Table Of ContentA Two Level Feedback System Design to Regulation Service
Provision
Bowen Zhang1 and John Baillieul2
Abstract—Demand side management has gained increasing simple as uniform. More recently, the possibility to integrate
importanceasthepenetrationofrenewableenergygrows.Based DLC into load smoothing or following is investigated. The
onaMarkovjumpprocessmodellingofagroupofthermostatic
notion of packetized DLC is proposed in [9] to smooth
loads,thispaperproposesatwolevelfeedbacksystemdesignbe-
electricity consumption with shortened appliances duty cycle.
tweentheindependentsystemoperator(ISO)andtheregulation
serviceprovidersuchthattwoobjectivesareachieved:(1)theISO Load following can also be found in [10]-[12] where various
can optimally dispatch regulation signals to multiple providers of control methods including discrete time directly on/off
3 in real time in order to reduce the requirement for expensive actuation control, set point shifting, as well as model based
1 spinningreserves,and(2)eachregulationprovidercancontrolits
predictive control are used. Results in these papers show that
0 thermostaticloadstorespondtheISOsignal.Itisalsoshownthat
loads, by acting as both a positive and negative generation
2 theamountofregulationservicethatcanbeprovidedisimplicitly
restrictedbyafewfundamentalparametersoftheprovideritself, source, is a promising resource to response against the load
n
such as the allowable set point choice and its thermal constant. request from the ISO to maintain supply demand balance.
a
J An interesting finding is that the regulation provider’s ability While the study of the DLC itself seems to be comprehen-
to provide a large amount of long term accumulated regulation
7 sive,therearetwoissuesremaintobeanswered.Thefirstone
and short term signal tracking restrict each other. Simulation
2 is the limitation of regulation provision, namely what is the
results are presented to verify and illustrate the performance of
the proposed framework. maximum regulation can be provided by a specific provider
]
Y given its implicit features such as population of appliances
pool, parameters in its own model. The second one is about
S I. INTRODUCTION the information exchange between the ISO and the providers.
.
s Wind energy, known as the most widely used form of Instead of the one way command-like signalling, can we
c
[ clean and renewable generation resource in power systems, design another framework such that the ISO can get feedback
is increasing its penetration around the world. Europe, Asia, information from providers? If so, how beneficial is it to the
1
and North America will have an exponential yearly increase real time operation?
v
6 in the amount of added wind power capacity by 2020 [1]. The objective of this paper is to answer these questions by
8 The increase in wind energy makes the generation side less developing a two layer feedback control system. We focus
3 controllable due to the intermittent feature of wind. As a primarily on smart buildings that are operated by a central
6 consequence, building level demand side management (DSM) operator who has the authority to shift the set point of
.
1 for thermostatic load becomes a crucial part that can be thermostaticapplianceswithinthebuilding.Consumersareas-
0 achieved generally through either a price-based signalling sumed to authorize the operator to control their set point once
3 control [2]-[5] or a non-priced direct load control (DLC). they provide the allowable set point range. It will be shown
1
Compared with pricing mechanism, the operator in DLC can that the operator in the lower feedback layer can design a
:
v have better understanding of the loads’ response in order to feedback control law to track the ISO regulation signal within
Xi provide short time scale regulation services [6]. a certain ramping rate. The higher feedback loop is designed
DLC approaches on thermostatic loads have been studied between the ISO and buildings where the former dispatches
r
a extensively during the past decade for difference purposes, real time signals and the latter send back information to
including load shifting, load smoothing, and load following. characterize their instant-by-instant capability to respond. We
A primitive investigation on load shifting has been studied also derive analytically the maximum possible regulation that
with a state queuing model to illustrate the effect of set-point can be provided from buildings based on two considerations
change [7]. It is shown that even a simple step change will – the capability to provide long term accumulated and short
resultinacomplicatedloadprofileduetotheinstantlychange term (one step) ramping regulation. It is further shown that
of loads’ status. Taking into account both the appliances two fundamental parameters determining the capability are
duty cycle as well as probabilistic consumer behaviour, a the building thermal constant τ and appliances temperature
steady state appliances population distribution during an load gain T . Large τ enables the building to provide larger long
g
shifting demand response is analysed in [8]. Results shows term, but smaller short term regulation service. Large T
g
that the steady state distribution exists and may not be as performs oppositely. The overall two level system has a good
performance in real time operation as it reduces the amount
*TheauthorsgratefullyacknowledgesupportoftheU.S.NationalScience
of spinning reserves required from high-cost generators.
FoundationunderEFRIGrant1038230
1Corresponding author. Division of Systems Eng., Boston University, 15 The paper proceeds as follows. Section II develops the
St.MarysSt.,Brookline,MA02446,email:[email protected] state space model for which the feedback system is designed.
2Dept. of Electrical and Computer Eng., Dept. of Mechanical Eng., and
Section III gives the overall design of the two layer feedback
Division of Systems Eng., Boston University, 110 Cummington St., Boston,
MA02215,email:[email protected] system in which we first introduce the lower layer controller
with feedback linearization, followed by the investigation of
maximum possible regulation provision, and then the higher
level feedback design that includes a quadratic program to
optimallydispatchrealtimeregulationtoindividualbuildings.
Section IV discusses an observer design to asymptotically
reducetheestimationerrorforunobservablestates.Simulation
results are given in section V to show the performance of the
proposedsystem.SectionVIconcludesthepaperandproposes
future work.
II. MARKOVJUMPTHERMALPROCESSMODELLING
We model the system as a continuous time, discrete state
Fig. 1. Markov jump process modelling. (a) Markov chain transition rate
Markov jump process where a state i is defined as a pair of
diagram. (b) Transition from state i to i+1. With temperature rise ∆T
temperaturevalueandthermostatvalue{T ,on(off)}.Previous (length of arrow), appliances whose temperatures inside the dotted area (A,
i
literaturehasconsideredsimilarstatedefinitions-forexample C,E)changestateandoutside(B,D)remaininthesamestate.
[7], [8], [10]. The departure of our modelling approach from
previous ones is that we consider state in a Markov setting
where the second equality follows from the relation between
and establish the implicit relations between thermal processes
warming rate and duty cycle. Since p = p , comparing
and the Markov jump processes. The Markov jump process i,i+1 dot
(1) with (3) we have α = N/t . Similarly, β = N/t for
model in this section is the fundamental model that we based off on
the duty off process.
to design the two level feedback system and to derive the
When control is applied to shift the set point at a rate of
limitation of providing regulation service.
r (the unit of r is the same as r or r ), there is also
Denotethefixedwidthcomfortbandaroundthesetpointas set set on off
a transition between adjacent states within ∆t. Intuitively,
[T ,T ]anddiscretizetemperatureinthebandintobinsof
min max when we change the set point, the absolute temperature in
widthδ,thenthenumberoftemperaturebinsisN =∆ /δ.
band eachindividualroomdoesnotchangeinstantly,butitsrelative
We say that an appliances is in state i for i = 1,...,N if
positioninthecomfortbandchanges.Whenthesetpointrises,
T ∈[T +(i−1)δ,T +iδ] with status off, and i for i=
i min min namely r >0, we can show that the resulting transition rate
N+1,...,2N ifT ∈[T −(i−N−2)δ,T −(i−N−1)δ] set
i max max isu=r /δ.Thetransitionrateisthesamewithr <0.The
with status on. set set
combined transition rate by the thermal process and set point
First let our control u = 0 and solve for the Markov
shifting process is the sum of these two individual processes
transition rate of the uncontrolled thermal process. Denote α
as shown in Fig.1(a). Note that the allowable control set is
the transition rate when the thermostat is off, and β the rate
L = {u|−β ≤ u ≤ α} to maintain a non-negative Markov
whenthethermostatison;seeFig.1(a).Inthedutyoffprocess, u
rate.Whenu>αoru<−β,thesystemfailstobeaMarkov
theprobabilitythatanapplianceswillbeinstateiattimet+1
chain.
given in state j at time t is given by,
Similar to (1), we can write the transition probability for
α∆t+o(∆t) if i=j+1 the duty off process with non-zero control u,
pi,j = 1−α∆t+o(∆t) if i=j (1) (α−u)∆t+o(∆t) if i=j+1
0 otherwise,
p = 1−(α−u)∆t+o(∆t) if i=j (4)
i,j
where o(∆t) means lim o(∆t) = 0. The above equation 0 otherwise.
∆t→0 ∆t
relates transition probability and transition rate for small ∆t. Let x(t) be a vector whose ith component is the probability
It can be interpreted that the transition probability between thatanapplianceisfoundintheith stateintheMarkovchain.
adjacent states linearly increases with small ∆t. The dynamics of x(t) for i=2,...,N is given by,
i
From a thermal point of view, the temperature rise ∆T
within a small ∆t is proportional to its warming rate roff, xi(t+∆t)=pi,ixi(t)+pi,i−1xi−1(t), (5)
∆T =r ∆t. (2) Substitute (4) into (5) we will have,
off
Assuming that an appliance’s actual temperature is uniformly xi(t+∆t)−xi(t)=(α−u)∆t[xi−1(t)−xi(t)]+o(∆t). (6)
distributed among its bins, then the probability that the state
Dividing ∆t on both sides and taking ∆t to 0 yields the
transits is equal to the probability that the appliance tempera-
dynamics for the ith component in the duty off process,
tureisinthedottedareainFig.1(b),namelyappliancesinthe
dotted area change their state from i to i+1, and appliances x˙ (t)=(α−u)[x (t)−x (t)], (7)
i i−1 i
outside remain in the same state.
for i=2,...,N. Similarly for the duty on process i=N +
The probability of being in the dotted area is given by,
2,...,2N,
r ∆t ∆ ∆t N∆t
p = off = band = , (3)
dot δ toffδ toff x˙i(t)=(β+u)[xi−1(t)−xi(t)], (8)
and reflection boundaries i=1,N +1,
x˙ (t)=−(α+u)x (t)+(β+u)x (t),
1 1 2N (9)
x˙ (t)=(α−u)x (t)−(β+u)x (t).
N+1 N N+1
From (7) to (9), the overall dynamics of x(t) is given by,
x˙(t)=[A+Bu(t)]x(t), (10)
where
−α 0 ... β
α ...
0 ... −α ...
A= α −β ... , Finidgi.v2id.ualTtwraockleinvgelcfoenetdrboallcekrsfyosrteamgiwvehnereregthuelaltoiownersifgeneadlb,aacnkddtehseighnisghtheer
... ... β ... fpereodvbidaecrksesnuacbhlesthtahteincfoomrmmautnioicnatiiosnexbcehtwanegenedthteo IrSeOachanbdetttheer rreegaullattiimone
... −β 0 regulationsignaldispatch.
0 ... 0 β −β
depends on the control u if
1 0 ... 1
−01 ...... 1 ... This meaxns(tt)he∈∈nu{{mxxb((tte))r∈∈ofRRc22oNNn||tCxroNBl(ltax)b(l+te)xa(cid:54)=p2Np0(l}ita,)n>ces0}a.round the
B= −1 −1 ... . comfortbandboundaryispositive,soshiftingthesetpointcan
... ... 1 ... change the aggregated consumption. The following change of
variables,
... −1 0 φ (x)
0 ... 0 1 −1 1
.. (cid:20) η (cid:21)
The output of the system, namely the aggregated consump- T(x)= . := ξ , (13)
tion, is φ2n−1(x)
Cx
y(t)=Cx(t), (11)
yields the internal-external state space representation,
where C =[0,...,0,N ,...,N ], and N is the total number
c c c η˙ = f (η(t),ξ(t))
(cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) 0
of appliances inNthe pool. N ξ˙ = C[A+Bu(t)]x(t) (14)
y = ξ(t),
III. TWOLEVELFEEDBACKSYSTEMDESIGN where η(t) stands for the internal and ξ(t) the external states
[13]. Then the output becomes the external variable ξ(t).
Based on the state space model developed in section II,
To design a controller for a relative degree 1 system such
we are able to design our two level feedback system as
that lim ξ(t)−R(t)=0 where R(t) is the required amount
showninFig.2.Inthebuildinglevelfeedbackloop,wedesign t→∞
of regulation needs to be provided, we need R(t),R˙(t) to be
a state feedback law such that the building can track the
available and bounded for all t > 0. This is obtainable since
required regulation signal. The higher level feedback is from
the ISO regulation requirement is bounded: R(t)∈[−R,R]
each individual building to the ISO where the former send r r
(R is maximum regulation sold to the market), and R(t) is
their information to the latter to characterize the building’s r
updated every 4 to 6 seconds prior to consumption. The time
capability of responding to regulation signals. The ISO, after
derivative of the tracking error,
receivingallinformationfromeachparticipant,dispatchesreal
time regulation signals after solving an optimization problem. e˙(t)=ξ˙(t)−R˙(t)=C[A+Bu(t)]x(t)−R˙(t), (15)
The next sub-section will shortly discuss the building level
will asymptotically approach 0 if u(t) is chosen such that
controller design by feedback linearzation.
e˙(t)=Ke(t),K >0. (16)
A. Building Level Feedback Linearization Design The controller
To solve the building level controller design for the non- u(t)=−CAx(t)+ 1 [−K(ξ(t)−R(t))+R˙(t)] (17)
linear system described in (10) and (11), the feedback lin- CBx(t) CBx(t)
earization method can be readily applied [13]. Note that the
satisfies the requirement because substituting (17) into (15)
system has relative degree 1 as the first derivative of y,
yields (16). For the discrete signal R(t), we can calculate the
R(t+∆t)−R(t)
y˙ =Cx˙(t)=C[A+Bu(t)]x(t), (12) first derivative of R(t) as R˙(t)= .
∆t
B. Long and Short Term Regulation Provision The third equation is based on the relation between transition
In this section we proceed to answer the first question rate and duty cycle. The last equation says that the sum of
proposed in section I, namely what is the regulation service cooling and warming rate is the rate that the cooling system
limitation that a building can provide given its model param- generates because roff = ramb and ron = rapp−ramb, see [9].
eters and user specified settings. This question has two parts: ramb is the warming rate caused by the ambient temperature
(1) in the long term, we need the accumulated shift of set that higher than room temperature, and rapp is the cooling
point to be within the allowable range to provide reserve, and ratecausedbytheoperationofairconditionercompressor.To
(2) in the short term, we need the signalling change to be maintainthesetpointshiftwithintheallowablesetpointband
within bounds such that the controller can track it. The first with T(0)=Tset, we need
propositionwilldiscussthelimitationonlongtermregulation. 1 1
T(t)∈[T − ∆ ,T + ∆ ], (27)
Proposition.1 For a given number of appliances Nc and set 2 set set 2 set
allowable set point range [Tset − 21∆set,Tset + 21∆set], the which yields
accumulated amount of reserve that a building can provide N τ∆
S(t)≤ c set. (28)
up to time t is given by 2T
g
t (cid:4)
(cid:88)
S(t)= R(i), (18)
Remark.1 According to proposition 1, the accumulated
i=0 long term provision capability is proportional to three pa-
and this is bounded by, rameters: N ,∆ , and τ/T . The intuition for the first two
c set g
N τ∆ parameters is that a large appliance population and allowable
S(t)≤ c set, (19)
2T set point shift range enables the operator to provide more
g
service. As for the parameter τ/T , we note that a large
where R(i) is the regulation signal at time i, T is the g
g value of τ impedes, and T facilitates, the change of room
temperature gain of air conditioner if it is on, and τ is the g
temperaturewhenthesamecontrolisapplied.Thustheallow-
effective thermal constant of the building.
able accumulated provision is proportional to τ and inversely
Proof. The total consumption up to time t is given by,
proportional to T .
g
t
(cid:88) When we consider short term regulation, the possible ramp
P =tR + R(i)=tR +S(t), (20)
cool b b of consumption is limited by the state x(t) because we
i=0
are shifting the set point rather than directly turning on/off
which is an average consumption level of
appliances. Aside from x(t), we are interested in finding a
S(t) few fundamental parameters that characterize the short term
P =R + . (21)
ave b t capability similar to those found in the first proposition. The
This is equivalent to choosing a constant non-zero u to second proposition below provides similar results.
maintain at a constant consumption level P . The relation Proposition.2 For a given number of appliances N and
ave c
between average consumption and non-zero control is, width of temperature band for a specific set point [T −
set
t N α−u 21∆band,Tset+ 12∆band], the amount of reserve that a building
P =N on =N β+u =N . (22) canprovideforoneperiodrampingislimitedbythefollowing
ave ct +t c N + N cα+β
on off α−u β+u bounds,
For an uncontrolled process at average consumption, N T N T
−Nx c g ≤∆R≤Nx c g , (29)
α 2Nτ∆ Nτ∆
R =N . (23) band band
b cα+β
where T is the temperature gain of air conditioner if it is on,
g
From the above three equations, we have, and τ is the effective thermal constant of the building.
S(t) −u Proof. In the dynamic operation when the tracking error is
=N (24)
t cα+β 0, the feedback controller is given by,
1
resulting a constant value of control
u= [N (αx −βx )−∆R]. (30)
N (x +x ) c N 2N
S(t)(α+β) c N 2N
u=− . (25)
tN Since the allowable control set is u ∈ [−β,α], then ∆R is
c
restricted by
The set point shift after time t by the constant u is given by,
−N (α+β)x ≤∆R≤N (α+β)x (31)
c 2N c N
T(t) =T(0)+tr
set
Using the relation between transition rate and duty cycle
=T(0)+tuδ developed in section II and the fact
S(t) N N T
=T(0)− Nc (toff + ton) Nband (26) ∆band =toffroff =tonron,
S(t)
=T(0)− (r +r ) the following inequality is seem to hold:
N on off
c
=T(0)− SN(t)Tτg. −Nx2NNc(r∆on+roff) ≤∆R≤NxNNc(r∆on+roff). (32)
c band band
Using the fact that r +r =T /τ yields (29). (cid:4) service R that a building can provide is given by,
on off g r,max
Remark.2 According to proposition 2, the short term pro- (cid:26) (cid:27)
N τ∆ kN kN
vision capability is proportional to Nc, and inversely propor- Rr,max =min 2c0Tset,min( t c, t c) . (33)
tional to ∆ and τ/T . The proportionality to N shares g on off
band g c
the same explanation as in the first remark. The two inverse Proof.Forlongtermregulationservice,themaximumvalue
proportionalitiescanbeexplainedbysayingthatsmallvalueof of S(t) for t∈[0,50] is S(t)=10R with t=30. According
r
∆band makes the width of individual temperature bin smaller, to proposition 1,
and large value of T /τ makes the thermal transfer faster. N τ∆
g 10R ≤ c set. (34)
These two factors in turn facilitate the state transition to r 2T
g
provide larger one-step service.
This yields
Remark.3Theadvantageoftakingcomfortband∆ →0
band N τ∆
is that we can provide large value of one step service and R ≤ c set. (35)
r 20T
at the same time make each room temperature stick to the g
set point, with the sacrifice that we shorten the appliance’s For short term regulation service, we need to track the rate of
duty cycle. This tradeoff between system performance and consuming up and down starting from steady state. It is easy
appliances functioning is consistent with what we find in [9] to verify that the steady state probability distribution satisfies,
where electricity consumption can be smoothed by shortening
β
appliances duty cycle. x1 =...=xN = N(α+β)
terRmemregaurkla.t4ioTnhperofvraiscitoionn. Bτa/sTegdaofnfec(1ts9)boanthdl(o2n9g),aintdcasnhobret xN+1 =...=x2N = N(αα+β). (36)
seenthatabuildingabletoprovidelargeamountofshortterm
Substitute the above equation into (29) and note that ∆R =
ramping regulation is more incapable of providing long term
R/k in a k minutes ramp process. Finally we have,
r
accumulatedregulation.Theoppositealsoholds.Thusthetwo
capabilities restrict each other. R ≤min(kNc,kNc). (37)
r t t
on off
C. Buildings in the Regulation Service Market Taking the minimum of (35) and (37) completes the proof. (cid:4)
The above propositions determine the maximum regulation
service that can be provided in the power market. To become D. Real Time Spinning Reserve
a qualified provider in the U.S., a building needs to pass the
When the real time consumption ∆P ramps up or down
T-50 qualifying test [14]. Fig.3 shows the test signal from
quicklyduetoshorttermstochasticconsumptionaggregation,
PJM. The dotted line is the ideal consumption response that
the ISO has to use spinning reserves to compensate for the
the building consumes according to the test signal. Two key
demand that cannot be covered by the regulation service.
requirements are needed to pass the test:
This follow from the limits of proposition 1 and 2. In such
• Rate of response: the building needs to able to reach the
cases, a feedback signal from the provider side to the ISO
maximum or minimum consumption level within 5 minutes.
is beneficial to the ISO so that it can determine the needed
•Sustainedresponse:thebuildingneedstoabletomaintain
spinning reserve P . The following proposition establishes
spin
atthemaximumorminimumconsumptionlevelfor5minutes.
the relation between P and ∆P when necessary building
spin
information is given.
Proposition.3 Denote ∆P as the stochastic demand ramp
inonestep,thenthespinningreserveP neededtomaintain
spin
grid balance is given by,
P = (∆P −∆R )1 +
spin max {∆P>∆Rmax} (38)
(∆P −∆R )1 ,
min {∆P<∆Rmin}
where 1 is the indicator function, and
{·}
(cid:26)
∆R = N (r x −r x )+
max c off N on 2N
T −Tmin (cid:27)
min( set set ,r )(x +x ) ,
∆tδ on N 2N
(cid:26) (39)
Fig. 3. T-50 test by PJM where regulation providers are obliged to ramp
theirconsumptionupanddowntomeetthetestrequirements. ∆Rmin = Nc (roffxN−ronx2N)−
Tmax−T (cid:27)
The following corollary gives the upper bound on the min( set set,r )(x +x ) ,
∆tδ off N 2N
regulation service that a building can provide.
Corollary. To pass the T-50 qualifying test with a rate of are the maximum and minimum reserve provision threshold
response within k minutes (k ≤ 5), the maximum regulation for the building.
Proof. The allowable control u is limited by two factors. Then the objective function is given by
The first is that the set point shift resulting from u should be
m
(cid:88)
within the allowable set point range, max Wi−MP2 , (44)
d spin
T +∆tδu∈[Tmin,Tmax]. i=1
set set set
namely we are maximizing the sum of m regulation service
The second is the requirement of maintaining non-negative
widths from each building at time t+1, minus the spinning
Markov rate,
generation penalty P with positive coefficient M. The
spin
u∈[−β,α].
maximization problem is subject to the following type of
From the above two inequalities, constrains:
(cid:20) Tmin−T Tmax−T (cid:21) State Dynamics:
u∈ max( set set,−β),min( set set,α) . (40)
∆tδ ∆tδ
xi(t+1)=xi(t)+∆t(αi−ui)(xi (t)−xi(t)),
N N N-1 N
From (30), the one period ramping can be expressed in terms
xi (t+1)=xi (t)+∆t(βi+ui)(xi (t)−xi (t)).
of u, 2N 2N 2N-1 2N
(45)
(cid:8) (cid:9)
∆R=Nc (αxN−βx2N)−u(xN+x2N) (41) Feedback Controller:
Substituting (40) into (41) yields ∆R ∈ [∆Rmin,∆Rmax], Nci∆t(xiN(t)+xi2N(t))ui+∆Ri =Nci∆t(αxiN(t)−βxi2N(t)).
where ∆Rmin and ∆Rmax take values in (39). For the grid (46)
balance, we have, Supply-demand Balance:
∆P =∆R+P , (42) m
spin (cid:88)
∆Ri+P =∆P. (47)
spin
andwewishtouseaslittlespinningreserveaspossible.Then
i=1
P will take value in (38). The indicator function in (38)
spin Non-negative Markov Rate:
gives the condition that ∆P is outside the range of ∆R.(cid:4)
IntheU.S.regulationservicemarket,theISOhaspurchased βi ≤ui ≤αi. (48)
regulation service a day ahead from multiple m providers.
Allowable Regulation Range:
Assuming that feedback signals are set up between all the
providersandtheISOsuchthatthelatterreceivesinformation R −R ≤Cx+∆Ri ≤R +R. (49)
b r b r
from all of them every 4 seconds. Upon receiving these
Allowable Set Point Range:
information, the ISO knows their individual instant capability
range [∆Rmiin,∆Rmiax],i=1,...,m, and then dispatches regula- Tsmetin,i ≤Tsiet(t+1)=Tsiet(t)+ui∆tδ ≤Tsmetax,i. (50)
tion signals to each of them. The signals are dispatched such
that the minimum spinning generation is used and such that The above optimization cannot be solved with standard tech-
it should be within the range of capability for the building nique because the objective function has the minimum opera-
to respond. Most of the time, the one period ramping is tor. We transform the original problem into a QP. Let
within the total regulation capability of the m buildings so Ti (t+1)−Tmin,i
no spinning generation is needed. In such cases, the ISO can mi1 =min( set ∆tδ set ,roin),
(51)
havemorethanonedispatchsolution.Thequestionarisesthat Tmax,i−Ti (t+1)
mi =min( set set ,ri ).
howtheISOshoulddispatchsignalsinanoptimized way.The 2 ∆tδ off
next subsection answers the question by solving a quadratic
Then the original objective function becomes
program (QP).
m
max (cid:80)Ni(xi(t+1)+xi (t+1))(mi +mi)
E. Optimization of Real Time Regulation Signal Dispatch i=1 c N 2N 1 2 (52)
When there is more than one solution to the regulation −MP2 .
spin
dispatch at time t such that no spinning generation is needed,
If we add the following constrains,
an optimal way to dispatch is that we maximize the regu-
lation provision capability at time t + 1. From proposition
3, the ith building can provide regulation service with range mi1 ≤ron,
r∆egRul∈atio[∆nRsemirivni,c∆eRismigaxiv].enThbeywidth of the closed interval of mi1 ≤ Tsiet(t+∆1)tδ−Tsmetin,i, (53)
mi ≤r ,
2 off
Wi =∆Ri −∆Ri Tmax,i−Ti (t+1)
d max min mi ≤ set set ,
2 ∆tδ
=Ni(xi(t+1)+xi (t+1))
(cid:26) c TNi (t+1)−T2Nmin,i (43) and solve the QP,
min( set set ,ri )+
∆tδ on
max (52)
Tmax,i(t+1)−Ti (cid:27) (54)
min( set set,ri ) .
∆tδ off s.t. (45)−(50),(53),
we will obtain the same optimal solution as solving the
original problem. This is because when reaching the optimal A˜=
solution, one of the inequality constraints for both mi and −α+u 0 ... β+u
1
mi2 in (53) will be strict, otherwise the optimal solution is α−u ...
not reached since we can increase the value of mi1 or mi2 0 ... −α+u ...
to increase the value of objective function due to positive
coefficient Nci(xiN(t + 1) + xi2N(t + 1)) in (52). Then (51) α−u −β−u ... +
itshesaotrisigfiiendalanpdrobolpetmim.izNinogteotvheart ((5544))ihsaesqquuivaadlreanttictoobsjoelcvtiinvge ... ... β+u ...
function with linear constraints, Matlab or CPLEX can solve ... −β−u 0
0 ... 0 ... β+u −β−u
this QP efficiently.
(cid:15)(t) 0 ... 0
0 ...
In controllerIVd.esSigTnAT(E17O),BiStEiRsVaEsRsuDmEeSdIGthNat x(t) can be 0 ... (cid:15)(t) ...
measured. If this is not true, especially when the temperature 0 (cid:15)(t) ... .
bcaannndotispfironveliydeditshcerertiezqeudi,reid.e.pδrecisisisomna,llwseonteheadt tthoe dseensisgonr ... ... 0 ...
an observer to estimate the state. Considering the following ... (cid:15)(t) 0
dynamics of the observer, 0 ... 0 −NcL2N ... −NcL2N (cid:15)(t)−NcL2N.
(61)
x˜˙ =Ax˜+Bx˜u+L(t)(y−Cx˜), (55)
DenoteA˜ asthei−by−isquarematrixfromtheupperleft
i
where L(t) is the time varying column vector to be designed. part of A˜. In order to show that A˜ is negative definite, it is
The last term is similar to the innovation term in Luenberger equivalent to show that
filter [15]. Define the estimation error, e=x−x˜, then (−1)idet(A˜ )>0,for i=1,...,2N. (62)
i
e˙ =(A+Bu−LC)e. (56) For i = 1,...,2N − 1, A˜ is a triangular matrix whose
i
determinantistheproductofitsdiagonalelements.From(61),
In the observer design, we restrict the allowable control set to
the diagonal elements A˜ is given by,
prevent control saturation with some small constant positive i,i
margin (cid:15)˜>0, (cid:26) −α+u+(cid:15)(t) if i=1,...,N
A˜ = (63)
i,i −β−u+(cid:15)(t) if i=N +1,...,2N −1
u(t)∈[−β+(cid:15)˜≤u(t)≤α−(cid:15)˜]. (57)
Since (cid:15)(t)=γmin[β+u(t),α−u(t)], we have A ≤(γ−
i,i
This guarantees an irreducible and ergodic Markov chain.
1)min[β +u(t),α−u(t)] < 0 for i = 1,...,2N −1. This
If the control signals is allowed to stay saturated, then the
yields
Markov chain breaks and some of the states become isolated,
i
see Fig.1(a) when you substitute u = −β or u = α into (−1)idet(A˜ )=(−1)i(cid:89)A˜ >0,for i=1,...,2N −1.
the model. Estimation error of those isolated states cannot be i j,j
j=1
guaranteed.
(64)
The following proposition states that we can select L(t) As for i = 2N, det(A˜) is a linear function of L as the
2N
such that the estimation error will approach zero as t→∞.
observer parameter only appears in the last row of the matrix,
Proposition.4 If the control is chosen according to (57), so there must exists a L such that det(A˜) > 0. This
2N
then the following holds: completes the proof of negative definiteness of the matrix A˜,
(1) There exists a time varying observer design L(t) =
and (58) holds.
[0,...,0,L (t)]T such that
2N (2). Let
(cid:124) (cid:123)(cid:122) (cid:125)
2N−1 (cid:82)t(cid:15)(τ)dτ
y(t)=e0 (cid:107)e(t)(cid:107)2, (65)
eT(A+Bu−LC)e<−(cid:15)(t)(cid:107)e(cid:107)2 (58)
then the time derivative of y(t) is given by,
for all e(t) with (cid:15)(t)=γmin[β+u(t),α−u(t)], γ ∈(0,1).
(2) The estimation error would converge to zero asymptot- (cid:82)t(cid:15)(τ)dτ (cid:82)t(cid:15)(τ)dτ
y˙(t) =(cid:15)(t)e0 (cid:107)e(t)(cid:107)2+2eT(t)A˜e(t)e0
ically
lim e(t)=0 (59) (cid:82)t(cid:15)(τ)dτ
t→∞ =e0 [(cid:15)(t)(cid:107)e(t)(cid:107)2+2eT(t)A˜e(t)]
Proof. (1). To prove (58), it is equivalent to show that (cid:82)t(cid:15)(τ)dτ
<e0 [(cid:15)(t)(cid:107)e(t)(cid:107)2−2(cid:15)(t)(cid:107)e(t)(cid:107)2]
eT(A+Bu−LC+(cid:15)(t)I)e<0. (60) (cid:82)t(cid:15)(τ)dτ
DenoteA˜=A+Bu−LC+(cid:15)(t)I,thenA˜(t)canbeexpressed =−e0 (cid:15)(t)(cid:107)e(t)(cid:107)2
bythefollowingequationbasedonthespecialstructureofthe =−(cid:15)(t)y(t),
state space matrices A,B and C derived in section II. (66)
Thetimederivativefromtheaboveequationindicatesthaty(t)
in (65) is stable at zero.
(cid:82)t(cid:15)(τ)dτ
On the other hand, from (65) we have lim e0 = ∞
t→∞
and lim y(t) = 0. Necessarily we need lim e(t) = 0 to
t→∞ t→∞
satisfy these two conditions, which indicates that the error
will approach zero asymptotically.(cid:4)
Remark.5 (cid:15)(t) = γmin[β +u(t),α−u(t)] ≥ γ(cid:15)˜ is the
lower (conservative) bound of the convergence rate we can
achieve. In real time operation when the control signal is
distributedacrosstheallowablesettobeawayfromsaturation,
the numerical convergence rate is larger than the conservative
(a) LongTermProvision
bound. It should be noted that as the duty cycle increases,
N N
the boundary value of α = and β = decreases,
t t
off on
which means we have smaller allowable control set. Then
the value of control signal would be closer to the boundary
than the scenario with small duty cycle appliances, therefore
it is anticipated that the numerical convergence speed of the
error vector will be slower for system having large duty cycle
appliances.
V. SIMULATION
In this section, we provide three simulation to verify the
theoretical framework proposed in previous sections. In the (b) ShortTermProvision
first example, we will show how building parameters derived
Fig.4. IndividualprovidersfailtopasstheT-50qualifyingtestduetoeither
in the first two propositions restrict the maximum delivery
aboundedallowablesetpointrangeoralimitedrampingcapability.
of regulation service. In the second one, we will verify the
effectiveness of the proposed QP optimization in regulation
signal dispatch by reducing real time spinning generation. In B. Optimal Regulation Signal Dispatch
thelastexample,wewillshowtheperformanceoftheobserver
that asymptotically reduces the estimation error. WeuserealtimePJMdata[16]toverifytheperformanceof
the overall two level control and optimization framework. We
can see that the regulation signal is stochastic and ramps up
anddownaccordingtotheareacontrolerrorwithnoapparent
A. Long and Short Term Regulation Service Limitation
probability distribution. Fig.5 shows the simulation results of
Fig.4 shows how the long and short term restrictions on the proposed system with lower level feedback linearization
regulation service affect the T-50 test. The first figure is controller design and higher level two-way communication
an example of the inability to provide enough accumulated feedbackdesign.Fig.6isthesimulationresultwithonlylower
regulation service due to limited allowable set point choice. level controller. We find that both systems can track the
In this figure, the set point shift hits the lower bound of the regulation signal when it ramps up or down with the design
allowablesetbetween15to20minutes.Asaconsequence,the of building level feedback controller, but they differ in the
buildingcannotprovideasustainedhighconsumptionlevelby maximum ability of tracking. Compared with Fig.6, where
further decreasing the set point. Similarly, the set point shift individual building regulation signals are proportional to their
hittheupperboundafter40minuteswhichrestrictsthesustain maximumprovisionlevel(ISOdoesnothaveknowledgeabout
provision of lower level aggregated consumption. realtimebuildinginformation,soitdispatchessignalspropor-
The second figure is an example to show the inability to tional to their a prior maximum commitment), the signals in
provide short term regulation service. Although the set point Fig.5 are determined by solving the QP. We find that the
shift is within the allowable set, the ramping speed to provide consumption trend of red curve and blue curve are different
positive and negative regulation service is smaller than the most of the time in Fig.5 that fully exploits their tracking
required speed for a given maximum regulation obligation. capability, while in Fig.6 individual building consumptions
This is because the thermostatic appliances in this simulation follow the same trend with proportionality. The accumulated
have large duty cycle. For example, the typical duty cycle generationofspinningreservesisreducedwhentheregulation
of refrigerators is around 20 minutes and has large effective dispatch is optimized by the QP, and the distribution of
thermalconstant[17].Suchalargedutycyclegreatlyrestricts spinning reserve is more concentrated around 0 with smaller
the maximum ramping rate in regulation service derived from variance. Tab.1 shows the statistics of both systems, the total
proposition 2. spinning generation is reduced by 50% after optimization.
The standard deviation, maximum, and minimum spinning
generation is reduced by 30%, 30%, and 15% respectively.
Furthermore, at times when the regulation signal ramps up
and down at a fast speed, for example at 15, 30, 35, 40,
and 50 minutes, the performance of the two level structured
system in Fig.5 is better than the system performance without
the design. This is because the required spinning reserves is
largelyreduceddespitethesharprampingregulationsignal.To
sum, the proposed two level feedback framework outperforms
the system with only building level controller design.
(a) BuildingOutputs
(a) BuildingOutputs
(b) SpinningGeneration
Fig. 6. Real time regulation dispatch when the ISO does not receive
information from individual providers. Spinning generation is more spread
outaround0withlargervariance.
norm converges to zero after 20 minutes when the duty cycle
is10minutes,andconvergestozeroafter35minuteswhenthe
duty cycle is 20 minutes. This is because smaller duty cycle
enables a larger allowable control set, i.e the system has large
values of α and β to prevent from saturation. In Fig.7(c) and
Fig.7(d) we compare the distribution of control signals across
(b) SpinningGeneration
the allowable control set, it can be seen that the control signal
Fig. 5. Real time regulation dispatch after the quadratic program is stays further away from saturation for system having smaller
solved.Trackingbecomesclosetothetargetandspinninggenerationismore
duty cycle. Such systems have a larger convergence speed of
concentratedaround0withsmallervariance.
the error vector.
TABLEI
COMPARISONOFSYSTEMSTATISTICS VI. CONCLUSIONSANDFUTUREWORK
Unit/kW Spin.Gen. Mean Std.Dev. Max Min Thispaperproposesaninnovativetwolevelfeedbackdesign
Opt.Alg. 10216 0.18 27.20 100.27 -210.59 system for the real time regulation provision. The lower level
Prop.Alg. 20525 -0.44 41.97 146.19 -247.65
building feedback controller allows the building operator to
track a given signal from the ISO, and the higher level infor-
mation feedback from individual buildings to the ISO allows
C. Observer Performance
the latter to optimally dispatch real time regulation signals to
We verify the performance of the proposed observer. multiple providers by solving a quadratic programming prob-
Fig.7(a) shows the real and estimated value of two sample lem such that the cost of spinning reserves is reduced. During
states, x and x , these are the two states around the the development of the system, we also derive analytically
N 2N
boundary of the comfort band that will affect the change in the building’s limitation on both long term and short term
electricity consumption. We can see that the estimation error regulation provision and provide intuitive explanations that
for these two boundary states approach zero after 20 minutes. match our theoretical results. To counter the problem with
Fig.7(b)showstheconvergencespeedofthenormdecayofthe effective measuring, we also proposed an observer design to
errorvectore(t)forapplianceswithdifferentdutycycles.The reduce the error of state estimation such that the feedback
controller can work well. Simulation results indicate that the
proposedframeworkissuperiorinreducingtheamountofreal
time spinning reserve.
Future work will include the modelling of similar two level
feedback system but with price based signalling between the
operatorandtheconsumers.Suchdesignwillfitintoproblems
where consumers do not authorize the operator to directly
control their set point. It is expected that price-based mecha-
nism has looser control compared with the direct load control
because consumers can choose their own utility function.
Consequently, it enables consumers to obtain different level
of desired comfort.
(a) StateEstimation
REFERENCES
[1] http://www.emerging-energy.com/
[2] I. Ch. Paschalidis, B. Li, and M. C. Caramanis, “A Market-Based
Mechanism for Providing Demand-Side Regulation Service Reserves”,
Proceedingsofthe50thIEEEConferenceonDecisionandControland
EuropeanControlConference,December12-15,2011,Orlando,Florida,
doi:10.1109/CDC.2011.6160541
[3] M. C. Caramanis, I. Ch. Paschalidis, C. G. Cassandras, E. Bilgin,
and E. Ntakou, “Provision of Regulation Service Reserves by Flexible
DistributedLoads”,toappearinthe51stIEEEConferenceonDecision
andControl,December10-13,2012,Maui,Hawaii
[4] C.Su,andD.Kirschen,“QuantifyingtheEffectofDemandResponse
on Electricity Markets”, IEEE Trans. Power Syst., Vol. 24, No. 3,
pp.1199−1208,Aug.2009,doi:10.1109/TPWRS.2009.2023259
[5] M.H.Albadu,andE.F.El-Saadany,“DemandResponseinElectricity
Markets: An Overview”, Power and Energy Society General Meeting,
(b) NormoftheErrorVector
June24-28,2007,Tampa,Florida,doi:10.1109/PES.2007.385728
[6] D. S. Callaway, “Can smaller loads be profitably engaged in power
system services?”, Power and Energy Society General Meeting, July
24-28,2011,Detroit,Michigan,doi:10.1109/PES.2011.6039890
[7] N.Lu,andD.P.Chassin,“AState-queueingModelofThermostatically
Controlled Appliances”, IEEE Trans. Power Syst., Vol. 19, No. 3,
pp.1666−1673,Aug.2004,doi:10.1109/TPWRS.2004.831700
[8] D. P. Chassin, and J. C. Fuller, “On the Equilibrium Dynam-
ics of Demand Response in Thermostatic Loads”, the 44th Hawaii
International Conference on System Sciences, January 4-7, 2011,
doi:10.1109/HICSS.2011.326
[9] B. Zhang, and J. Baillieul, “A Packetized Direct Load Control Mech-
anism for Demand Side Management”, to appear in the 51st IEEE
Conference on Decision and Control, December 10-13, 2012, Maui,
Hawaii
[10] D. S. Callaway, “State Estimation and Control of Heterogeneous
Thermostatically Controlled Loads for Load Following”, the 45th
HawaiiInternationalConferenceonSystemScience,January4-7,2012,
(c) ControlDistributionwithDutyCycle10Minutes
doi:10.1109/HICSS.2012.545
[11] D.S.Callaway,“Tappingtheenergystoragepotentialinelectricloadsto
deliverloadfollowingandregulation,withapplicationtowindenergy”,
Energy Conversion and Management, Vol. 50, pp. 1389−1400, May
2009,doi:10.1016/J.ENCONMAN.2008.12.012
[12] D. S. Callaway, “Using load switches to control aggregated elec-
tricity demand for load following and regulation”, Power and En-
ergy Society General Meeting, July 24-28, 2011, Detroit, Michigan,
doi:10.1109/PES.2011.6039039
[13] H.K.Khalil,“Non-linearSystems”,PrenticeHall,UpperSaddleRiver,
NewJersey,2002
[14] “PJM Manual 12: Balancing Operations”, available online at
http://www.pjm.com/∼/media/documents/manuals/m12.ashx
[15] B.Friedland,“ControlSystemDesign:AnIntroductiontoState-Space
Methods”,DoverPublications,Mineola,NewYork,2005
[16] http://pjm.com/markets-and-operations/ancillary-services/mkt-based-
regulation.aspx
[17] J. Cavallo, and J. Mapp, ”Targeting Refrigerators for Repair or Re-
(d) ControlDistributionwithDutyCycle20
placement”, Proceedings of 2000 ACEEE Summer Study on Energy
Efficiency in Buildings, Wisconsin, Mar. 2000. On-line available at:
Fig. 7. Observer performance validation. (a) Real and estimated value of
http://www.ipd.anl.gov/anlpubs/2000/04/35529.pdf
twosamplestatesxNandx2N.(b).Thenormoftheestimationerrorgradually
approacheszeroastheestimationproceeds.Convergencerateoftheestimation
error vector depends on the duty cycle value. (c) and (d) the distribution of
control signal across the allowable set when different duty cycle values are
used
