC) to realize energy and monetary
savings [3]. Additional considerations include: measured
building data often contains low information content;
engineering models contain designer’s intent instead of
actual construction; and building’s usage evolves over time
[9]. Unfortunately, in practice there has yet to be
demonstrated a scalable, low-cost method to readily acquire
these much needed accurate models of individual buildings’
unique thermal envelopes.
For continuous commissioning and lifetime adaptability a
low-cost scalable method to acquire control-oriented
building models must: learn both the dynamics and the
disturbance patterns quickly, provide stable extrapolation, be
adaptable to future changes in building structure or use, and
use existing available data. White-box, first-principles,
forward modeling approaches are often inaccurate, not
robust to changes, and take extensive engineering or
research effort to build [2], [10]. Black-box approaches take
up to 6-months to train and cannot be safely extrapolated [5],
[6]. Recently gray-box methods have begun to show
potential as a scalable option for learning control-oriented
building models in some limited specific studies [11], [12],
[13], [14].
We propose a multi-mode Unscented Kalman Filter
(UKF) as a generalizable on-line gray-box data-driven
method to learn the building’s multi-zone thermal dynamics
and detect unknown time varying thermal loads. By coupling
known building information and simple physics models with
existing measurable building data we demonstrate how a
probabilistic estimation framework can overcome
shortcomings of many previously attempted specialized
solutions. Our method adapts over time to continually learn
both dynamics and disturbances while providing stable
prediction performance.
Continuing our work in [11], this paper aims to generalize
our findings and method with the following contributions:
 literature survey on control-oriented thermal
modeling for buildings,
 development of minimal parameterization for
dynamics estimation,
 generalized thermal disturbance pattern estimation,
 multi-mode heuristic for simultaneous parameter
and disturbance estimation.

Abstract—This study proposes a general, scalable
method to learn control-oriented thermal models of
buildings that could enable wide-scale deployment of
cost-effective predictive controls. An Unscented Kalman
Filter augmented for parameter and disturbance
estimation is shown to accurately learn and predict a
building’s thermal response. Recent studies of heating,
ventilating, and air conditioning (HVAC) systems have
shown significant energy savings with advanced model
predictive control (MPC). A scalable cost-effective
method to readily acquire accurate, robust models of
individual buildings’ unique thermal envelopes has
historically been elusive and hindered the widespread
deployment of prediction-based control systems.
Continuous commissioning and lifetime performance of
these thermal models requires deployment of on-line
data-driven system identification and parameter
estimation routines. We propose a novel gray-box
approach using an Unscented Kalman Filter based on a
multi-zone thermal network and validate it with
EnergyPlus simulation data. The filter quickly learns
parameters of a thermal network during periods of
known or constrained loads and then characterizes
unknown loads in order to provide accurate 24+ hour
energy predictions. This study extends our initial
investigation by formalizing parameter and disturbance
estimation routines and demonstrating results across a
year-long study.
I. INTRODUCTION
A. Overview
Significant energy savings in buildings’ heating,
ventilating, and air-conditioning (HVAC) systems could be
realized with advanced control systems [1], but deployment
of these control systems requires a method to readily acquire
low cost models of buildings’ unique thermal envelopes [2],
[3]. Previous studies have investigated several methods but
generally arrived at non-scalable specialized solutions [4],
[5], [6], [7].
This work was supported by the Department of Defense (DoD) through
the National Defense Science & Engineering Graduate Fellowship
(NDSEG) Program.
P. Radecki and B. Hencey were with the Sibley School of Mechanical
and Aerospace Engineering, Cornell University, Ithaca, NY, 14850 USA.
Currently Radecki is a Transmission Development Engineer at General
Motors Powertrain in Detroit, Michigan, email: ppr27@cornell.edu.
Currently Hencey is with the Air Force Research Laboratory in Dayton,
Ohio, email: bhencey@gmail.com.
Copyright © 2015 Peter Radecki. All Rights Reserved.

1

A comparison of UKF and EKF applied estimation
techniques is included for the benefit of practicing engineers.
The robustness of the UKF estimation technique learning
both parameters and disturbances is demonstrated against a
multi-zone high-fidelity EnergyPlus simulation in a yearlong study.
A short explanation of our proposed method follows.
Using a simple first order heat transfer model with multiple
zones, the UKF estimates model parameters of the thermal
dynamics during periods which have small or wellcharacterized thermal loads. After learning the dynamics
during low disturbance periods, such as nighttime, the UKF
is augmented to track unknown disturbances while
continuing to improve its dynamics model. The UKF is
simpler than an adaptive control technique to implement
because it internally maintains a covariance quality metric
which only adjusts parameter estimates if the incoming data
provides new thermal information. Fig. 1 shows how the
proposed UKF enables rapid deployment of advanced
predictive controllers for BAS.
The paper is organized as follows. First, a background
section initially examines the scope of the problem and
previous approaches before proposing and analyzing an
extensible on-line data-driven approach. After deriving the
thermal model and parameterization, we formulate and
compare performance of the EKF and UKF. True utility of
the UKF is then demonstrated across a year-long study.
Based on data generated from our simple passive 5-zone
thermal model plus a more complex passive 5-zone
EnergyPlus simulated model, less than 2 weeks of training
data is shown to make reliable 24-hour predictions. Based on
testing and performance, a discussion of how the UKF fits
into the building thermal modeling problem and an
identification of areas of future research conclude the paper.

Despite these significant advances, most modern buildings
have realized only minimal energy savings [3]. BAS modelbased predictive controllers are rarely implemented and
typically underperform [15]. Larger buildings’ successes in
realizing energy savings has been generally limited to
automated lighting control, changing nighttime temperature
set-points, and load shedding during peak demand [15]. The
data available from BAS is underutilized—generating trend
plots instead of enabling intelligent energy management
decision and control tasks [4]. It is not uncommon for
buildings to simultaneously run cooling and heating
components throughout all 12 months of the year [9]. Load
shedding, reducing peak usage by duty cycling off portions
of a system, is often based on some arbitrary component
order and can significantly impact occupant comfort—if one
portion of the system that is already at capacity sheds, it may
not recover until nighttime [16]. Certain sites, such as Drury
University and UC Merced have a human operator regularly
check weather forecasts and vary temperature set points
based on personal experience and intuition [16], [17]. These
techniques are labor and expertise intensive. Usually,
however, the cost of the human operator doesn’t justify the
energy savings and limits widespread deployment.
Optimal building thermal control is a multi-objective
optimization problem involving user comfort, air quality,
energy cost, smart grid demand response, and thermal
dynamics, which generally cannot be performed optimally
by a human operator without expertise. Realistic widespread
improvement in building controls requires a scalable method
to accurately learn unique thermal models.
C. Building Controls and Modeling Survey
Many researchers have evaluated and implemented
custom one-off BAS demonstrating energy savings with
MPC [18], but widespread implementation has been illusive
because a scalable method to learn thermal models has not
been available. Both forward and inverse modeling efforts
have traditionally taken months of a researcher’s or
engineer’s time to accurately generate. Given that the
majority of buildings which will be in use by 2030 are
currently over 10 years old, the retrofit problem is
significant. A scalable method must handle new
construction, existing structures, and remodeling projects
and continue to function if a building is repurposed.
White-box, or forward models can be generated by
analyzing blueprints, building materials, and expected use
patterns to generate a sophisticated simulation, but are
generally not cost effective for retrofit or existing buildings.
Furthermore, some studies have shown that predicted and
actual energy consumption can differ by up to 40% using
forward models from the design stage [10].
MPC can perform adequately given simple, accurate first
or second order heat transfer models, so there is viability for
inverse models [9], which use building data recorded over
time to infer the dynamics. Today’s networked sensors and
available computation power make data-driven models
feasible. The inverse modeling paradigm can use either pure
numerical methods in a black-box environment or physics,
first-principles based methods in a gray-box environment.

B. Building Automation Systems Background
Buildings use 39% of the total US energy supply, a
significant fraction of which is used to provide people with a
comfortable indoor working and living environment by
operating HVAC systems. It is estimated that 25% to 30% of
building energy usage or around 10% of the total US energy
consumption could potentially be reduced with component
and controls upgrades [1], [2]. Reducing this energy
consumption would also help significantly reduce CO2
generation [9].
BAS and their connected components advanced
tremendously over the past few decades and have started to
be widely deployed in modern buildings and retrofit into
remodeled buildings. Utilizing technologies such as wireless
sensing, LonTalk, and BACnet, BAS provide networked
infrastructure making it easier for sensor information and
control signals to be distributed throughout and between
buildings. Component advancements include sensors that
can detect occupants, CO2 level and light in addition to
traditional temperature and humidity measurements.
Furnaces, water heaters and air conditioners have all started
to approach their maximum theoretical efficiency. The
computational power necessary to run demand-responsive
and predictive control algorithms is cheaply available.
2

Current Methodology:
Up to 1 year of time per building

Proposed Methodology:
Weeks of time per building

Analyze building
plans/geometry

Discover/Utilize networked
sensors in building

Manually forward model
and simulate or perform
custom study of building
using recorded data.

Recursive online gray‐box
learning algorithms
determine dynamics of
building

Apply advanced controls

Apply advanced controls

Fig. 1. Comparison of existing typical thermal modeling process left and proposed method right . Picture Credits: EnergyPlus: DOE, R.
L. Smith Building center : ThermoAnalytics, and floor plan: 3 .

gain wide acceptance because zone interactions affect BAS
control systems [18].
In [11] we demonstrated the first published study of a
scalable modeling and online estimation framework for
multi-zone building states, parameters, and unmodeled
dynamics. Since our study in 2012 several other researchers
have validated our initial claims and highlighted new
challenges.
Studies using real and simulated data demonstrated
individual aspects of the proposed scalable modeling and
estimation framework. Massoumy in 2013 [13]
demonstrated the applicability of gray-box estimation with
real data collected from Michigan Tech’s new Lakeshore
Center building–he used an off-line batch parameter
estimation routine with online state estimates, validating the
results of our EKF versus UKF comparison. Fux [12]
demonstrated the relevance of 1R1C models to individual
rooms or entire zones by generating accurate predictions
from a simple single-zone model of an entire building using
real data in an EKF. The study by Fux [12] also validated the
concept of multi-mode learning and the importance of
characterizing disturbances. Martincevic [14] used simulated
data from IDA-ICE in a year-long study to demonstrate a
50-zone model that learned parameters without disturbances
from a constrained UKF. The model had a prediction error
of less than 1oC RMS error if no unmodeled disturbances
were present.
Other researchers demonstrated the extensibility of the KF
as an online estimation framework and presented important
insight. Studies showed ground coupling was not necessary
in RC models for certain scenarios [8], extended an EKF for
fault diagnosis and monitoring with real-data from a
supermarket [20], used an Ensemble KF to constrain
parameters to physically realistic values [21], and showed
the applicability of RC models as a design tool using offline
parameter estimation [22]. Lin [23] showed the need for
meaningful inter-zone excitation in order to guarantee
satisfactory information content is present in measured data.
Lin’s result corresponded with our follow-up study using an
UKF learned model for MPC [24]. Lin further noted that one
simple plot demonstrating prediction accuracy versus

Many researchers from present day to those who participated
in the 1990’s ASHRAE Energy Predictor Shootouts have
taken data-driven approaches, but their generated methods
and models have typically been too specialized to scale to
other buildings [2], [4], [5], [6]. A brief comparison of
previously tried data-driven methods is provided before
proposing our solution.
Black-box models such as Artificial Neural Networks
(ANN) are often impractical for modeling thermal dynamics
in building systems because of the large amount of training
data (6 months to a year) that must be analyzed before
getting an accurate model [6]. Because ANN create an
arbitrary representation of the system, they are sensitive to
the quality of the data collected, may meld together effects
from loads and dynamics, are not robust to component
failures, and are not adaptable when building use and
configuration patterns change [2], [18]. Pure numerical
methods may have better applications where simple heuristic
and physical models are impractical such as pattern
recognition of occupancy, lighting, or thermal component
loads.
Gray-box methods, which use pre-existing knowledge of
the dynamical structure, have been used off-line with genetic
algorithms (GA) [5] and recursively on-line (in real-time).
Off-line methods may be good for initial model acquisition
but on-line methods are desirable—buildings age and change
physically over time due to deterioration or reconfiguration
and change temporally as occupant usage evolves. On-line
methods can also be dual-purposed as process monitoring,
analysis, and fault detection devices.
Examination of traditional online gray-box techniques in
buildings with adaptive control [2] and Extended Kalman
Filters (EKF) [19] shows the need for further development.
Historically, adaptive control techniques have fallen short
because they require autonomous tuning of a complex
forgetting factor, which requires actively monitoring the
excitation level of incoming data as an information criterion
used to enable or disable thermal model learning [2]. Many
on-line methods [2], [4], and [5] have used single zone
models for demonstrating utility, but in reality multi-zone
simulations are necessary before any modeling method will

3

measured data is meaningless in that it doesn’t say anything
about the model’s robustness for control.
In summary, recent studies have shown RC models are
applicable for thermal modeling of buildings and Kalman
filtering can learn parameters for models of buildings using
both simulated and real data. Our paper builds upon recent
advancements
with
an
explanation
of
unique
parameterization and formalizes the multi-mode estimation
technique with a generalized algorithm for learning any
disturbance pattern. We conclude with a year-long study of
simultaneous state, parameter, and disturbance estimation
showing robust, meaningful prediction accuracy.

Thermal radiation does play a significant role in the
heating and cooling of many buildings. For the purposes of
evaluating UKF parameter estimation and thermal load
detection for a passive building, the radiation between
surfaces and zones is linearly approximated and lumped with
convection and conduction [25]. The proposed framework
could readily be augmented for nonlinear radiation effects at
the cost of increasing the number of associated parameters to
learn.
Thermal disturbances significantly affect most buildings
but are often overly complex to model requiring information
about building geometry and neighboring foliage [26]. Solar
gain is treated as an unmodeled external disturbance. This
simplification removes complexities of modeling diffuse and
direct sunlight, shading, and night sky radiation temperature,
and allows for simple disturbance generation in EnergyPlus
by turning on or off environmental radiation transfer. The
solar gain provides us with a specific periodic disturbance to
estimate with patterns. In practice this technique could
estimate any number of disturbances if one has some
information about the disturbance frequency, intensity, or
timing such as dusk and dawn times or building occupancy
times. Common examples amenable to disturbance pattern
estimation include occupant body-heat, equipment,
computers, electrical loads, lighting, and HVAC.
Using the 2-node example, a state space representation
can be derived where is a vector of temperatures; A is a
matrix of RC values; and
is a vector of additive,
independent, time-varying disturbances such as solar
radiation.

II. PARAMETER ESTIMATION FORMULATION
A. Thermal Model
A standard thermal network captures the dominant
convection and conduction heat transfer modes and mass
transfer occurring between zones inside and outside the
building, while solar gain is treated as an unmodeled
disturbance. Internal zone radiation is linearly approximated
and lumped with convection and conduction [25]. The
thermal network matches that commonly used in the
community and provides a simple mechanism to explore
simultaneous
model
and
disturbance
estimation.
Subsequently developed filters use the thermal model but are
not restricted to it. One could select a non-linear model
incorporating HVAC dynamics and use it in the proposed
estimation framework.
Convection, conduction, and mass transfer heat flux
(watts) into zone
is contributed from the temperature
differential to connected adjacent node(s) divided by the
(degree/watt) plus an additive term
thermal resistance
(watts) representing disturbances. (Note: unless otherwise
mentioned, subscripts denote zones.)

1

1
1

1

2

/
and thermal capacity
(joule/degree)
The heat flux
affects the time-based temperature rate of change .

Based on [27], an n-node thermal network can be formalized
by defining a simple undirected weighted graph with: nodes
and temperatures
≔ 1,2, … ,
assigned capacitances
; edges
⊂
that connect adjacent nodes with
weights
∀ , ∈ such that
∀ , ∈
that
are assigned resistances. For a general thermal network with
n nodes, the A matrix is
0
if
, , ∉

Substituting for , the temperature rate of change of node i
due to connection(s) with node(s) j and disturbance
becomes
/

/ .

1

The derived representation for temperature change due to
heat transfer is mathematically analogous to voltage change
due to current flow in a resistor-capacitor network. For
visualization a simple 2-node example with two capacitances
and one resistance is shown in Fig. 2.

if
∑

, ,

∈ .

3

if

B. Parameterization
A minimal set of independent parameters must be
specified for filters to enforce the system dynamics during
parameter estimation [28]. Over-parameterization causes
unidentifiable parameter manifolds or extra degrees of

Fig. 2. Two node example thermal network.

4

freedom and can result in violation of dynamics constraints
and physics laws such as conservation of energy. In machine
learning and system identification, indeterminate degrees of
freedom can cause overfitting where the model learns the
noise instead of the dynamics of interest. In estimation
theory, parameter observability requires that the Fisher
information matrix is invertible—redundant parameters or
over parameterization breaks this observability criterion
resulting in an unobservable subspace [29].
Efficient and reliable parameter estimation requires
estimating a minimal number of parameters [28]. From
Equation 2 there are only two unique parameters required
to describe the A matrix despite it containing three
variables—two resistances and one capacitance. The extra
parameter acts as a scaling factor and can be quantified only
if the heat flux q is provided in addition to the temperature
histories. Without the scaling factor only a time-constant can
be inferred. Suppose the time-constant for our system was 1.
Then
1 and we get the plot in Fig. 3 of possible
values for
and , many of which are violations of
physics first principles such as a negative thermal
capacitance and resistance. Unfortunately, removing the
negative-valued parameter space does not resolve the
ambiguity in selecting the true resistance and capacitance
values for the provided time constant. This ambiguity
generally makes the estimation problem numerically
unstable, theoretically unobservable, or practically
unreliable. Rectifying the ambiguity could be done with
actual heat flux information which is generally unavailable
in practice, so for this study, selecting a minimal set of
parameters mitigates the problem.

Now we present methods based on a careful graph study
to obtain a minimal parameter set for thermal network
estimation. Because diagonal terms in A are linear
combinations of the off-diagonal terms, parameter
estimation is only performed for off-diagonals. RC products
are estimated together in order to reduce the non-linearity of
the estimation problem. Parameterization of trees, graphs
with no cycles of which Fig. 2 is an example, with combined
RC products automatically guarantees a minimal
representation of the system.
Unfortunately this minimal guarantee does not extend to
graphs containing closed cycles. Fig. 4 is an example of a
graph containing a cycle whose state space A matrix is
shown.

4

We arbitrarily selected R13C1 to show that one of the six RC
products is redundant and can be eliminated by multiplying
and dividing the other RijCi parameters by each other around
the cycle:

In a graph, each cycle which uses at least one unique edge
and passes through no nodes with infinite capacitance may
be used to eliminate one redundant RC product from the
estimation problem by multiplying and dividing around the
loop. For any thermal network the total number of unique
parameters should be one less than the sum of the number of
resistances and capacitances.
In general unique edges should be selected for
elimination. Eliminating a shared edge between two cycles
joins the two cycles mathematically through multiplication
in the estimation routine which can negatively impact
numerical stability. Selecting multiple redundant parameters
to prune from a graph estimation problem should be done
such that each redundant RC parameter lies on a globally
unique edge for its respective cycle, and the shortest
available cycle should be chosen for calculation in order to
guarantee minimal parameter cross-sensitivity.
Two nodes that have no shared conduction or convection
are considered independent, and any edge directly
connecting them is pruned from the graph to give the
simplest representation. Independent ambient nodes such as
external temperatures have infinite capacitance in the
thermal network. External nodes may have unique update
functions depending on the simulation and weather desired
for the modeling exercise. As an example look back at Fig.
2, if
were an external temperature, setting
∞ would
give the following state space representation.
1
1

Indeterminate Parameterization: Rij *Ci =1
4

Ci

2
0
-2
-4
-4

-2

0
Rij

2

4

Fig. 3. Non‐minimal parameterization gives rise to estimation
ambiguity. The curve satisfying
1
demonstrates the
unobservable subspace which includes physically meaningless
and cannot be uniquely identified.
negative quantities.

C1

R12

R13

C2

R23
C3

0

Fig. 4. Three‐node graph with one loop.

5

0

5

drawn from distribution . The measurement noise terms
̅
are drawn from distribution .
The measurement function of the thermal network is
linear so H is simply an n row identity matrix (provided all
temperatures are measured) padded with columns of zeros
for the parameters. The update portion of the filter therefore
can be written as a simple linear Kalman Filter. However in
the prediction step, calculation of the Jacobian matrix F
depends whether one chooses to estimate RC or 1/RC
parameters. A comparison of both cases demonstrates that
RC parameters would be a poor choice, especially during the
initial acquisition stage, because poor parameter estimates
will be squared and could easily cause the filter to diverge
and blowup.

C. Extended Kalman Filter (EKF)
For all tests the state space system is integrated at one
minute interval time-steps with Euler integration to allow
discrete-time filter implementation. Parameter estimation
with the Kalman Filter is achieved by augmenting the
temperature states
…
with unique parameters
… 1/
and
disturbances
̅
1/
/ … /
together in the state representation
, ̅ ,
. For the purposes of estimation, the full
discrete-time stochastic system is
1
1
1
̅

̅

6

̅

̅

Case 1:

represents process noise,
represents
where
represents
estimation uncertainty in RC parameters,
process noise for disturbances, and ̅
represents
measurement noise. All noise terms are assumed zero mean,
Gaussian, white, and stationary. Note that in 6 , the matrix
is actually a vector of functions composed from ̅
as
defined by the parameterization of the RC terms. This
representation results in multiplication and division of
estimated parameters through the dynamics function.
Specifically, temperature is being multiplied by RC
parameters necessitating non-linear estimation techniques.
For baseline comparisons, an EKF and UKF are
formulated. In order to define notation, the prediction and
update steps of the discrete EKF are shown in 7 , but for
proper treatment of the derivation and background of the
Kalman Filter, please consult [29], [30], [31]. (Note: For
brevity in the following Kalman Filter formulations, notation
deviates from the modeling section: subscripts denote time
rather than node indices.)

Case 2:

|

|

|

,

State Estimate
State Covariance
Update:

|
|
|
|
|

|
|

Innovation
Innov. Covariance
Optimal Kalman Gain
State Estimate
State Covariance

f
x xˆ k 1|k 1 , u k 1 ,

Hk 

8

D. Unscented Kalman Filter (UKF)
Unlike the EKF, which uses a Jacobian first order
linearization evaluated at the current estimate to propagate a
probability distribution through a non-linear transform, the
UKF uses the Unscented Transform to pass a distribution
through a nonlinear transform. Specifically the UKF samples
(2n+1) points in the distribution, evaluates each point
through the non-linear transform and then recombines these
points to generate a transformed mean and covariance which
is oftentimes more accurate and stable than that obtained
from the single point EKF linearizations [32]. The samples,
called sigma points, are evenly spaced to capture at least the
first and second order moments of the distribution and are
weighted such that the covariance and mean of the samples
matches that of the original distribution. After being
mapped through the non-linear transform the resulting points
are multiplied by their assigned weights to determine the
transformed mean and covariance. For linear systems both

7

Jacobians:

Fk 

1

1

Thus, for numerical stability the EKF must follow Case 2
and estimate 1/RC parameters.
Measurement noise is specified based on the accuracy of
the temperature sensors. Process noise is specified for the
temperature states based on the level of zone aggregation
used while the RC and disturbance process noise is set to an
artificial value greater than zero in order to allow the filter to
vary its estimate of these parameters through time.
Increasing process noise level for any parameter indicates
that the model isn’t confident of its ability to describe the
process evolution of that parameter. Disturbances, which by
their nature the model is not explicitly capturing, are biased
and vary with time. In order to estimate the disturbances
over time; their noise level is set to be non-zero. Because the
RC values should be fairly constant while the disturbance
bias may change throughout the course of a day, the noise
level for RC parameters should be much smaller than
disturbances.

Predict:
|

1

h
x xˆ k |k 1

For the filter dynamics function
, temperature state
dynamics follow the previously derived thermal model while
the RC and disturbance parameters are modeled as constants.
Artificial process noise for the constant parameters, denoted
, allows the filter to change its estimate of these values
through time and allows the filter to track the true time
,
varying disturbance. The set of process noise terms
, and
are stacked as defined by the state and
6

the EKF and UKF perform identically to a traditional
Kalman Filter but for certain non-linear systems a UKF can
provide higher accuracy with the same order of calculation
complexity.
The UKF was implemented with the same augmented state
vector and used the same measurement and process noise
values as the EKF. Removing the requirement to design and
calculate a Jacobian, the UKF is amenable to either RC or
1/RC parameter representations—both are evaluated in the
results section. The standard values of
10 ,
0,
2, typical for a Gaussian distribution, were used to
generate the following samples and weights .
,

,

|

For:

|

1, … ,

|

|

For:
,

A 10,000 run Monte Carlo simulation was conducted to
compare the EKF and UKF performance for the simple twoparameter search problem. Table 1 shows the distributions
for all parameter and filter values which were randomly
sampled at the beginning of each test run. The distributions
of parameter and temperature values were chosen such that
some runs will have high levels of excitation while others
will have no excitation—effectively T(0) equal to Text.
Results of the simulation, shown in Table 2, demonstrated
that the UKF outperformed the EKF for non-linear
estimation of parameter p1 but had statistically similar
performance for estimation of linear parameter p2. The UKF
showed filter stability—resilience to exponential tracking
divergence—while 2% of the EKF runs went unstable and
did not complete execution. These results agree with
published studies comparing the filters’ general performance
in other application areas [33], [34]. The EKF could estimate
linear additive disturbances, which is in agreement with
[19], but when estimating coefficient parameters such as
weights in a thermal network, the UKF is a more viable
solution.
Further testing showed that increasing the number of
temperature zones or trying to directly estimate RC instead
of its reciprocal, 1/RC, further degraded the EKF stability
and performance. However for the UKF, comparison of
estimating RC products and their reciprocals showed that
direct RC parameters are more robust. For zones with little
thermal connection, the filter estimating RC parameters will
continue increasing estimates but will be finitely decreasing
the covariance, so the estimate will eventually converge.
When 1/RC parameters are utilized for the same zones, the
estimate will be driven to zero, but the covariance will not

|

1, … ,2

|

|

|

1
For:

1, … ,2

1
2∗

The samples were then recombined to give the a priori state
and covariance estimates.
|

|

|

f

|
|

TABLE I
MONTE CARLO SAMPLED PARAMETER DISTRIBUTIONS

|

|

|

Variable
|

Range

0,1
0,20
̂ 0
0,1
̂ 0
0,20
0
0,100
0
0,10
0
| .5, .5 |
| .5, .5 |
1,0,0
1, 0,10 , …
Est. Process Variance
0,10
1, 0,5 10 , …
Initial Est. Variance
0
0,5 10
= Uniform Distribution,
= Normal Distribution.
Truth
Truth
I.C. Estimation
I.C. Estimation
Truth
I.C. Estimation
True Meas. Error
Est. Meas. Error
True Process Variance

Note that λ and W can be reused so only the χ terms need to
be recalculated each iteration. Because the measurement
function is linear, the unscented transform is only used in the
prediction step; the measurement step is simply calculated as
a linear Kalman filter update.
E. EKF vs. UKF
In order to compare an EKF and UKF the 2-node thermal
network from Fig. 2 was chosen. Looking back to 5 , the
system can be parameterized as Ap.
0

Quantity

0

TABLE 2
10,000 RUN MONTE CARLO RESULTS

In order to compare both the parameter estimation and bias
detection capabilities only T was sensed—Text was treated as
was used
a constant latent variable. A second parameter
to estimate Text as a linear additive disturbance as shown.

Test
EKF
EKF
EKF
UKF
UKF
UKF

9

7

Statistic
Numerical Instability Count
p1 Average Estimation Error
p2 Average Estimation Error
Numerical Instability Count
p1 Average Estimation Error
p2 Average Estimation Error

Result
175
3.16
60.5
0
0.18
58.0

decrease as quickly, which can result in a zero crossing that
violates conservation of energy by estimating a negative
parameter. As a result, RC parameters were estimated by the
UKF for all remaining tests.

R.C. Disturbance

Solar Gain
1000

0.1
0.05
0
-0.05

F. Disturbance Estimation and Pattern Recognition
Direct sensing of disturbance heat flux is rarely practical in
building systems. However, timing information for
disturbances is typically available, so a practical method is
presented to learn disturbances given only timing
information with no prior disturbance quantification. This
method could readily be augmented if additional disturbance
heat flux data was available.
Looking at thermal model 1 , a change in zone
temperature may be explained away using either connected
zones with their respective temperatures or additive
disturbances. In order to get satisfactory estimation
performance this ambiguity must be considered in the filter
design lest it manifest problems akin to non-minimal
parameterization. The engineering solution selected to
rectify the problem, splits the estimation problem based on
the presence of disturbances and manipulates the process
covariance for estimated disturbance parameters based on
timing of expected unquantified disturbances. From a
control theory perspective the system does not have timeinvariant observability. However, buildings are a timevarying system that have some periodicity. Looking over a
horizon, for example one day for solar disturbances, we can
learn constant parameters when no disturbances are present
and learn disturbances after having estimated the constant
parameters. This partitioning enables time-varying
observability.
The presented method is not claimed to be optimal, rather
it is a practical solution based on engineering judgment of
typical scenarios common in buildings. Typically, a system
can sense if people are using a building but cannot measure
their heat flux or the equipment they use, likewise it can
sense if the HVAC is on but not the exact heat flow
delivered to a specific room. The approach attempts to use
commonly available timing knowledge to quantify and infer
disturbances that are not directly measurable and only
partially predictable.
Learning of disturbances was done in a Markov fashion:
the estimator assumed no knowledge of previous historical
disturbance patterns and estimated a new disturbance value
at each time step based on the previous time step’s
estimate, the dynamic model, and the current measurement.
The disturbance states in the UKF were modeled as
constants which have zero-mean Gaussian additive noise. A
characteristic change in the disturbance such as a heater
turning on or the sun coming up at dawn violates the zeromean assumption causing a bias in the disturbance. In order
to track these sudden bias changes using a simple UKF, the
variance(s) correlating to those specific zone(s) disturbances
were inflated to allow the filter to acquire and track the new
value. This artificial tuning of the covariance is similar to
tuning a forgetting factor in adaptive control frameworks.
In the Matlab-based simulation, heating or cooling was
arbitrarily added to individual zones from 10am to noon, so

500

0

6

12

18

24

0

-6

1

0

0

-1

-1
6

6

12

18

24

12 18
Hour

24

x 10 Variance

1

0

0

-4

x 10 Variance

12 18
Hour

24

0

6

Fig. 5 Disturbance Parameter Process Variance Tuning

the covariance was increased around those times. In the
EnergyPlus simulation, the primary unmodeled disturbance
was solar radiation, so the covariance was increased around
dusk and dawn. This variance tuning is visually depicted in
Fig. 5.
Because the UKF can explain temperature swings by either
tuning RC or disturbance values, a multi-mode approach was
chosen as a uniform method to split the estimation problem.
In order to acquire good estimates, the UKF was operated in
two modes: A) Acquisition Mode: initially only RC products
were estimated by running the filter at night when solar
gains were at a minimum, and B) Monitoring Mode: both
RC products and disturbances were estimated
simultaneously after RC product estimates had started to
converge to constant values. This splitting proved critical to
obtaining good estimates from EnergyPlus data but
unnecessary for the simple Matlab based simulation due to
its consistently high level of external temperature excitation.
Disturbance estimates from the entire multi-day learning
period were heuristically combined in order to generate a 24hour pattern. This disturbance pattern was then used when
predicting the building’s thermal response.
The heuristic pattern recognition algorithm was a simple
weighted average. For thermal network simulated data the
true disturbances were identical every day. Thus the bias
values learned at each time step were equally averaged to
generate one 24-hour pattern for each bias term.
Mathematically this can be written as summation over
days of learning with minute time steps to generate a
disturbance pattern
correlating to zone based on the
as shown, where the value inside
UKF estimated bias
brackets notates the minute-based time step index.
,

∀ ∈ 1

24

60

10

The predominant unmodeled disturbance from EnergyPlus
data was radiation from the sun, which is dependent on the
cloud cover, time of day, season of year and other factors.
For the purpose of engineering a robust simple solution we
made a realistic assumption that we had a measurement of
the average solar intensity for morning and afternoon and
used this for both pattern recognition and simulation
8

predictions. The solar intensity reading was calculated as the
summation of the Environment Direct Solar and
Environment Diffuse Solar variables from EnergyPlus.
Algorithm 1 contains the weighted average and prediction
steps used for EnergyPlus simulations.
ALGORITHM 1
♦ INITIALIZATION
-Max solar radiation:
-Indices: zone , day , time of day
-Dawn, midday, and dusk times:
,
-Morning & afternoon avg solar radiation:
,
-Estimated bias per zone :
-Measured solar radiation intensity: ,
♦ WEIGHTED DISTURBANCE PATTERN
-Disturbance Pattern:
for
1 to
if
/2 0.35
for
1 to 24 60
,
if
then
elseif
then
,
elseif
then
,
else
end
end
end
/
♦ PREDICTED DISTURBANCE
-Bias for Prediction:
for
1 to …
for
1 to 24 60
if
then
,
,
elseif
then
elseif
then
,
,
else
end
end

III. SIMULATION RESULTS
A. UKF 5-room Simulated Performance
A six-node thermal network, corresponding to five internal
zones and one external temperature shown in Fig. 6, was
used to evaluate the UKF parameter estimation and thermal
disturbance detection. For this first evaluation, measurement
data was generated from a model whose dynamics were
structurally identical to the dynamics used in the UKF. The
5-room models shown here are based on that in [11], but
feature extended explanations, derivations, and simulation
results.
Given five finite capacitances and thirteen resistances,
there were a total of seventeen unique RC products for the
UKF to estimate in this model. A first test was run in
acquisition mode, so disturbance estimation was disabled.
The external temperature forcing function was composed
from the sum of a 40 degree peak-to-peak sinusoid with
period of one day, a 10 degree peak-to-peak sinusoid with
period of 4 hours, and random noise which would allow the
temperature to drift from day to day. Resistance and
capacitance values were chosen such that thermal lag in the
simulation would be similar order as thermal lag in a small
to medium sized building. Temperature states were
initialized with less than a degree of error while RC
estimates were all arbitrarily initialized to 1000 with a
standard deviation of 500. Using four days of recorded data,
RC parameters were learned by the UKF. At the end of the
four day simulation, a 48 hour prediction of the five zones’
temperatures was made using the acquired RC parameters.
The true 48 hour external temperature profile was provided
to both the true dynamics and UKF dynamics in order to
establish a fair comparison baseline for evaluation of the
UKF. In a real system inaccuracies in the weather forecast
would degrade the prediction quality.

,
,

then
, /
/

By averaging the solar intensity before midday and after
midday, two weights were determined for each day. In order
to ensure sufficient signal to noise ratio for disturbance
pattern estimation, a day’s disturbance estimates were
discarded if the day’s total solar intensity averaged below
35% of the maximum solar intensity possible for that
location. The remaining days had their bias estimation
values multiplied by the ratio of maximum solar radiation to
respective half day average measured intensity in order to
normalize the bias values. Then the normalized bias values
were averaged using the same equation for to estimate the
repeated daily disturbance profile. When used for
predictions, the disturbance profile was scaled by the
predicted solar intensity weights to predict each zone’s
unique solar disturbance quantity at each timestep
throughout the day. Thus predictions utilized the final
estimated RC parameter, 24-hr disturbance patterns,
predicted external temperature profile, and half-day average
predicted solar intensity. The pattern recognition shown in
Algorithm 1 could be applied to any cyclic disturbance. The
variables ,
,
, and
would be
adapted to whatever period and timing knowledge a designer
had of other loads or disturbances present in the
environment, and the arrays
and
would be
modified to contain the pattern array.

Fig. 6. Left: Five‐room building used
for simulations, Right: Top view of node labeled internal thermal
network representation. White lines represent building internal walls.
A sixth unlabeled node acts as an external temperature

4000

4000

3000

3000

Truth
Estimate

2000

2000

3  variance

1000

1000

0

0

-1000

0

0.5
Time (Day)

-1000
1
0

0.5
Time (Day)

Fig. 7. Example estimation of two parameters from 5‐room building
using RC‐thermal model data.

9

1

Prediction Performance

Bias Estimates
Temperature (deg/min)

Temperature

80
70
60
Truth
Prediction

50
40

0

0.5

1
Time (Day)

1.5

0.1
0.05
0

0

6

12

18

24

12
Time (hrs)

18

24

0
-0.05
-0.1

2

0

Truth
6
Estimate

Fig. 9. The actual disturbances introduced into system for the two
zones shown in Fig. 10 are labeled above as “Truth” and were
repeated on a 24 hour cycle. Dashed lines represent the UKF 4‐day
average estimate of the disturbances which was used for
predictions in Fig. 10. Deg F

Fig. 8. Example 48‐hour prediction window in a disturbance free
environment based on RC estimates. External temperature forcing
function was composed from two sinusoids plus white noise. Deg F

Running the described simulation showed that some
parameters are estimated very well while others are not; two
characteristic examples of this are shown in Fig. 7. The
external temperature does not fully excite all of the node-tonode thermal connections which limits the filter’s ability to
precisely determine all of the parameters. Despite this
numerical estimation error, the 48-hour prediction
demonstrated excellent matching between the UKF
estimated model and the true model. From a poorly
performing filter one would expect predictions to have
increasing divergence or lag as the prediction window is
increased. Characteristic temperature predictions of the
highest and lowest capacitance rooms are shown in Fig. 8 for
visualization. In our chosen model the poorly estimated
parameters correlate with higher order dynamics that neednot be accurately estimated for good model fitting. For
practical applications this is analogous to having multiple
zones that are always excited together such that their relative
interaction need not be known for useful predictions.
For a second evaluation daily repeating disturbances were
introduced uniquely into each zone of the house. The
disturbance states in the UKF were modeled as constants
which have zero-mean Gaussian additive noise; in order to
track a sudden bias change using a simple UKF, the variance
correlating to that specific bias must be inflated to allow the
filter to acquire and track the new value. This inflation must
occur anytime the disturbance changes significantly enough
that the zero mean Gaussian assumption is severely violated,
such as when an HVAC system turns on or at dusk and dawn
due to solar radiation. For the combined RC and disturbance
estimation, 4 days of data were again used for training.
Disturbances turned on at 10:00 and off at 12:00 each day,
so the estimate covariance P was boosted at those times and
the process variance was also temporarily increased when
the disturbance was cycled on and off as previously shown
in Fig. 5. With this variance tuning method, the filter had
excellent disturbance tracking and similar RC estimation
accuracy. Example tracking of two disturbances is shown in
Fig. 9. Using the described simple heuristic pattern
recognition with a 24 hour period shown in Equation 10 ,
the estimated biases from all four days were averaged to
generate a single day estimate of the repeating disturbance
pattern. A 48-hour prediction was then made using the
average disturbance and final RC estimates along with the

Prediction Performance
80

Temperature

75
70
65
Truth
RC Only Pred
RC + Dist Pred

60
55

0

0.5

1
Time (Day)

1.5

2

Fig. 10. Example 48‐hour prediction derived from RC estimates
and estimated daily cyclic disturbances. Gray boxes denote times
where external disturbances were present in the true system. An
external temperature forcing function was composed from two
sinusoids plus low amplitude white noise. Deg F

exact external temperature profile. Again, excellent
predictions were made with the estimated model. Fig. 10
plots temperatures of two zones comparing predictions from
the RC only estimation model and the RC plus disturbance
estimation model to the truth model. This evaluation
provides good indication of the applicability of the UKF to
thermal network parameter and disturbance estimation.
B. UKF EnergyPlus Performance
Given the excellent performance of the UKF on data
generated by the thermal network model, the UKF was
tested on data generated from an EnergyPlus simulation.
Individual data traces show typical estimation performance
which is validated in an aggregated year-long demonstration.
This more realistic EnergyPlus model, shown in Fig. 6, had
five rooms correlating to the five zones; realistic data for the
floor, wall, and ceiling composition; and windows on the
four exterior walls, which pointed in the cardinal compass
directions. The structure was simulated with weather and
solar radiation for Elmira, NY. In order to acquire good
estimates, the UKF was operated in two modes: A)
Acquisition Mode: night-only estimation of RC products,
10

Prediction Performance
40

Temperature

35
30
25
20
15
Fig. 11 Screenshot from Sustain showing afternoon sun on West Wall

10 Truth
0 RC Only Pred
0.5
RC + Dist Pred

and B) Monitoring Mode: simultaneous estimation of both
RC products and disturbances. Distinguishing learning
modes was less of a concern for the thermal network
simulated data because the disturbances were only on for
short periods of time and the external temperature had a
consistently high level of variation to excite the system.
Because the primary unmodeled thermal disturbance was
solar radiation, covariance tuning was done for the
Monitoring Mode by increasing the bias states’ process
variance for two hours starting at dawn and another two
hours ending 20 minutes after dusk as previously shown in
Fig. 5.
Fig. 11 shows the path of the sun on March 9th and the
variability of the sun path over the course of the year that
will be simulated by EnergyPlus. The graphic was generated
by Sustain, a front-end for EnergyPlus developed by
researchers at Cornell University Program of Computer
Graphics [35]. Due to the axis-inclination of the Earth, the
sun’s path and solar gain varies over the year. Fig. 13 shows
the resulting average disturbances from a four day test where
biases were only estimated for the last two days and then
combined into a 24-hour pattern. Notice how the East room
is heated in the morning, the West room is heated in the
afternoon, and the South room is heated all day, which
correlates nicely with expected heat from solar radiation.
Plots of the 48-hour predictions are shown in Fig. 12.
Predictions which utilize solar disturbances have much
higher accuracy than the RC only predictions. The accuracy
of these predictions ground assumptions made in the thermal
network formulation and more importantly demonstrate the
utility of the UKF for system identification of buildings’
thermal envelope.

1

1.5
Time (Day)

2

Fig. 12. Top South Zone, Bottom West Zone of 48‐hour EnergyPlus
predictions Deg C

Temperature (deg/min)

-4

2
1
0
-1
2
1
0
-1
2
1
0
-1
2
1
0
-1

x 10

0x 10-4

East

5

10 North 15

20

5

10 West 15

20

0x 10

5

10 South 15

20

0

5

10

-4

0x 10

-4

15

Time (hrs)
20

Fig. 13. Estimated 24‐hour cyclic disturbances from EnergyPlus data
Deg C .

calculation failure. From further analysis, simple estimation
monitoring by a human or addition of heuristic rules to the
existing framework would fix all 43 estimation routine
failures. For example over 10 of the failures occurred
because 4 consecutive days had less than 35% solar radiation
causing no disturbance pattern to be learned. Fixing these
sorts of numerical issues to guarantee 100% reliability in an
automated algorithm is outside the scope of the current
study. Results suggested the algorithm could easily be
matured for practical application by adding a number of
heuristics.
Using the 314 successful estimation runs, we compared the
24-hour prediction simulations against the buildings’ truth
simulation from EnergyPlus and found good accuracy—the
models have enough fidelity to be used for control. Over a
12 hour prediction horizon the root mean square (RMS)
temperature prediction error was 1.16oC and over 24 hours
the RMS temperature prediction error was 1.48oC. ASHRAE
standards mandates that vertical temperature stratification in
an occupied zone should be less than 5.4oF (3oC) [36]. Home
and office thermostats often use a dead-band of 4oF (2.2oC)
to 8oF (4.4oC). The model’s prediction errors are well within
these design bounds for the 24 hour prediction horizon. In
Fig. 14 the RMS error for the prediction is shown over time

C. UKF EnergyPlus Year Study
Further analysis of the EnergyPlus generated data was
conducted by analyzing a total year of data. A unique UKF
instance was initialized each day on the first 357 days of the
year and run for 7 days, 3 days in acquisition mode and 4
days in monitoring mode. Then the UKF was used to predict
the building’s response on the eighth day and compared
against the buildings actual response from EnergyPlus. This
generated 357 sets of learned parameters, bias estimates, and
24-hour prediction simulations.
Of the total set, 43 simulations resulted in estimation
routine errors such as negative parameter estimates or
covariance shrinking to zero causing an UKF matrix inverse
11

Standard Deviation of Prediction Error

East Zone Prediction

2.5

Temperature (deg C)

Temperature Deg.

20

2
1.5
1
7 day
14 day
28 day

0.5
0

0

0.2

0.4
0.6
Time (Day)

0.8

10
0

5

0

5

10
15
20
West Zone Prediction

25

30

25

30

20
10

1

10

15
20
Time (days)

Fig. 15 Month prediction using a model learned from 1 week of data.

Fig. 14. Root Mean Square error of temperature predictions of all 5
zones over the year for different lengths of estimation learning time.
Deg C

depending on the target application and available
computation and sensing hardware. For example, Dobbs [37]
compared accuracy of thermal models across different levels
of RC zone aggregation—leveraging such a tool may aid
control-oriented model creation for the UKF. Extensions to
the UKF may offer new opportunities for fault detection and
monitoring [38].
One outstanding challenge remaining with this online
estimation technique is a demonstration of the learned
models’ performance with model predictive controllers in
practice. Some studies [13], [24], [23] have begun
investigating how the quality of the learned model affects
the performance of predictive controllers that use the model.
The consensus to date is that intra-zone excitation is
necessary in order to learn a building’s internal coupling.
Before controlling the building in a novel way to maximize
energy savings, a buildings internal thermal coupling must
be known. Deriving methods to monitor the quality of the
measured data and better learn the building’s thermal
dynamics on-demand, by experimentally exciting the
building, are the subject of a future paper by the authors.

demonstrating good performance. Increased learning periods
of 14 and 28 days further reduced prediction error but did
not drive it to zero—likely because the simple RC model
could not capture the entire fidelity of the EnergyPlus truth
simulation.
Additionally Fig. 15 shows a month-long stable prediction
of the East and West zones based on a model learned from 7
days of data. This long prediction used the same data types
as that in Fig. 12: correct zone initial temperature conditions,
correct external temperatures over the horizon and half day
average solar intensity values over the horizon. The longhorizon prediction accuracy demonstrates the learned model
is unbiased, stable, and robust. To the author’s knowledge,
this is the first year-long study of an online UKF estimating
disturbances with parameters and states for a building.
IV. DISCUSSION
Results of the UKF estimation and model prediction
capabilities have demonstrated the method as a powerful
tool for thermal modeling of building systems. The
simplicity with which a thermal network can be described
combined with the numerical stability and robustness of the
UKF are important factors which could enable its
deployment as a scalable system identification routine for
buildings thermal envelopes.
No physics constraints were applied to ensure RC
parameters were positive, or to inform the bias and
disturbance estimation. Realistic estimation of values was
solely dependent on the quality of the chosen thermal model
representation, estimation technique, and measured data. The
authors expect that good results obtained from this paper’s
simulations would reflect realistic expectations of good
perfomance in real world applications. Accurate bias
tracking was achieved though covariance tuning, but this
might not be scalable to certain buildings where disturbances
occur on erratic schedules, so multiple hypothesis estimation
or constraints may be augmented with a UKF to provide a
more powerful solution.
Further investigation into model selection and fidelity
could lead to performance improvements for the UKF

V. CONCLUSION
A multi-mode implementation of a multi-zone UKF was
presented as a scalable and rapidly deployable system
identification routine for building thermal dynamics. Using a
passive 5-room model, the UKF demonstrated the ability to
learn both dynamics parameters for a thermal network and
unknown disturbances. 24-Hour predictions from UKF
estimated parameters yielded accurate results which were
validated with EnergyPlus simulations using a full year of
data. The UKF, a data-driven, model-based approach,
amenable to augmentation with numerical methods, provides
a promising step towards a scalable framework to realize
advanced BAS predictive controllers.
ACKNOWLEDGMENT
We would like to thank Dave Bosworth for his help
generating an EnergyPlus 5-room building dataset for
evaluation of parameter estimation methods.

12

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