# 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 time-scale 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.

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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 non-dispatchability 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 generation-balancing 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 (Fokker-Planck equation) model. The model

was derived by first assuming a homogeneous group of

thermostats (all thermostats having the same parameters),

andthenextendedusingperturbationanalysistoobtainthe

model for a non-homogeneous group of thermostats. In

ROWTH in renewable power generation brings with

it concerns for grid reliability due to the inter-

[5], a discrete-time 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 hysteresis-based 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 steady-state 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 cooling-load (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 22-26, 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 ON-loads

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

OFF-loads 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 22-26, 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 OFF-state and ON-state 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) OFF-state 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) ON-state 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. a-d 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

θ-0

θ+0

θ-θ+

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 22-26, 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 steady-state forms. Accordingly,

the steady-state 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 22-26, 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)

steady-state 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 low-noiseand 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 on-line) 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 state-space 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 steady-state value. The state-space 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 open-loop 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 pre-compensator 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 state-space

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 ON-state and OFF-state 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 22-26, 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) OFF-state 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) ON-state 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 plug-in electric vehicles, an-

other candidate for compensating the variability in renew-

able generation.

17th Power Systems Computation ConferenceStockholm Sweden - August 22-26, 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.

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Transactions on Power Systems, 3(3):1213-1219, 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”, Prentice-Hall, 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 square-wave g(t).

A.1Derivation of Ga(s,τa)

We note that the waveform ga(t,τa) shown in Figure 6

is a time-shifted square-wave. 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 unit-step function, defined as

{

The Laplace transform of the square-wave 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 square-wave 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 22-26, 2011

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