2012 45th Hawaii International Conference on System Sciences
Development and Validation of Aggregated Models for Thermostatic
Controlled Loads with Demand Response
Karanjit Kalsi
Marcelo Elizondo
Jason Fuller
Shuai Lu
Pacific Northwest National Laboratory
karanjit.kalsi@pnnl.gov marcelo.elizondo@pnnl.gov jason.fuller@pnnl.gov shuai.lu@pnnl.gov
david.chassin@pnnl.gov
Abstract
real-time prices combined with enabling technology
to "signal" loads when supply is scarce (prices are
high) or supply is plentiful (prices are low). If prices
are updated frequently and load controls are highly
interactive, then loads can become as effective at
following intermittent generation as standard
generation is at following changing load, without
adversely affecting consumers.
The development of advanced smart grid
simulation tools, such as GridLAB-D [4], provides a
platform to design and test these smart grid asset
models. While many of the smart grid technologies
proposed involve assets being deployed at the
distribution level, most of the significant benefits
accrue at the transmission level. However, the size
and complexity of typical distribution systems makes
direct integration with transmission models
computationally intractable. It is, therefore, important
to be able to represent distributed smart grid assets
using reduced-order models. The developed models
will, for the first time, provide industry with a suite
of analysis tools and results that can be used to
design, improve, and validate smart grid control
strategies and ensure bulk power system stability,
reliability and security. For example, aggregated load
models can be used to quickly predict load reduction
obtained by a control signal from the utility control
center to compensate for a change in renewable
generation. An understanding of load population
dynamics through adequate modeling is essential to
extract the potential of load control for a successful
implementation of the smart grid.
Dynamic modeling of thermostatic controlled loads
was first studied in [5] and [6]. In [5], aggregate load
models are designed to study the effects of cold load
pickup after a service interruption. Functional models
of devices, which account for factors such as weather
and human behavior, are developed in [6]. A model
of a large number of similar devices is then obtained
through statistical aggregation of the individual
component models. In [7] and [8], aggregated
dynamic homogenous and non-homogeneous models
are developed for thermostatic loads. The main
One of the salient features of the smart grid is the
wide spread use of distributed energy resources
(DERs) like small wind turbines, photovoltaic (PV)
panels, energy storage (batteries, flywheels, etc),
Plug-in Hybrid Electric Vehicles (PHEVs) and
controllable end-use loads. The affect of these
distributed resources on the distribution feeder and
on transmission system operations needs to be
understood. Due to the potentially large number of
DERs that are expected to be deployed, it is
impractical to use detailed models of these resources
when integrated with the transmission system. This
paper focuses on developing aggregated models for a
population of Thermostatic Controlled Loads (TCLs)
which are a class of controllable end-use loads. The
developed reduced-order models are validated
against simulations of thousands of detailed building
models using an open source distribution simulation
software (GridLAB-D) under both steady state and
dynamic conditions (thermostat setback program as a
simple form of demand response).
1. Introduction
Large-scale deployment of intermittent nondispatchable generation by the year 2030 will require
a transformation in the control of smart-grid assets
such as demand response, distributed generation,
energy storage, and distribution automation. The
development of the smart grid concept has given rise
to the notion that instead of trying to regulate the
system by controlling generation, the focus should be
to control demand to the extent possible. Direct load
control programs can be effective in providing peak
load management and are used by many utilities as
shown in [1]. Another approach that has been tested
in [2] and [3] and that shows promise is the use of
Pacific Northwest National Laboratory is operated by Battelle
Memorial Institute for the US Department of Energy under
Contract DOE-AC06-76RLO 1830.
978-0-7695-4525-7 2012
U.S. Government Work Not Protected by U.S. Copyright
DOI 10.1109/HICSS.2012.212
David Chassin
1959
contribution of [9] is to develop a finite dimensional
aggregated state space model for air conditioning
loads based on solving bilinear PDEs model. It is
shown that an order of 200 is needed for the
aggregated state space to accurately capture the
detailed model dynamics. In [10], a survey is
presented of different load control programs for
providing various power system services.
The focus of this paper is to design and validate
aggregated models for a class of thermostatic
controlled end-use loads. One approach to develop
reduced-order models for TCLs like Heating,
Ventilation and Cooling (HVAC) and water heaters is
to use state transition diagrams to represent the
behavior of a population of devices as in [11]. The
first step is to construct a physical model representing
the dynamics of different end use loads. A state
transition diagram can then be created from the
physical model with the transition rates being a
function of the physical and environmental
parameters and control actions. After having obtained
the state transition diagram, a state-space model
representing the dynamics of device occupancy in
each state can then be constructed. Based on the
number of devices in each state, the total power
consumption of the devices can be estimated. The
developed models are shown to capture both the
steady-state and dynamic (with demand response)
behavior of the device population. The state space
models are validated against simulations of 10,000
building models in GridLAB-D both for the case
without demand response, and with a thermostat
setback program as a simple form of demand
response. An illustration of the computational
advantages of the aggregated load model is also
given.
In this paper, Section 2 will discuss the physical
model of thermostatic loads. In Section 3, aggregate
models for heating/cooling loads will be derived, and
in Section 4, validated against GridLAB-D
simulations. In Section 5, conclusions and future
research are discussed.
air-conditioning. Thermostatic loads at the level of a
single device give rise to pulse sequences where the
pulse width and frequency arise from the control
hysteresis and the heat balance in the load, while the
power drawn changes between two or more relatively
fixed values. The thermal model, adapted from [4],
representing such individual device loads is shown in
Figure 1.
Figure 1. Thermal model of home heating/cooling
system
The formulation for this particular model is given as
follows:
dTa
1
=
[Tm H m − Ta (U a + H m ) + Qa + ToU a ]
dt Ca
(1)
dTm
1
=
[H m (Ta − Tm ) + Qm ]
dt
Cm
Here Ua is the conductance of the building envelope,
To is the outdoor air temperature, Ta is the indoor air
thermostat setpoint, Qa is the heating/cooling
capacity of the HVAC unit, Ca is the thermal mass of
the air, Cm is the thermal mass of the building
materials and furnishings. For the purposes of this
paper it is assumed that the thermal coupling, Hm,
between air and mass is perfect, i.e. Ta = Tm and total
heat capacity is given by C = Ca + Cm. Taking these
assumptions into account, (1) is simplified as follows
2. Physical Model of an Individual
Thermostatic Controlled Load
Ta − c1Ta = c2
with
This section discusses the physical model of an
individual load device. This individual device model
is the base to develop an aggregated load model
representing the behavior of a population of devices
in section 3. Residential electric end-use loads can be
divided into two principal classes, one for all nonthermostatic loads, such as lights and plugs, and one
for all thermostatic loads, such as water heating and
c1 =
−Ua
C
c2 =
(2)
Qa + U aTo .
C
Integrating the above and using the initial condition
of Ta0 gives
1960
Ta (t ) = k1e r1t −
steady state (section 3.1) and the transient behavior
introduced by demand response (section 3.2).
The aggregated load model developed here is valid
for a time scale from seconds to hours. The model
captures the on/off transitions of devices and
temperature dynamics of the population.
The aggregated model can be used by system
operators to help design and implement demand
response programs on a population of thermostatic
controlled loads. The aggregated state space load
model can be used to simulate collective load
behavior without the need of computationally
expensive simulations of each individual load.
c2
c1
where
k1 = Ta 0 +
c2
c1
r1 = c1 .
and
(3)
Differentiating the above gives
Ta (t ) =
Ua
−
t
1
[U a (To − Ta 0 ) + Qa ]e Cm +Ca .
Ca + Cm
(4)
3.1. Steady state model
The rates of the device turning ON and OFF are
given by ron and roff where,
roff =
1
[U a (To − Ta 0 ) + Qi + Qs ]e
Ca + Cm
The state dynamics of a population of
heating/cooling units can be derived from a state
transition diagram. The units can be in either the on
or off state. Each of these states can be further
divided into two sub-states, where the loads are
satisfied or unsatisfied. The device population can
therefore occupy any one of these four states. The
objective of the heating unit, for example, is to keep
the room temperature at a certain value. The satisfied
state is when the room temperature is above the
setpoint temperature. When the temperature falls
below the setpoint, the unit is said to be in the
unsatisfied state.
The heating units, in the on state, heat at a certain
rate, ron, and move from the unsatisfied to the
satisfied state. When the upper control point of the
thermostat is reached, the units move from the on
satisfied to the off satisfied state. The units then coast
at a rate roff to the off unsatisfied state after which
they turn on and continue the cycle. The state
diagram illustrating the heating cycle of a HVAC
system is shown in Figure 2.
The dynamics of device occupancy in each state
based on the state transition diagram in Figure 2 are
given as
us
us
s
N off
= −ϕN off
+ ϕN off
Ua
−
t
C m + Ca
(5)
Ua
−
t
1
ron =
[U a (To − Ta 0 ) + Qi + Qs + Qh ]e Cm +Ca
Ca + C m
where Qi is the heat gain from the internal load, Qs is
the solar heat gain and Qh is the heat gain from the
heating/cooling system. The rates computed are given
in oF/h. Given the thermostat deadband, L, needed for
adequate control hysteresis, the time for the on and
off cycles is
toff =
The duty cycle
ϕ
L
roff
; ton =
L
ron
(6)
can then be computed as
ϕ=
roff
ton
=
ton + toff ron + roff
(7)
In (7), without loss of generality, it can be assumed
that ron+ roff = 1, in which case ϕ = roff and 1 - ϕ = ron,
a condition necessary to ensure numerical stability in
certain finite differences implementation of state
models. In the following sections, state space models
will be constructed based on this “normalized”
physical model.
s
s
N off
= −ϕN off
+ (1 − ϕ ) N ons
us
us
us
N on
= ϕN off
− (1 − ϕ ) N on
(8)
us
N ons = (1 − ϕ ) N on
− (1 − ϕ ) N ons
Using (4), the state space representation of the steady
state dynamics without demand response is given by
(9)
x = Ax
where,
3. Aggregated models for a population
of HVAC units
Based on the physical model of an individual load
discussed in section 2, this section develops a state
space model to represent the behavior of a population
of HVAC units. The state space model is valid for
1961
us
⎤
⎡ N off
⎢ s ⎥
N
x = ⎢ offus ⎥ ;
⎢ N on ⎥
⎢ s ⎥
⎣⎢ N on ⎦⎥
ϕ
−ϕ
0
0
roff
0 ⎤
0
1 − ϕ ⎥⎥
ϕ −1 0 ⎥
⎥
1 − ϕ ϕ − 1⎦
.
time (t = t2, see Figure 3) the TCL population will be
working at a new steady state with corresponding
variables ton3, toff3, ron3, roff3, and φ3. Between t0 and t2,
most of TCLs switch to one statte (“on” state in the
case of the figure), in order to reeach the new steady
state, for the period t0 < t < t2; therefore,
t
the steady
state model is not valid for this transient period. A
time varying model that swittches characteristics
when a temperature setpoint is applied to the
population of TCL is proposed.
0
s
Noff
(10)
off
roff
ron
us
Noff
⎡− ϕ
⎢ 0
A=⎢
⎢ϕ
⎢
⎣ 0
us
Non
ron
unsatisfied
Nons
on
satisfied
d
Tset
Indoorr air temperature
Figure
2.
State
transition
d
diagram
heating/cooling units (heating cycle)
Figure 3.
Transient dynamics of indoor air
temperature in three individual HVAC with a
change in setpoint
for
The time varying model that captures the transient
a described by (11).
behavior consists of three parts as
A steady state model is used beffore the temperature
setpoint change is applied (t<t0). A transient model is
used between t0 and t1 (t1 loweer than t2, where t2
marks the end of the transien
nt period). For t>t1
(passing by t2), the TCL populatiion is described by a
second steady state model.
3.2. State space model with deman
nd response
In this section, a time variant modeel to represent
TCL population dynamics when a tempperature setting
adjustment is applied to the populatioon is proposed.
This model uses the state diagram developed in
section 3.1 and notions of the phhysical model
discussed in section 2. A populattion of three
individual TCLs is considered to explain
conceptually the development the time variant model.
The time variant model is valid for a laarge population
as shown with simulations in section 4.
When a step change in temperatuure setting is
applied, the population exhibits a trannsient behavior
that is not captured by the steadyy state model
discussed in Section 3.1. Figure 3 illustrates this
transient behavior, displaying the evoluution of inside
ambient temperature, Ta, for three indivvidual HVACs
with the same temperature settings annd deadbands.
To explain the problem conceptually, llet us assume a
population of household TCLs in steaddy state with a
temperature setting, Tset1, in all loadss, as shown in
Figure 3. The inside temperature,, Ta, at each
household is within one deadband of Tset1from time
t = 0 to t = t0. During this period, the ton1 and toff1 are
average steady state values corresponnding to steady
state rates (ron, roff) and duty cycle φ1, aas explained in
Section 3.1. Let us further assume thaat at t = t0 the
temperature setting is changed to Tseet2. After some
⎧ A1 x
⎪
x = ⎨ A2 x
⎪A x
⎩ 3
if
if
t <= t 0
t 0 < t <= t1
if
t1 < t <= t f
(11)
where,
0
0 ⎤
⎡− ϕ1 ϕ1
⎢ 0 −ϕ
0
1 − ϕ1 ⎥⎥
1
A1 = ⎢
⎢ ϕ1
0 ⎥
0 ϕ1 − 1
⎢
⎥
0 1 − ϕ1 ϕ1 − 1⎦
⎣ 0
⎡− t on 2
⎢ 0
A2 = ⎢
⎢ t on 2
⎢
⎣⎢ 0
1962
t on 2
0
− t on 2
0
0
0
− t off 2
t off 2
0 ⎤
t off 2 ⎥⎥
0 ⎥
⎥
− t off 2 ⎦⎥
(12)
⎡ −φ3
⎢ 0
A3 = ⎢
⎢ φ3
⎢
⎣ 0
In (12),
φ3
−φ3
0
0
space aggregation model. The HVAC on and off
states were aggregated over time for each of the
building models, and then compared against the
predicted value from the state space aggregate model.
The following results should be considered
exemplary, but not complete, as work in on-going to
further validate the aggregate model under various
population and system configurations and demand
response programs.
0 ⎤
1 − φ3 ⎥
⎥
0 ⎥
φ3 − 1
⎥
1 − φ3 φ3 − 1⎦
0
0
ϕ1 and ϕ3 are the duty cycles of the HVAC
unit corresponding to the original and new
temperature setpoints and are computed using (2) –
(7). The corresponding state transition matrices, A1
and A3, represent the steady state behavior of the
HVAC unit before and after the temperature setpoint
change. Parameters ton2 and t1 define the duration and
peak of the transient and need to be estimated. These
parameters can be derived from (3) where Ta = Tset2,
Ta0 = Tset1, and Δt = t2 - t0, which is the estimated
duration of the transient. The expression for Δt is
given as
Δt =
⎛T +c c ⎞
1
log ⎜ set 2 2 1 ⎟
r1
⎝ Tset1 + c2 c1 ⎠
4.1. Steady state model
A comparison between a simulated population and
the aggregate model when operating at or near a
steady state condition (no external air temperature
changes and set points are held constant) is shown in
Figures 4 - 5. The aggregated load model was
developed in section 3.1, while GridLAB-D solves
the individual loads described in section 2. The on
and off states, whether satisfied or not, are summed
together (only the on states are shown). It can be seen
that while the aggregate model produces a more
smooth response, but in general, the number of units
in the on state settles to approximately the same
values in both simulations. It should also be noted
that in Figures 4 – 5, the GridLAB-D population does
not start out exactly at equilibrium, but approaches
equilibrium after approximately one hour. This
transient is caused by the difficulty in initializing the
agent-based simulation with perfectly distributed
parameters. This results in a settling time to reach the
final steady state.
(13)
The values of ton2 and t1 can be estimated from Δt as:
tˆ1 − t0 = Δt
L
(14)
tˆon 2 = Δt
A fraction of Δt (Δt / L), where L is the deadband, is
used to estimate t1 because the system needs to
switch to model A3 in (11) and (12) before the end of
the transient (notice that Δt marks the end of the
transient).
3000
GridLAB-D
Aggregated
2500
4. Validation of aggregated model
against GridLAB-D
N
ON
2000
To provide for initial validation of the aggregate
model, an open source tool, GridLAB-D, was used.
GridLAB-D is capable of simultaneously simulating
thousands of unique buildings using an equivalent
thermal model [12-15] to create a diverse population
of building and HVAC system parameters; however
the simulations can often take a substantial amount of
time and computing resources, especially as the
model size increases. In the following experiments,
10,000 building models were created and simulated
for a single day. Parameters such as Ua, Cm, Ca, and
Hm were randomly varied across the population to
provide a certain amount of diversification, and
outside weather and internal setpoints were modified
at various times to test the performance of the state
1500
1000
500
0
1
2
3
Time (hr)
4
5
6
Figure 4. State dynamics of HVAC units starting at
equilibrium with Tset=75 and Tout=77
1963
“saturates”, and all of the HVAC units are forced into
the on state. This could be considered an extreme
case, where the control signal synchronizes all of the
loads and decreases system diversity leading to an
extreme rebound in power demand. In both cases, the
aggregate model reasonably captures the peak
rebound effect and follows the envelope of the
population response.
5000
GridLAB-D
Aggregated
4500
4000
NON
3500
3000
2500
2000
2500
GridLAB-D
Aggregated
1500
1000
0
1
2
3
Time (hr)
4
5
2000
6
Figure 5. State dynamics of HVAC units starting at
equilibrium with Tset=75 and Tout=95
NON
1500
1000
4.2. State space model with demand response
To test the ability of the state space model to
transition between multiple equilibrium points, a
simple form of demand response, a thermostat
setback program, is explored. The aggregated load
model that captures transient behavior was developed
in section 3.2. Figures 6 - 7 show a comparison of the
number of HVACs in the on state between the state
space model and the population model. In these
cases, all of the HVACs are participating in a
thermostat setback program where the thermostats
setpoints are simultaneously shifted up from 75 to 76
º F and 75 to 77 º F, respectively. After the stored
thermal energy in the buildings is exhausted, in
roughly one hour, the system begins to return to a
new steady state with a slightly lower consumption
than at the original steady state. Notice that the time
frame for which all the units are in the off position is
not two times longer for the 2 degree shift versus the
1 degree shift, highlighting some of the non-linear
characteristics of the system that are captured by the
aggregate model. This also highlights the fact that
thermostatically controlled loads are only capable of
providing load reduction for a limited amount of time
defined by the stored amount of thermal energy
within the building.
A commonly observed phenomenon associated
with demand response programs is often called the
“rebound” effect, or an immediate spike in power
consumption when the thermostat setback is released
back to normal operation. Figures 8 - 9 compare the
responses in the two models when a thermostat
setback program releases the thermostats back to
their natural state, in this case from 75 to 74 ºF in
Figure 8 and 75 to 70 ºF in Figure 9. Figure 8 shows
the “unsaturated” case, where only a portion of the
population immediately switches to the on state.
Figure 9 shows the case where the state model
500
0
0
2
4
Time (hr)
6
8
Figure 6. State dynamics of HVAC units with
setpoint change from 75 to 77 degrees
2500
GridLAB-D
Aggregated
2000
NON
1500
1000
500
0
0
2
4
Time (hr)
6
8
Figure 7. State dynamics of HVAC units with
setpoint change from 75 to 76 degrees
In the unsaturated case (Figure 8) however, the
aggregate model is not able to capture the oscillations
set up by the setpoint change. This is due to the fact
that the control signal in the population model again
synchronizes all of the loads, resulting in a loss of
diversification, which takes some time to be
reestablished. One of the key assumptions of the state
space model is that the thermostatic loads always
maintain diversity. Figure 10 demonstrates the effects
when the control signal releasing the thermostats
back to normal operation is randomly applied over a
period of approximately 45 minutes to the different
1964
buildings within the population (e.g. each building
receives the release signal at a different time). The
inability to capture the effects of loss of diversity is
one of the shortcomings of the aggregate model, and
will be addressed in future work.
5. Conclusions
In this paper, aggregate load models have been
developed for thermostatic controlled loads by
considering the state dynamics of a large
homogenous population of such devices. These
models have been initially validated against
GridLAB-D simulations. Results show acceptable
agreement between the aggregated load model and
detailed simulation of 10,000 individual loads.
Additionally, the computation time is tremendously
reduced when using the aggregated load model as
opposed to using detailed numerical simulations of
each TCL.
Future work will continue to validate the aggregate
model under various system parameters and
conditions, while employing various control and
demand response strategies Additional transition
rules and time varying parameters will also be
addressed.
10000
GridLAB-D
Aggregated
8000
NON
6000
4000
2000
0
0
2
4
Time (hr)
6
8
Figure 8. State dynamics of HVAC units with
setpoint change from 75 to 74 degrees
5000
10000
GridLAB-D
Aggregated
GridLAB-D
Aggregated
4500
8000
4000
NON
NON
6000
3500
3000
4000
2500
2000
0
0
2000
1500
0
2
4
Time (hr)
6
8
2
4
Time (hr)
6
8
Figure 10. State dynamics of HVAC units with
setpoint change from 75 to 74 degrees and
random initial start points
Figure 9. State dynamics of HVAC units with
setpoint change from 75 to 70 degrees
4.3. Computation time
6. References
The aggregated load model presents a tremendous
improvement in computation time, as compared with
the detailed model of a large population of TCLs.
The simulations of the TCL population shown in this
paper took approximately 20 to 25 minutes. On the
other hand, it takes only a fraction of a second for the
aggregated load model to run in Matlab. This type of
speed improvement is required for applications where
simulation of thousands to tens of thousands of
distribution system feeder circuits is required.
1965
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1966