Modeling and control of thermostatically controlled loads
ABSTRACT As the penetration of intermittent energy sources grows substantially, loads
will be required to play an increasingly important role in compensating the
fast timescale fluctuations in generated power. Recent numerical modeling of
thermostatically controlled loads (TCLs) has demonstrated that such load
following is feasible, but analytical models that satisfactorily quantify the
aggregate power consumption of a group of TCLs are desired to enable controller
design. We develop such a model for the aggregate power response of a
homogeneous population of TCLs to uniform variation of all TCL setpoints. A
linearized model of the response is derived, and a linear quadratic regulator
(LQR) has been designed. Using the TCL setpoint as the control input, the LQR
enables aggregate power to track reference signals that exhibit step, ramp and
sinusoidal variations. Although much of the work assumes a homogeneous
population of TCLs with deterministic dynamics, we also propose a method for
probing the dynamics of systems where load characteristics are not well known.

Conference Paper: Load modeling and calibration techniques for power system studies
[Show abstract] [Hide abstract]
ABSTRACT: Load modeling is one of the most uncertain areas in power system simulations. Having an accurate load model is important for power system planning and operation. Here, a review of load modeling and calibration techniques is given. This paper is not comprehensive, but covers some of the techniques commonly found in the literature. The advantages and disadvantages of each technique are outlined.North American Power Symposium (NAPS), 2011; 09/2011  SourceAvailable from: Konstantin Turitsyn[Show abstract] [Hide abstract]
ABSTRACT: We introduce and analyze Markov Decision Process (MDP) machines to model individual devices which are expected to participate in future demandresponse markets on distribution grids. We differentiate devices into the following four types: (a) optional loads that can be shed, e.g. light dimming; (b) deferrable loads that can be delayed, e.g. dishwashers; (c) controllable loads with inertia, e.g. thermostaticallycontrolled loads, whose task is to maintain an auxiliary characteristic (temperature) within predefined margins; and (d) storage devices that can alternate between charging and generating. Our analysis of the devices seeks to find their optimal pricetaking control strategy under a given stochastic model of the distribution market.03/2011;  SourceAvailable from: Borhan M. Sanandaji[Show abstract] [Hide abstract]
ABSTRACT: Recently it has been shown that an aggregation of Thermostatically Controlled Loads (TCLs) can be utilized to provide fast regulating reserve service for power grids and the behavior of the aggregation can be captured by a stochastic battery with dissipation. In this paper, we address two practical issues associated with the proposed battery model. First, we address clustering of a heterogeneous collection and show that by finding the optimal dissipation parameter for a given collection, one can divide these units into few clusters and improve the overall battery model. Second, we analytically characterize the impact of imposing a noshortcycling requirement on TCLs as constraints on the ramping rate of the regulation signal. We support our theorems by providing simulation results.10/2013;
Page 1
MODELING AND CONTROL OF
THERMOSTATICALLY CONTROLLED LOADS
Soumya Kundu
University of Michigan
Ann Arbor, USA
soumyak@umich.edu
Nikolai Sinitsyn
Los Alamos National Laboratory
Los Alamos, USA
sinitsyn@lanl.gov
Scott Backhaus
Los Alamos National Laboratory
Los Alamos, USA
backhaus@lanl.gov
Ian Hiskens
University of Michigan
Ann Arbor, USA
hiskens@umich.edu
Abstract  As the penetration of intermittent energy
sources grows substantially, loads will be required to play an
increasingly important role in compensating the fast time
scale fluctuations in generated power.
modeling of thermostatically controlled loads (TCLs) has
demonstrated that such load following is feasible, but analyt
ical models that satisfactorily quantify the aggregate power
consumption of a group of TCLs are desired to enable con
troller design. We develop such a model for the aggregate
powerresponseofahomogeneouspopulationofTCLstouni
formvariationofallTCLsetpoints. Alinearizedmodelofthe
response is derived, and a linear quadratic regulator (LQR)
has been designed. Using the TCL setpoint as the control
input, the LQR enables aggregate power to track reference
signals that exhibit step, ramp and sinusoidal variations.
Recent numerical
Keywords  Load modeling; load control; renewable
energy; linear quadratic regulator.
1 INTRODUCTION
G
mittency and nondispatchability associated with such
sources. Conventional power generators have difficulty
in manoeuvering to compensate for the variability in the
power output from renewable sources. On the other hand,
electrical loads offer the possibility of providing the re
quired generationbalancing ancillary services. It is fea
sible for electrical loads to compensate for energy im
balance much more quickly than conventional generators,
which are often constrained by physical ramp rates.
A population of thermostatically controlled loads
(TCLs) is well matched to the role of load following. Re
search into the behavior of TCLs began with the work of
[1] and [2], who proposed models to capture the hybid dy
namics of each thermostat in the population. The aggre
gate dynamic response of such loads was investigated by
[4], who derived a coupled ordinary and partial differen
tial equation (FokkerPlanck equation) model. The model
was derived by first assuming a homogeneous group of
thermostats (all thermostats having the same parameters),
andthenextendedusingperturbationanalysistoobtainthe
model for a nonhomogeneous group of thermostats. In
ROWTH in renewable power generation brings with
it concerns for grid reliability due to the inter
[5], a discretetime model of the dynamics of the temper
atures of individual thermostats was derived, assuming no
external random influence. That work was later extended
by [6] to introduce random influences and heterogeneity.
Although the traditional focus has been on direct load
control methods that directly interrupt power to all loads,
recent work in [3] proposed hysteresisbased control by
manipulating the thermostat setpoint of all loads in the
population with a common signal. While it is difficult to
keep track of the temperature and power demands of in
dividual loads in the population, the probability of each
load being in a given state (ON  drawing power or OFF 
notdrawinganypower)canbeestimatedratheraccurately.
System identification techniques were used in [3] to obtain
an aggregate linear TCL model, which was then employed
inaminimumvariancecontrollawtodemonstratetheload
following capability of a population of TCLs.
In this paper, we derive a transfer function relating the
aggregate response of a homogeneous group of TCLs to
disturbances that are applied uniformly to the thermostat
setpoints of all TCLs. We start from the hybrid temper
ature dynamics of individual thermostats in the popula
tion, and derive the steadystate probability density func
tions of loads being in the ON or OFF states. Using these
probabilities we calculate aggregate power response to a
setpoint change. We linearize the response and design a
linear quadratic regulator to achieve reference tracking by
the aggregate power demand.
2 STEADY STATE DISTRIBUTION OF LOADS
2.1 Model development
The dynamic behavior of the temperature θ(t) of a
thermostatically controlled coolingload (TCL), in the ON
and OFF state and in the absence of noise, can be modeled
by [5],
mal capacitance, R is the thermal resistance, and P is the
power drawn by the TCL when in the ON state. This re
sponse is shown in Figure 1.
˙θ =
−
1
CR(θ − θamb+ PR),
ON state
−
1
CR(θ − θamb),
OFF state
(1)
where θamb is the ambient temperature, C is the ther
17th Power Systems Computation ConferenceStockholm Sweden  August 2226, 2011
Page 2
θ+
θ
P
time
0
Tc
Th
Figure 1: Dynamics of temperature of a thermostatic load.
In steady state the cooling period drives a load from
temperature θ+to temperature θ−. Thus solving (1) with
initial condition θ(0) = θ+gives
(
From (2) we can calculate the steady state cooling time Tc
by equating θ(Tc) to θ−,
(PR + θ+− θamb
A similar calculation for the heating time gives,
(θamb− θ−
In general, the expressions for the times tc(θf) and th(θf)
taken to reach some intermediate temperature θf during
the cooling and heating periods, respectively, are,
(PR + θ+− θamb
(θamb− θ−
It follows immediately that tc(θ−) = Tcand th(θ+) =
Th.
For a homogeneous1set of TCLs in steady state, the
number of loads in the ON and OFF states, Ncand Nh
respectively, will be proportional to their respective cool
ing and heating time periods Tcand Th. In the absence
of any appreciable noise, which ensures that all the loads
are within the temperature deadband, Nh+ Nc= N, we
obtain,
θ(t) = (θamb− PR)1 − e−
t
CR
)
+ θ+e−
t
CR.
(2)
Tc= CRln
PR + θ−− θamb
)
.
(3)
Th= CRln
θamb− θ+
)
.
(4)
tc(θf) = CRln
PR + θf− θamb
)
(5)
th(θf) = CRln
θamb− θf
)
.
(6)
Nc=
Tc
Tc+ ThN
Th
Tc+ ThN
(7)
Nh=
(8)
By analogy, it follows that the number of ONloads
within a temperature band [θ, θ+] is proportional to the
time taken tc(θ) to cool a load down from θ+to θ ≥ θ−,
and is given by
nc(θ) = tc(θ)Nc
Tc
= tc(θ)
N
Tc+ Th,
(9)
where (7) was used to obtain (9). Likewise, the number of
OFFloads with temperature in the band [θ−, θ] is given
by
nh(θ) = th(θ)
N
Tc+ Th
(10)
where th(θ) is the time taken for a load’s temperature to
rise from θ−to θ ≤ θ+.
We will denote the ON probability density function by
f1(θ) and the OFF probability density function by f0(θ),
while the corresponding cumulative distribution functions
are denoted F1(θ) and F0(θ), respectively. Because F0(θ)
is the probability of a load being in the OFF state, with
temperature in the range [θ−, θ], we obtain directly that
F0(θ) = nh(θ)/N. Inestablishingtheequivalentrelation
ship between F1(θ) and nc(θ), we must keep in mind that
F1(θ) is defined relative to the temperature band [θ−, θ],
whereas nc(θ) refers to the band [θ, θ+]. Consequently,
we obtain F1(θ) = (Nc− nc(θ))/N.
We can therefore write,
f0(θ) =dF0(θ)
dθ
1
Ndθ
1
Tc+ Th
=
d
dθ
(nh(θ)
N
Tc+ Th
dth(θ)
dθ
CR
N
)
=
dth(θ)
=
=
(Tc+ Th)(θamb− θ)
(11)
and
f1(θ) =dF1(θ)
dθ
=
d
dθ
CR
(Nc− nc(θ)
N
)
=
(Tc+ Th)(PR + θ − θamb).
(12)
2.2 Simulation
Figure 2 shows a comparison of the densities calcu
lated using (11) and (12) and those computed from ac
tual simulation of the dynamics of a population of 10,000
TCLs that included a small amount of noise. The result
suggests that the assumptions underlying (11) and (12) are
realistic.
θa=32oC, θs=20 oC, ∆=2oC
18.51919.52020.52121.5
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
θ (oC)
Probability densities
f1 (actual)
f0 (actual)
f1 (analytical)
f0 (analytical)
Figure 2: Steady state densities.
1All loads share the same values for parameters θamb, C, R and P.
17th Power Systems Computation ConferenceStockholm Sweden  August 2226, 2011
Page 3
3SETPOINT VARIATION
Control of active power can be achieved by making a
uniform adjustment to the temperature setpoint of all loads
within a large population [3]. It is assumed that the tem
perature deadband moves in unison with the setpoint. Fig
ure 3 shows the change in the aggregate power consump
tion of a population of TCLs for a small step change in
the setpoint of all devices. The resulting transient varia
tions in the OFFstate and ONstate distributions for the
population are shown in Figure 4.
747678
Time (hrs.)
808284
21
22
23
24
25
26
27
Total Power Variation
Power (MW)
74 76 78
Time (hrs.)
80 82 84
19.99
19.995
20
20.005
20.01
20.015
20.02
20.025
20.03
Temperature Set−point Variation
Temerature Setpoint (degC)
Figure 3: Change in aggregate power consumption due to a step change
in temperature setpoint.
19.5
20
20.5
21
70
75
80
85
90
95
0
0.5
1
1.5
2
2.5
Temperature (oC)
Probability density function (OFF state)
Time (hrs.)
PDF (OFF)
(a) OFFstate distribution.
19.6
19.8
20
20.2
20.4
20.6
70
75
80
85
90
95
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Temperature (oC)
Probability density function (ON state)
Time (hrs.)
PDF (ON)
(b) ONstate distribution.
Figure 4: Variation in distribution of loads due to setpoint disturbance.
The aggregate power consumption at any instant in
time is proportional to the number of loads in the ON state
at that instant. The first step in quantifying the change in
power due to a step change in setpoint is therefore to an
alyze the behavior of the TCL probability distributions.
Figure 5 depicts a situation where the setpoint has just
been increased. The original deadband ranged from θ0
to θ0
tive step change, the new deadband lies between θ−to θ+,
with the deadband width ∆ = θ0
ing unchanged. The setpoint is shifted by δ = θ−− θ0
θ+−θ0
consider four different TCL starting conditions immedi
ately after the step change in setpoint, i.e. ad in Figure 5.
Using Laplace transforms, we compute the time depen
dence of the power consumption for each of these loads
(shown in Figure 6) and then compute the total power con
sumption by integrating over the distributions f0and f1.
At the instant the step change in applied, the temperatures
of loads at points a, b, c and d are θa,θb,θcand θd, respec
tively.
−
+, with the setpoint at (θ0
−+ θ0
+)/2. After the posi
+−θ0
−= θ+−θ−remain
−=
+. To solve for the power consumption, we need to
a
b
c
d
OFF
ON
θ0
θ+0
θθ+
δ
δ
Δ
Figure 5: Different points of interest on the density curves.
The power consumption ga(t,τa) of the load at a start
ing from the instant when the step change in setpoint is
applied is shown in Figure 6(a). All the loads in the OFF
state and having a temperature between θ−and θ0
instant when the deadband shift occurs will have power
waveforms similar in nature to ga(t,τa). Thus the load at
a typifies the behavior of all the loads lying on the OFF
state density curve between θ−and θ0
ment applies for loads at points b, c and d. Figures 6(a)
6(d) illustrate the general nature of the power waveforms
of the loads in all four regions, marked by a, b, c and d in
Figure 5.
+at the
+. The same argu
time
P
τa
Tc
Th
{θ }{θa}{θ+}
{θ}
ga(t)
0
(a) Power waveform at point a.
17th Power Systems Computation ConferenceStockholm Sweden  August 2226, 2011
Page 4
time
P
τb
Tc
Th
{θ+ }{θb} {θ}
{θ+}{θ}
gb(t)
0
(b) Power waveform at point b.
time
P
τc
Tc
Th
{θ }{θc}{θ+}
{θ}
gc(t)
{θ}
Th
0
(c) Power waveform at point c.
time
P
τd
Tc
Th
{θ+ }{θd}{θ+}
{θ}
gd(t)
{θ}
Th
0
(d) Power waveform at point d.
Figure 6: Power waveforms at four different points marked in Figure 5.
It is shown in the appendix that the Laplace transform
of ga(t,τa) is given by
Ga(s,τa) = e−sτaG(s)
where
G(s) =
P(1 − e−sTc)
s(1 − e−s(Tc+Th))
and τa = Th− th(θa), with th(θa) given by (6). Aver
aging over all such loads (represented by a) on the OFF
density curve between temperatures θ−and θ0
the Laplace transform of the average power demand,
+, we obtain
Pa(s) =
∫θ0
+
θ−
f0(θa)Ga(s,τa)dθa
(13)
where f0(θa) can be computed from (11).
In Figure 6(b), a load at point b on the ON density
curve in Figure 5 has power consumption gb(t,τb), where
τb= Tc− tc(θb), and tc(θb) is given by (5). The Laplace
transform is derived in the appendix as
(
We can compute the average power demand of all the
loads represented by b as
Gb(s,τb) =es(Tc−τb)G(s) −P
s
(es(Tc−τb)− 1))
.
Pb(s) =
∫θ0
+
θ−
f1(θb)Gb(s,τb)dθb.
(14)
In Figure 6(c), the power consumption gc(t,τc) of a
load at point c on the OFF density curve in Figure 5 has
the Laplace transform
Gc(s,τc) = e−s(Th+τc)G(s),
(θamb−θ−
mand of the loads represented by the point c is then given
by
∫θ−
Figure 6(d) depicts the situation of a load at point d on
the ON density curve, that suddenly switches to the OFF
state as the deadband is shifted (for now we assume the
deadband is shifted to the right, i.e., there is an increase
in the setpoint). The power consumption gd(t) has the
Laplace transform
where τc = CRln
θamb−θc
)
. The average power de
Pc(s) =
θ0
−
f0(θc)Gc(s,τc)dθc
(15)
Gd(s,τd) = e−s(Th+τd)G(s),
where the dynamics in (1) can be solved for τd
CRln
θamb−θ−
loads characterized by point d in Figure 5 is then given
by
∫θ−
The average power demand of the whole population
becomes,
=
(
θamb−θd
)
. The average power demand of the
Pd(s) =
θ0
−
f1(θd)Gd(s,τd)dθd.
(16)
Pavg(s) = Pa(s) + Pb(s) + Pc(s) + Pd(s).
(17)
Using (13), (14), (15) and (16) we obtain an expression
for Pavg(s) that is rather complex. It is hard, and perhaps
even impossible, to obtain the inverse Laplace transform.
However, with the assistance of MATHEMATICA?,
Pavg(s) may be expanded as a series in s. We also make
use of the assumptions,
∆ ≪ (θs− θamb+ PR)
∆ ≪ (θamb− θs)
δ ≪ ∆
where θsis the setpoint temperature. Note that the first
two assumptions require that the deadband width is small,
while the third assumption requires that the shift in the
deadband is small relative to the deadband width. This
latter assumption ensures that the load densities are not
perturbed far from their steadystate forms. Accordingly,
the steadystate power consumption is given by
Pavg,ss≈(θamb− θ+)N
ηR
,
where η is the electrical efficiency of the cooling equip
ment and N is the population size. The deviation in power
response can be approximated by
(d
Ptot(s) ≈ −
s+
ωA∆
s2+ ω2
)
δ
(18)
17th Power Systems Computation Conference Stockholm Sweden  August 2226, 2011
Page 5
where
A∆=
5√15C(θamb− θ+)(PR − θamb+ θ+)
η(P2R2+ 3PR(θamb− θ+) − 3(θamb− θ+)2)3/2
×(3PR − θamb+ θ+)N
(Tc0+ Th0)
2√15(θamb− θ+)(PR − θamb+ θ+)
CR∆√P2R2+ 3PR(θamb− θ+) − 3(θamb− θ+)2,
d =N
ηR.
,
ω =
and Tc0and Th0are the original (prior to the setpoint shift)
steadystate cooling and heating times, respectively, given
by (3) and (4). The transfer function for this linear model
is,
T(s) =Ptot(s)
δ/s
Dueto theassumptions of lownoiseand homogeneity,
our analytical model is undamped. The actual system, on
the other hand, experiences both heterogeneity and noise,
and therefore will exhibit a damped response. In order to
capture that effect, we have chosen to add a damping term
σ (to be estimated online) into the model, giving
(
Figure 7 shows a comparison between the response
calculated from the model (19) and the true response to
a step change in the setpoint obtained from simulation. A
damping coefficient of 0.002 min−1was added, as that
value gave a close match to the decay in the actual system
response.
Total Power Variation
= −
(
d +
A∆ωs
s2+ ω2
)
.
T(s) = −
d +
sωA∆
(s + σ)2+ ω2
)
.
(19)
74 767880 8284
21
22
23
24
25
26
27
Time (hrs.)
Power (MW)
actual power
calculated from model
Figure 7: Comparison of the approximate model with the actual simula
tion, for the same setpoint disturbance as in Figure 3.
4CONTROL LAW
The TCL load controller, described by the transfer
function (19), can also be expressed in statespace form,
˙ x = Ax + Bu
y = Cx + Du
where the input u(t) is the shift in the deadband of all
TCLs, and the output y(t) is the change in the total power
demand from the steadystate value. The statespace ma
trices are given by
[−2σ
C =[−1
Our goal is to design a controller using the linear
quadratic regulator (LQR) approach [7] to track an ex
ogenous reference yd. We observe that the system has
an openloop zero very close to the imaginary axis (d ≪
ωA∆) and hence we need to use an integral controller.
Considering the integral of the output error e = (y − yd),
where yd is the reference, as the third state w(t) =
∫t
˙ x = Ax + Bu + Eyd
y = Cx + Du
A =
−ω
0
σ2+ω2
ω
]
,B =
[ωA∆
0
]
,
0],D = −d.
0(y(τ) − yd(τ))dτ of the system, the modified state
space model becomes
where x = [xw]⊤and,
A =
[A02×1
0
0],
C
]
,B =
[B
D
]
,
C =[C
Minimizing the cost function
∫∞
where Q ≥ 03×3and R > 0 are design variables, we
obtain the optimal control law u(t) of the form
D = D,E =
[02×1
−1
]
.
J =
0
(x(t)⊤Qx(t) + u(t)2R)dt
u = −(Kx + Gyd),
with G a precompensator gain chosen to ensure unity DC
gain. Since we can only measure the output y(t) and the
third state w(t), the other two states are estimated using
a linear quadratic estimator [7] which has the statespace
form,
˙ˆ x = A ˆ x + Bu + L(y − yd)
ˆ y = C ˆ x + Du
[ˆ x
The plots in Figure 8 show that the controller can be
usedtoforcetheaggregatepowerdemandoftheTCLpop
ulation to track a range of reference signals. The transient
variations in the ONstate and OFFstate populations are
shown in Figure 9. In comparison with the uncontrolled
response of Figure 4, it can be seen that the controller
suppresses the lengthy oscillations. Figure 9 shows that
in presence of the controller, the distribution of loads al
most always remains close to steady state, justifying an
assumption made during the derivation of the model.
u = −K
w
]
+ Gyd.
17th Power Systems Computation ConferenceStockholm Sweden  August 2226, 2011
Page 6
0 10 203040
10
15
20
25
30
35
40
Time (hrs.)
Power (MW)
Total Power Variation
actual power,y
reference power,yd
010203040
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
Temperature Set−point Variation
Temerature Setpoint (degC)
Time (hrs.)
(a) Response to step reference and the control input.
010203040
20
25
30
35
Time (hrs.)
Power (MW)
Total Power Variation
actual power,y
reference power,yd
010203040
−1.6
−1.4
−1.2
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
Temperature Set−point Variation
Temerature Setpoint (degC)
Time (hrs.)
(b) Response to ramp reference and the control input.
510
Time (hrs.)
1520
15
20
25
30
35
Power (MW)
Total Power Variation
actual power,y
reference power,yd
510
Time (hrs.)
1520
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
Temperature Set−point Variation
Temerature Setpoint (degC)
(c) Response to sinusoidal reference and the control input.
Figure 8: Reference tracking achieved through setpoint shift.
19.5
20
20.5
45
50
55
60
65
70
0
0.5
1
1.5
Temperature (oC)
Probability density function (OFF state)
Time (hrs.)
PDF (OFF)
(a) OFFstate distribution.
19.5
20
20.5
21
45
50
55
60
65
70
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Temperature (oC)
Probability density function (ON state)
Time (hrs.)
PDF (ON)
(b) ONstate distribution.
Figure 9: Variation in distribution of loads under the influence of the
controller.
5CONCLUSION
In this paper we have analytically derived a transfer
function relating the change in aggregate power demand
of a population of TCLs to a change in thermostat setpoint
applied to all TCLs in unison. We have designed a linear
quadratic regulator to enable the aggregate power demand
totrackreferencesignals. Thissuggeststhederivedaggre
gate response model could be used to allow load to track
fluctuations in renewable generation. The analysis has
been based on the assumptions that the TCL population
is homogeneous and that the noise level is insignificant.
Further studies are required to incorporate the effects of
heterogeneity and noise into the model. Those extensions
are important for determining the damping coefficient.
Similar analysis can be used to establish the aggregate
characteristics of groups of plugin electric vehicles, an
other candidate for compensating the variability in renew
able generation.
17th Power Systems Computation ConferenceStockholm Sweden  August 2226, 2011
Page 7
ACKNOWLEDGEMENT
We thank Dr. Michael Chertkov of Los Alamos Na
tional Laboratory, USA for his support and useful insights
throughout this work. We also thank Prof. Duncan Call
away for many helpful discussions.
REFERENCES
[1] S. Ihara and F.C. Schweppe, “Physically based mod
elling of cold load pickup”, IEEE Transactions on
Power Apparatus and Systems, 100(9):41424150,
September 1981.
[2] C.Y. Chong and A.S. Debs, “Statistical synthesis of
power system functional load models”, Proceedings
of the 18th IEEE Conference on Decision and Con
trol, 264269, December 1979.
[3] D.S. Callaway, “Tapping the energy storage poten
tial in electric loads to deliver load following and
regulation, with application to wind energy”, Energy
Conversion and Management, 50(5):13891400, May
2009.
[4] R. Malham´ e and C.Y. Chong, “Electric load model
synthesis by diffusion approximation of a highorder
hybridstate stochastic system”, IEEE Transactions
on Automatic Control, 30(9):854860, September
1985.
[5] R.E. Mortensen and K.P. Haggerty, “A stochastic
computer model for heating and cooling loads”, IEEE
Transactions on Power Systems, 3(3):12131219, Au
gust 1988.
[6] C. Uc ¸ak and R. C ¸a˘ glar, “The effects of load parameter
dispersion and direct load control actions on aggre
gated load”, POWERCON’98, 1998.
[7] B.D.O. Anderson and J.B. Moore, “Optimal Control:
Linear Quadratic Methods”, PrenticeHall, 1990.
AAPPENDIX
Thisappendixprovidesanoutlineofthestepsinvolved
in deriving the expressions for Ga(s,τa) and Gb(s,τb).
Derivation of the expressions for Gc(s,τc) and Gd(s,τd)
is similar to that of Ga(s,τa) and hence is not included.
time
P
Tc
Th
(θ+)
(θ)(θ+)
g(t)
Figure 10: The reference squarewave g(t).
A.1Derivation of Ga(s,τa)
We note that the waveform ga(t,τa) shown in Figure 6
is a timeshifted squarewave. Considering the waveform
g(t) in Figure 10, where g(t) = 0 for t < 0, we can ex
press ga(t,τa) as ga(t,τa) = g(t−τa)1(t), where 1(t) is
the unitstep function, defined as
{
The Laplace transform of the squarewave g(t) is
G(s) =
of ga(t,τa) is
∫∞
=
0
∫∞
=
−τa
where t∗= t−τa. Therefore, becauseg(t∗) = 0, ∀t∗< 0,
∫∞
=e−sτaG(s).
1(t) =
1,
0,
t ≥ 0
t < 0.
(20)
P(1−e−sTc)
s(1−e−s(Tc+Th)). Hence, the Laplace transform
Ga(s,τa) =
0
ga(t,τa)e−stdt
∫∞
g(t − τa)1(t)e−stdt
=
0
g(t − τa)e−stdt
∫∞
g(t∗)e−s(t∗+τa)dt∗
Ga(s,τa) =
0
g(t∗)e−s(t∗+τa)dt∗
(21)
The expressions for Gc(s,τc) and Gd(s,τd) can be de
rived similarly.
A.2Derivation of Gb(s,τb)
The squarewave gb(t,τb) can be expressed as
gb(t,τb) = g(t + Tc− τb)1(t) and its Laplace transform
as
∫∞
=
0
∫∞
=
Tc−τb
where t∗= t + Tc− τb. Therefore,
Gb(s,τb) =es(Tc−τb)
Tc−τb
=es(Tc−τb)
0
Gb(s,τb) =
0
gb(t,τb)e−stdt
∫∞
g(t + Tc− τb)1(t)e−stdt
=
0
g(t + Tc− τb)e−stdt
∫∞
g(t∗)e−s(t∗−Tc+τb)dt∗
∫∞
∫∞
g(t∗)e−st∗dt∗
g(t∗)e−st∗dt∗
− es(Tc−τb)
∫Tc−τb
0
g(t∗)e−st∗dt∗
=es(Tc−τb)G(s)
− es(Tc−τb)
∫Tc−τb
0
Pe−st∗dt∗
=es(Tc−τb)G(s)
− es(Tc−τb)P
(1 − e−s(Tc−τb))
s
(es(Tc−τb)− 1)
s
=es(Tc−τb)G(s) − P
. (22)
17th Power Systems Computation Conference Stockholm Sweden  August 2226, 2011
Powered by TCPDF (www.tcpdf.org)
View other sources
Hide other sources
 Available from Soumya Kundu · Jun 10, 2014
 Available from ArXiv