Proceedings of the 44th Hawaii International Conference on System Sciences - 2011
On the Equilibrium Dynamics of Demand Response in Thermostatic Loads
David P. Chassin and Jason C. Fuller
Pacific Northwest National Laboratory, Richland, Washington, USA
[email protected] and [email protected]
Abstract
2. Introduction
Demand response is expected to play a crucial
role in many aspects of “smart grid” technologies
being examined in recently undertaken demonstration
projects. The behavior of load as it is affected by
load control strategies is key to understanding the
degree to which various classes of end-use load can
contribute to demand response programs at various
times. This paper examines a simple case in an effort
to illustrate how subtle changes in the consumer
behavior that drives demand can have dramatic
impacts on the diversity of load and consequently the
effectiveness of demand response programs. The
example studied here uses electric water heaters to
illustrate not only the complexity of the problem, but
also the significance of the effect. This example can
be readily extended to other end-use devices, such as
air-conditioners, heat pumps, and electric vehicle
chargers.
1. Nomenclature
L
N
Noff
Non
noff
non
roff
ron
t
toff
ton
x
ϕ
Φ
η
ρ
thermostat control range (K)
total number of devices (unitless)
number of devices that are “off” (unitless)
number of devices that are “on” (unitless)
“off” devices density per unit temperature (K-1)
“on” devices density per unit temperature (K-1)
cooling rate of thermostatic devices (K.h-1)
heating rate of thermostatic devices (K.h-1)
total cycle time of a thermostatic device (h)
total “off” time of a thermostatic device (h)
total “on” time of a thermostatic device (h)
device temperature (K)
duty cycle of a device (unitless)
diversity of a population of devices (unitless)
consumer demand rate for “off” devices (h-1)
probability of persistent demand (unitless)
The authors thank Jeffry V. Mallow of Loyola University
Chicago and Forrest S. Chassin of Washington State
University for their contributions to this paper. Pacific
Northwest National Laboratory is operated by Battelle
Memorial Institute for the US Department of Energy under
Contract DOE-AC06-76RLO 1830.
Demand response programs have been used
widely for many years to reduce electricity
consumption in order to mitigate the effects of load
growth [1]. The costs of these programs are often
justified on the basis of deferred capacity expansion
costs, improved asset utilization, and customer
savings potential [2] [3].
Price-based programs, such as time-of-use rates
and critical peak rates are effective at reducing peak
loads [4]. However to be effective, price-base
demand response must employ significant price
differentials to provide sufficient incentive to
response. Large price differentials can adversely
affect consumers to such an extent that these
programs are often voluntary, entry fees are imposed
to create barriers to customers who are not well
suited, and yet consumer resistance persists.
Direct load control programs can also be effective
in providing peak load management and are used by
many utilities [1]. But the infrastructure costs
associated with direct load control remains an
ongoing issue. In addition, consumer resistance to
ceding control of appliances and services to the
utility limits the adoption of these strategies.
Finally, a common anecdotal concern is that “freeloaders” subscribe to programs in the hopes that it
will never be called, and when it is they drop out,
wasting the infrastructure investment made.
While none of the extant demand response
approaches is ideal, they do provide a measure of
effective means to address the problems caused by
load growth and escalating peaks when insufficient
resources are invested in capacity expansion.
But other more insidious problems persist. In
particular, rebound effects from excessive
deployment of demand response are commonly seen
[5]. In the past, this problem has been addressed by
employing strategies such as rotating schedules for
direct load control, and stepped or shoulder pricing
rates. These methods have been effective for the
most part. But one could argue that they address the
symptoms of flawed demand response program
design rather than directly addressing the core
1530-1605/11 $26.00 © 2011 IEEE
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Proceedings of the 44th Hawaii International Conference on System Sciences - 2011
question of how load actually behaves when demand
response programs are active.
The objective of this paper is to open the
discussion on how load behaves at a fundamental
level, particularly when it is participating in demand
response programs. Knowledge of the actual behavior
should enable a more precise discussion of how these
programs could, or should be improved.
The
assumption is that if we understood the behavior of
loads and its impact on demand response better, then
we could design better programs that can fully
exploit the inherent nature of load, rather than
fighting it. At the very least we could avoid some of
the more subtle but significant adverse aspects of
load behavior.
3. State dynamics
We begin by considering the aggregate behavior
of cycling energy consuming devices, and in
particular thermostatic heating devices such as
electric water heaters. Such devices exhibit a duty
cycle, which is defined as the ratio of the on-time ton
to the total cycle time ton + toff.
In the case of a water heater operating in standby
regime the duty-cycle is determined by its thermal
and control properties, i.e., the rate roff at which it
cools when it is off, the rate ron at which it heats
when it is on, and the control band L of the
thermostat, which are related by
t on =
L
ron
; t off =
L
roff
.
(1)
The duty cycle of a water heater is therefore
ϕ=
roff
t on
=
.
t on + t off roff + ron
(2)
In utility analysis, diversity is defined as “The
ratio of the sum of the individual non-coincident
maximum demands of various subdivisions of the
system to the maximum demand of the complete
system. The diversity factor is always 1 or greater.
The (unofficial) term diversity, as distinguished from
diversity factor refers to the percent of time available
that a machine, piece of equipment, or facility has its
maximum or nominal load or demand […]” [6].
Therefore, the diversity factor of a population of N
identical water heaters is the inverse of the duty cycle
of a single water heater within that population. When
the water heaters are not identical, the diversity is a
maximum power weighted average of the duty-cycles
of the individual water heaters within that population.
We will refer to diversity Φ when we consider a
population of devices, and duty cycle ϕ when we
speak of a single device. The load arising from ϕ is
usually called the stand-by load or base demand.
When a consumer draws hot water, no matter how
short the duration of the water draw, the water heater
immediately turns on if it was off, creating a
consumer demand event in addition to the stand-by
load. This happens because the cold water that is
admitted to the bottom of the tank to replace the hot
water drawn from the top enters in such a way that it
does not mix with the hot water in the tank. This
creates a layer of cold water in the bottom separated
from the hot water by a thermocline. The thermostat
and the primary heating coil are placed at the bottom
as well so that the heating process is enabled as soon
as cold water is present. While the coil is on,
convection lowers the thermocline created by the
newly added cold water.
If the consumer continues to draw hot water, then
depending on the relative rate of water draw and
heating, the thermocline will rise or fall. For
purposes of this analysis, we will assume the
thermocline stalls when the flow of heat in from the
coil roughly equals the flow of heat out in hot water.
The question for demand response program
analysts is whether and how the diversity of a
population of water heaters is affected by changes in
consumer demand and utility load control. We define
the consumer demand η as the fraction of devices per
unit of time that are subjected to consumer demand
events, as described above. This can be thought of as
the rate at which devices turn on if off. If the water
draw persists, then the water heater becomes part of
the population of stalled water heaters, with a
probability ρ. The total demand given the base
demand and the consumer demand is the combined
effect of these two kinds of demand.
Consider the thermostatic cycling model shown in
Figure 1. According to this model, the relative rates
of stand-by losses and reheating of the individual
water heaters give rise to an aggregate base demand
for the population of the devices shown in Figure
2(a). In addition hot water use causes consumer
demand events that occur randomly at the rate η in
the population and persist with some probability ρ.
The overall effect is to produce the consumer demand
seen in Figure 2(b).
An equilibrium condition occurs when the
number of devices on or off at any given temperature
is invariant. During the time interval dt, the number
of devices that are caused to turn on as a result of
consumer demand is η Noff dt; the number of devices
that turn on because they get too cold is
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Proceedings of the 44th Hawaii International Conference on System Sciences - 2011
Pon
n0
n1
n2
… off states …
nL-2 nL-1
nL
η
0
n0
n1
n2
… on states …
Power
roff = ϕ
L
nL-2 nL-1
Poff
nL
Temperature
Ton
(a)
Probability Density
ron = (1-ϕ)
(a)
roffnoff(x)
roffnoff(x)
noff
ηNoff
roffnoff(0)
non
(1-ρ)ronnon(L)
(1-ρ)ronnon(x)
Temperature
Ton
(b)
Figure 1. (a) The flow of a population of
thermostatic devices within a control band 0 to L
as time passes during a load curtailment event.
The fractional rate η at which devices turn on is
between 0 and +1 and turn off is between –1
and 0.
(b) The state flow diagram for a
particular state x when η < 0, under a
curtailment event with persistent demand ρ.
roff noff(0) dt when η ≥ 0; and the number of devices
that turn off is (1–ρ) ron non(L) dt.
Using the population density curves, we know
that the total number of devices that are off and on is
(3)
respectively. Because the system is at equilibrium,
the total populations in each state, off and on, are not
changing over time. So
ηN off = (1 − ρ )ron non (L ) − roff noff (0 ) .
Toff
(b)
-ηNon(x)
(1-ρ)ronnon(x)
⎧ N = L n ( x )dx
⎪ off ∫0 off
,
⎨
L
(
)
N
n
x
dx
=
⎪⎩ on ∫0 on
Toff
(4)
We also know that the number of devices exiting the
on state at the control point L must be the same as the
number of devices entering the off state at L, and the
number of devices existing the off state at the control
point 0 must be the same as the number of devices
entering the on state at 0. Therefore
Figure 2. (a) The power consumption of a
thermostatically controlled heating device when
operating in a stand-by regime. The x dimension
represents temperature between the control
points Ton = 0 and the Toff = L. (b) The probability
density curve of a single (solid line) and control
transitions (dashed) of N devices, where the n
dimension represents the density of devices per
unit of temperature x. The flow rates of devices
around the on and off probability density curves
is shown, including the flow η resulting from
demand.
N off =
=
(1 − ρ )ron
η
roff
η
[n
off
[non (L ) − non (0 )]
,
(L ) − noff (0 )]
(5)
We can combine equations (3) and (5) to find
∫
L
0
noff ( x )dx =
roff
[n (L) − n (0)].
off
η
off
(6)
The only solution of equation (6) for noff(x) where its
integrand and integral are the same is
noff ( x ) = noff (0 )e
ηx roff
.
(7)
By similar reasoning from equation (5), we also find
that
non (x ) =
roff
(1 − ρ )ron
noff ( x ) .
(8)
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Proceedings of the 44th Hawaii International Conference on System Sciences - 2011
To determine the value of noff(0), we must
evaluate the total number of devices
N = N off + N on .
(9)
From equation (3), we find
roff
roff ηL roff
⎡
⎤
N = ⎢1 +
e
−1 ,
⎥noff (0 )
η
⎣ (1 − ρ )ron ⎦
(
)
(10)
(1 − ρ )ron
⎧
⎪ N off = N (1 − ρ )r + r
⎪
on
off
,
⎨
roff
⎪ N on = N
(1 − ρ )ron + roff
⎪⎩
respectively. We can now see that time-independent
solution for the population diversity is a function of
only the natural demand and persistent demand:
which allows us to find
Nη (1 − ρ )ron
.
noff (0 ) =
ηL r
roff (1 − ρ )ron + roff e off − 1
](
[
)
(11)
Therefore, the number of devices in the off and on
states at a given temperature x is
Nη(1 − ρ )ron
ηx roff
⎧
⎪noff (x ) = r (1 − ρ )r + r eηL roff − 1 e
off
on
off
⎪
.(12)
⎨
Nη
ηx roff
⎪non (x ) =
e
ηL r
⎪
(1 − ρ )ron + roff e off − 1
⎩
](
[
[
](
)
)
The result of equation (12) shows that in the
presence of non-zero consumer demand the
distribution of the devices states at a given
temperature follows an exponential distribution that
rises toward the thermostats' on temperatures.
The significance of this result cannot be
overstated. Without such a detailed understanding of
equilibrium dynamics, we are tempted to describe the
probability density function over the temperature
domain as perhaps normally distributed or uniformly
distributed. It is certainly nothing of the kind when
consumer demand is non-zero. (Although when
consumer demand is zero we observe a special case
of (12) for η = 0, and the distribution is indeed
uniform.) The exponential distribution is a unique
and essential feature of thermostatic devices subject
to premature cycling of the control regime by
processes such as consumer demand and direct load
control. The exponential rise from the off end toward
the on end of the regime becomes more pronounced
as the persistent demand increases until, as consumer
demand approaches saturation at unity, all the
devices are found to be at the control point L. (Note
that when the demand exceeds device capacity, the
net flow of heat is negative, and the devices
population moves below the control point L.)
The distribution given by (12) allows us to
determine that the total number of devices in the off
and on states are
(13)
Φ=
N on
ϕ
.
=
N ϕ + (1 − ρ )(1 − ϕ )
(14)
This result shows that as the probability of persistent
demand ρ approaches unity, so does the diversity of
the devices. There is a loss of diversity and demand
grows, as we have come to expect.
4. Numerical analysis
Computationally efficient numerical models of
the time-dependent solution to the state dynamics
using state queueing models [7] are readily developed
based on state transition diagrams such as is shown in
Figure 1. Thus we can identify the finite state
difference equations of the system when η ≤ 0, which
corresponds to a load curtailment event where for the
sake of clarity we have normalized the bin size
Δx = 1 K, the time step Δt = Δx/(ron+roff) h, the rates
roff +ron=1 such that roff = ϕ Δx/Δt K/h and ron = (1–ϕ)
Δx/Δt K/h, and the persistent demand probability
ρ = η Δt:
Δn on (x , t + Δt ) = −(1 − ϕ )n on (x , t )
+ (1 + ρ )(1 − ϕ )n on (x + Δx , t )
Δn off (x , t + Δt ) = −ϕn off (x , t )
− ρn on (x , t )
+ ϕn off (x − Δx , t )
.(15)
Δn on (L , t + Δt ) = −(1 − ϕ )n on (L , t )
+ ϕn off (L , t )
Δn off (0, t + Δt ) = −ϕn off (0, t )
+ (1 − ϕ )n on (0, t )
The advantage of such a model is that it can easily
describe the three principal kinds of demand response
programs that are used by utilities:
1) Direct load control in which some fraction of the
loads are turned off. This is modeled by
adjusting the value of η between –1 and 0.
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Proceedings of the 44th Hawaii International Conference on System Sciences - 2011
2) Thermostat setback in which the position and/or
size of the control band 0 to L is adjusted.
3) Duty cycle limits, in which the value of the ϕ is
reduced.
The finite state equations can be implemented
easily in numerical applications to produce results
such as those shown in Figure 3, in which a direct
load control signal to drop 50% of load is sent to a
10MW demand response program with (a) a stand-by
duty cycle of 0.10 and a cycling period of 90
minutes, and (b) a stand-by duty cycle of 0.75 and a
cycling period of 45 minutes.
The result shows that a well-regulated demand
response call can be sustained well beyond the time
of first load cycling period, but an over-subscribed
call is unsustainable much beyond the first cycle. In
the latter case, the direct load control initially
responds quite quickly, but very soon begins to decay
as the population diversity is restored and the
ultimate equilibrium load is reached. However, when
the load control is released, the recovery rebound is
clearly visible and can indeed be larger than the
original curtailment. A slight “ringdown” effect can
also be observed in the latter case.
(a)
5. Discussion
The most immediate and practical consequence of
equation (12) is that any agent-based simulation or
numerical model that includes thermostatic devices
subject to demand and curtailment can and should be
initialized using the appropriate distribution when
demand is non-zero. This will avoid the initialization
transient that is often observed.
Initialization
transients in load models are typically mitigated by
running the simulation for a few days until all the
equilibrium dynamics are established before
introducing any perturbations to the system, as shown
in Figure 4.
The state statistics that emerge from the analysis
in the previous section are applicable only to
equilibrium conditions. Thus any attempt extend
these results to situations where the consumer
demand, the thermal properties or the control limits
change over time needs to be considered very
carefully. Such a generalization is only warranted if
it can be shown that the distribution of states and the
density of devices over the range of temperatures
remain relatively close to the equilibrium distribution
described by equation (12). This assumption, which
is analogous to an adiabatic approximation, would be
violated by any significant changes in the parameters.
Such changes would perturb the system enough to
cause a non-trivial redistribution of states and devices
(b)
Figure 3. A numerical of the rebound effect after
a 10 MW direct load control program is actuated
for 4 hours with (a) a load cycling period of 90
minutes at 0.1 duty cycle and (b) a period of 45
minutes at 0.75 duty cycle.
densities. Large fluctuations in any of the parameters
can be expected to cause perturbations in the density
functions non and noff that are likely to take time to
dissipate.
Furthermore, when consumer demand is very
low, we should expect that it will take longer for a
perturbation in the device densities to dissipate than
when consumer demand approaches unity. This can
be readily understood by observing the manner in
which consumer demand preferentially “scatters”
devices from any overpopulated regions of the off
regime into under-populated regions of the on
regime. In the limit, if consumer demand is 0, no
scattering occurs and the only way in which diversity
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Proceedings of the 44th Hawaii International Conference on System Sciences - 2011
6. Conclusions
We have examined the behavior of a specific kind
of load to elucidate the difficulties and importance of
doing so. We have gained a better understanding of
how load behavior impacts demand response
programs. We have seen that some, but not all,
device control regimes result in the expected
behavior of load diversity and that when deviations in
load from typical behavior occur, they can be very
significant. These results have led us to conclude
that load diversity cannot be assumed to have normal
or uniform state statistics, nor can it be assumed to be
insensitive to demand response programs in the
expected ways.
Finally, we have observed that the evolution of
the diversity of device states, when perturbed by
changes in the device control parameters, thermal
properties, or consumer demand can result in
significant and prolonged deviations from their
normal aggregate behavior, leading to large and
potentially difficult to predict fluctuations in total
load; a phenomenon that has been observed
previously and modeled numerically but has yet to be
studied analytically in any detail.
7. References
(a)
Figure 4. The initialization transient often seen in
numerical simulation of loads (a) can be typically
be eliminated by using equation (12) to properly
diversify the initial states of devices so that the
temperature states of devices are correctly
distributed within the control band (b).
is restored is through the variations in the thermal and
control properties of the individual devices, which
cause the duty-cycles to slowly diversify the
population. Consequently, we must expect load
diversity to be restored more quickly during times of
high consumer demand and more slowly during times
of low consumer demand. This phenomenon is one
that could be considered by utilities when scheduling
demand response programs and forecasting the
duration and amplitude of the post-curtailment
rebound.
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