Optimal Decentralized Protocol for Electric Vehicle Charging
Lingwen Gan Ufuk Topcu
Steven Low
Abstract— Motivated by the power-grid-side challenges in
the integration of electric vehicles, we propose a decentralized
protocol for negotiating day-ahead charging schedules for
electric vehicles. The overall goal is to shift the load due to
electric vehicles to fill the overnight electricity demand valley. In
each iteration of the proposed protocol, electric vehicles choose
their own charging profiles for the following day according
to the price profile broadcast by the utility, and the utility
updates the price profile to guide their behavior. This protocol
is guaranteed to converge, irrespective of the specifications
(e.g., maximum charging rate and deadline) of electric vehicles.
At convergence, the l
2
norm of the aggregated demand is
minimized, and the aggregated demand profile is as “flat” as it
can possibly be. The proposed protocol needs no coordination
among the electric vehicles, hence requires low communication
and computation capability. Simulation results demonstrate
convergence to optimal collections of charging profiles within
few iterations.
I. INTRODUCTION
Electric vehicles (EVs) offer great potential for increas-
ing energy efficiency, reducing greenhouse gas emissions,
and relieving reliance on foreign oil for transportation [1].
Currently, several types of EVs are either already in the
U.S. market, or about to enter [2], and electrification of
transportation is at the forefront of many research and
development agendas [3]. On the other hand, the potential
comes with a multitude of challenges including those in
the integration with the electric power grid. EV charging
increases the electric loads, and potentially amplifies the
peak demand or creates new peaks [4]. It also increases
the demand side uncertainties, and potentially reduces the
distribution circuit and transformer lifespan [5]. Moreover,
power losses and voltage variations become more likely [6].
The simulation-based study in [7] suggests that, if no
regulation on EV charging is implemented, even a 10% pene-
tration of EVs may cause unacceptable variations in the volt-
age profiles. On the other hand, many studies demonstrate
that adopting “smart” charging strategies can mitigate some
of the integration challenges, defer infrastructure investment
needed otherwise, and even stabilize the grid. For example,
scheduling EV charging so that aggregated EV load fills
the overnight demand valley may reduce daily cycling of
power plants and operational cost of the electricity utility
[8]. Furthermore, the energy stored in EVs may be utilized
as an alternative ancillary service resource [9] for regulating
L. Gan is with the Department of Electrical Engineering, California
Institute of Technology, e-mail: lgan@caltech.edu.
U. Topcu is with Control and Dynamical Systems, California Institute of
Technology, e-mail: utopcu@cds.caltech.edu.
S. Low is with the Department of Computer Science and the Depart-
ment of Electrical Engineering, California Institute of Technology, e-mail:
slow@caltech.edu.
voltage profiles, ride-through support for fault protection, and
even compensating fluctuating renewable energy generation
[10].
Studies on EV charging control fall into three cate-
gories: effect of time-of-use rates [11], coordinated charging
scheduling [6], [10], [12], and decentralized scheduling [13]
[14]. Reference [11] explores the effect of higher price
during peak hours on shifting EV load, but does not provide
strategies for setting the price. Reference [6], [10], [12]
study centralized control strategies that minimize power
losses, load variance, or maximize load factor, allowable
EV penetration level. These strategies require a centralized
structure to collect all information of all EVs and optimize
over the charging profiles of all EVs, hence incur prohibitive
communication and computation requirements. Reference
[13] demonstrates, through a simulation-based study, that EV
load can be smoothed by introducing load-side participation
into the power market. They propose a decentralized strategy,
but do not provide any analytical optimality guarantees. Ref-
erence [14] proposes another decentralized charging strategy,
and proves its optimality in the case where all EVs plug in
at the same time with the same state-of-charge (SOC), and
have the same deadline and allowable charging rates. For
future reference, we call this setting homogeneous.
In this paper, we partially alleviate some of the restrictions
imposed in [14]. First, we define optimal charging profiles of
EVs explicitly (this definition generalizes the implicit defi-
nition in [14]). Second, we propose a decentralized charging
strategy that guarantees optimality in both homogeneous and
non-homogeneous cases, where EVs can plug-in at different
times with different SOC, have different maximum charging
speed and deadlines. Third, we remove the artificial tracking
error penalty in EV owners’ objective in [14] by introducing
another penalty term that vanishes at convergence.
The rest of the paper is organized as follows. Section
II formulates EV charging control as a global optimization
problem. Section III explores properties of optimal charging
profiles, proposes a decentralized algorithm and proves its
convergence to optimal charging profiles. Numerical simu-
lations are used to illustrate these results in section IV, and
conclusions and future works are presented in section V.
II. PROBLEM FORMULATION
We consider a scenario where an electric utility negotiates
with N electric vehicles (indexed 1, . . . , N) for a day-ahead
charging schedule over T time slots (indexed 1, . . . , T ),
each of length ∆T . Let D(t) denote the base demand
(aggregated non-EV demand) at time t, r
n
(t) denote the
charging rate of EV n at time t, for all t ∈ {1, . . . , T }.
2011 50th IEEE Conference on Decision and Control and
European Control Conference (CDC-ECC)
Orlando, FL, USA, December 12-15, 2011
U.S. Government work not protected by U.S.
copyright
5798
TABLE I
LIST OF SYMBOLS
T time horizon
t time, t ∈ {1, . . . , T }
N number of EVs
n EV n, n ∈ {1, . . . , N}
D(t) base demand at time t.
D(·) base demand profile.
r
n
(t) charging rate of EV n at time t
r
n
(·) charging profile of EV n
r(·) charging profile of all EVs
R
r
(·) aggregated charging profile corresponding to r
r
n
(t) charging rate lower bound for EV n at time t
r
n
(t) charging rate upper bound for EV n at time t
R
n
charging rate sum of EV n
D
n
set of feasible charging profiles for EV n
D set of feasible charging profiles for all EVs
S set of optimal charging profiles
if a, b ∈ R
n
, a b ⇔ a
i
≤ b
i
for all i ∈ {1, . . . , n}.
Let r
n
:= (r
n
(1), . . . , r
n
(T )) denote the charging profile of
vehicle n, for all n ∈ {1, . . . , N }. Then, roughly speaking,
a charging profile r := (r
1
(·), . . . , r
N
(·)) is “valley-filling”,
if it minimizes
T
X
t=1
D(t) +
N
X
n=1
r
n
(t)
!
2
.
A more rigorous discussion on valley-filling will be given in
section III-A. The overall goal is to achieve a valley-filling
charging profile, subject to the following constraints.
For EV n, due to battery specification, its charging rate
should be within some interval [r
n
, r
n
]. Taking deadline into
account, we let r
n
and r
n
be time-dependent and set r
n
(t) =
r
n
(t) = 0 for t outside the allowable charging period of EV
n. Hence
r
n
(t) ≤ r
n
(t) ≤ r
n
(t)
for all n ∈ {1, . . . , N} and all t ∈ {1, . . . , T }. Let
B
n
, s
n
(0), and η
n
denote the battery capacity, initial SOC
and charging efficiency of EV n. By its deadline, EV n
should be fully charged, this is captured by the total energy
stored over the whole time horizon η
n
P
T
t=1
r
n
(t)∆T =
B
n
(1 − s
n
(0)). Define R
n
:= B
n
(1 − s
n
(0)) /η
n
/∆T , it
is equivalent to
T
X
t=1
r
n
(t) = R
n
.
for all n ∈ {1, . . . , N }. Reference [15] summarizes some of
the recently announced EV models, and typical values of r
n
and R
n
can be derived from these models. r
n
is usually set
to be 0.
Definition 1: Let D ∈ R
T
be the base demand. A charging
profile r = (r
1
, . . . , r
N
), where r
n
∈ R
T
for all n ∈
{1, . . . , N}, is optimal, or valley-filling, if it solves
minimize
r
1
,...,r
N
T
X
t=1
D(t) +
N
X
n=1
r
n
(t)
!
2
(1)
subject to r
n
r
n
r
n
, n = 1, . . . , N,
T
X
t=1
r
n
(t) = R
n
, n = 1, . . . , N,
where r
n
, r
n
∈ R
T
and R
n
∈ R for all n ∈ {1, . . . , N}.
Definition 2: A charging profile is feasible if it satisfies
the constraints in (1).
III. MAIN RESULTS
In this section, we first explore properties of optimal
charging profiles, and then propose a decentralized algorithm
for solving (1). In the end, we prove that the algorithm we
propose converges to the set of optimal charging profiles.
Given a charging profile r := (r
1
, . . . , r
N
), let
R
r
:=
N
X
n=1
r
n
denote the aggregated charging profile corresponding to r.
A. Optimal Charging Profiles
Property 1: Let r be a feasible charging pro-
file. If there exists A ∈ R, such that R
r
(t) =
max
n
P
N
n=1
r
n
(t), A − D(t)
o
for all t ∈ {1, . . . , T }, then
r is optimal.
Proof: Note that R
r
is the unique solution to the problem
minimize
R
T
X
t=1
(D(t) + R(t))
2
(2)
subject to
T
X
t=1
R(t) =
N
X
n=1
R
n
R
N
X
n=1
r
n
.
For any r
0
feasible for (1), R
r
0
is feasible for (2). Further-
more, the objective function in (2) evaluated at R
r
0
is equal
to the objective function in (1) evaluated at r
0
. Hence, the
optimal value d
∗
of (2) is a lower bound for the optimal
value p
∗
of (1). The aggregated profile R
r
attains d
∗
, so r
attains d
∗
≤ p
∗
. Since r is feasible, it is optimal.
Let D
n
:=
n
r
n
|r
n
r
n
r
n
,
P
T
t=1
r
n
(t) = R
n
o
denote
the set of feasible charging profiles of EV n. Then D :=
D
1
× · · · × D
N
is the set of feasible charging profiles of all
EVs.
Property 2: If the set D of feasible charging profiles is
non-empty, then optimal charging profiles exist.
Proof: D is the feasible set of (1) by Definition 2. Since
D
n
is compact for each n, D as a product of N compact
sets is also compact. Furthermore, D is non-empty, and the
objective function of (1) is continuous. Hence, its optimal
value is attained at some r ∈ D.
5799
20:00 0:00 4:00 8:00 12:00 16:00
0.4
0.5
0.6
0.7
0.8
0.9
1
time of day
normalized aggregated demand (kW)
base demand
flat valley−filling
non−valley−filling
20:00 0:00 4:00 8:00 12:00 16:00
0.4
0.5
0.6
0.7
0.8
0.9
1
time of day
normalized aggregated demand (kW)
base demand
non−flat valley−filling
non−valley−filling
Fig. 1. Base demand curve is the average residential load in the service area
of SCE from 20:00 on Feb. 13th to 19:00 on Feb 14th, 2011 [16]. Valley-
filling curve corresponds to the outcome of Algorithm ODC, and can be
flat or non-flat. A hypothetical non-valley-filling curve is shown with the
marker o.
Property 1 is directly related to our intuitive notion of
valley-filling. As shown in Figure 1 (top), a valley-filling
charging profile has the same aggregated demand during
periods in which some EVs charge (0:00-17:00). And the
aggregated demand in this period is no higher than that
outside this period. We call such aggregated demand profile
completely flat. However, under certain conditions, complete
flatness may not be possible. For example, the “valley” may
be so deep at time t that even if every vehicle charges at its
maximum rate, the valley may still not be completely filled,
e.g., at 4:00 in Figure 1 (bottom). Moreover, vehicles may
have such stringent deadline constraints that the flexibility
of scheduling is limited, and complete flatness cannot be
achieved. Definition 1 relaxes these restrictions as a result
of Property 2. In cases where complete flatness is possible,
Property 1 guarantees that our constructive definition of
valley-filling agrees with complete flatness. In cases where
complete flatness is not possible, Property 2 guarantees
existence of an optimal charging profile. Furthermore, as
shown in Figure 1 (bottom), a valley-filling charging profile
indeed gives a “smoother” aggregated demand, compared to
non-valley-filling charging profiles.
We now establish an equivalence relation between charg-
ing profiles, and use it to describe an important property of
the set of optimal charging profiles.
Definition 3: Feasible charging profiles r and r
0
are
equivalent, denoted as r ∼ r
0
, if R
r
= R
r
0
.
That is, r and r
0
are equivalent if their aggregated charging
profiles are the same. It is easy to check that this is indeed an
equivalence relation. Now, we can define equivalence classes
based on this equivalence relation. Given base demand D,
define
S := {r ∈ D|r optimal w.r.t D}
as the set of optimal charging profiles.
Theorem 1: If feasible charging profiles exist, then the
set S of optimal charging profiles is non-empty, compact,
convex, and an equivalence class.
Proof: Property 2 implies that S is nonempty when
feasible charging profiles exist. Let r be an optimal charging
profile, then r ∈ S. Define S
0
:= {r
0
∈ D|r
0
∼ r}. It is easy
to show that S
0
is closed and convex. Since S
0
⊆ D, which
is compact, S
0
is also compact. Hence S
0
is non-empty,
compact, convex, and an equivalence class (by construction).
It is easy to see that S
0
⊆ S. In order to prove S = S
0
,
we now show that S ⊆ S
0
. For all r
0
∈ S, from the first
order optimality condition for (1),
hD + R
r
, R
r
0
− R
r
i = 0,
hD + R
r
0
, R
r
− R
r
0
i = 0.
Reverse R
r
0
−R
r
in the first equation and subtract the second
one to obtain hR
r
− R
r
0
, R
r
− R
r
0
i = 0. Hence R
r
= R
r
0
,
r
0
∼ r, r
0
∈ S
0
, S ⊆ S
0
. Consequently, S = S
0
is non-empty,
compact, convex, and an equivalence class.
Corollary 1: Optimal charging profile is generally not
unique.
Corollary 1 is a direct consequence of the fact that S is
an equivalence class. In general, two EVs can exchange their
charging profiles in the opposite direction without affecting
the objective value. For instance, EV 1 raises its charging
rate at time t
1
by ∆r and decreases its charging rate at t
2
by ∆r, while not violating the constraints in (1). Conversely,
EV 2 decreases its charging rate at time t
1
by ∆r and
increases its charging rate at t
2
by ∆r, while not violating the
constraints in (1). If the original charging profile is optimal,
the new profile, which is equivalent to the original one, is
also optimal, due to Theorem 1.
B. Algorithm ODC
We develop a decentralized algorithm to solve (1). System
diagram of the algorithm is shown in Figure 2. Given broad-
cast electricity price profile, each EV makes an independent
decision and chooses its charging profile. The utility guides
their behavior by setting the prices.
Algorithm ODC (optimal decentralized charging): Given
T , N , K, D(t), R
n
, r
n
(t), r
n
(t), for all n ∈ {1, . . . , N}
and all t ∈ {1, . . . , T }, define α := N/2.
5800
TABLE II
LIST OF SYMBOLS IN ALGORITHM
p
k
(·) price profile in iteration k
r
k
n
(·) charging profile of vehicle n in iteration k
K maximum number of iterations
Fig. 2. Schematic view of the information flows in Algorithm ODC. Given
predicted price profile, vehicles make independent decisions and choose
their own optimal charging profiles. The utility guides their behavior by
setting the price based on the aggregated demand.
1) Initialization:
p
0
(·) := D(·), r
0
n
(·) := 0, k ← 0.
2) Utility broadcasts p
k
(·).
3) EV n calculates a new charging profile r
k+1
n
(·) as
1
argmin
r
n
(·)
T
X
t=1
p
k
(t)r
n
(t) + α
r
n
(t) − r
k
n
(t)
2
subject to r
n
(·) r
n
(·) r
n
(·),
T
X
t=1
r
n
(t) = R
n
.
4) Utility collects new charging profiles of all EVs, and
updates the price according to
p
k+1
(·) := D(·) +
N
X
n=1
r
k+1
n
(·). (3)
5) k ← k + 1, go to (2) until k = K or convergence.
In iteration k + 1, the algorithm can be split into two
parts. In the first part, EV n chooses a charging profile that
minimizes its objective function
C
n
(r
n
) =
T
X
t=1
p
k
(t)r
n
(t) + α
r
n
(t) − r
k
n
(t)
2
. (4)
The first term is the electricity cost and the second term
penalizes deviations from the charging profile computed
in the previous iteration. This extra penalty term ensures
convergence of Algorithm ODC. In section III-C, we prove
1
Scaling factors used to cancel the units are omitted for brevity.
that r
k+1
n
(t) − r
k
n
(t) → 0 for all n and all t as k → ∞.
Hence, the objective function of each EV boils down to its
electricity cost at convergence.
In the second part of iteration k +1, the utility updates the
price according to (3). It sets high prices for the periods with
high aggregated demand. Intuitively, in the next iteration, ve-
hicles are given the incentive to shift their charging profiles.
As a result, aggregated demand may be smoothened.
As proved in section III-C, Algorithm ODC converges
to optimal charging profiles. Furthermore, simulation results
demonstrate fast convergence (usually within a few itera-
tions) of Algorithm ODC.
C. Analysis of Algorithm ODC
We now analyze the convergence properties of Algorithm
ODC. Since optimal charging profile is generally not unique,
we need to establish an appropriate notion of convergence
to optimal charging profiles. Let superscript k denote the
respective value of in iteration k of Algorithm ODC. For
example, r
k
n
denotes the charging profile of vehicle n in it-
eration k. Similarly, R
k
:=
P
N
n=1
r
k
n
denotes the aggregated
charging profile in iteration k, d
k
:= (D + R
k
)/N denotes
the normalized aggregated demand profile in iteration k.
Lemma 1: If feasible charging profiles exist, then the
inequality
d
k
+ r
k+1
n
− r
k
n
2
2
≤
d
k
2
2
−
r
k+1
n
− r
k
n
2
2
(5)
holds for all n ∈ {1, . . . , N } and all k ≥ 1.
Proof: Since feasible charging profiles exist, feasible
charging profiles for EV n exist, and r
k
n
∈ D
n
for all
n ∈ {1, . . . , N} and all k ≥ 1. Further,
r
k+1
n
= argmin
r
n
∈D
n
T
X
t=1
p
k
(t)r
n
(t) + α
r
n
(t) − r
k
n
(t)
2
= argmin
r
n
∈D
n
T
X
t=1
2d
k
(t)r
n
(t) +
r
n
(t) − r
k
n
(t)
2
= argmin
r
n
∈D
n
T
X
t=1
d
k
(t) + r
n
(t) − r
k
n
(t)
2
= argmin
r
n
∈D
n
d
k
+ r
n
− r
k
n
2
2
. (6)
From the first order optimality condition for (6),
hd
k
+ r
k+1
n
− r
k
n
, r
n
− r
k+1
n
i ≥ 0 (7)
for all r
n
∈ D
n
. Choose r
n
= r
k
n
to obtain
hd
k
+ r
k+1
n
− r
k
n
, r
k
n
− r
k+1
n
i ≥ 0.
Split the first term to obtain
hd
k
, r
k+1
n
− r
k
n
i ≤ −
r
k+1
n
− r
k
n
2
2
.
Hence,
d
k
+ r
k+1
n
− r
k
n
2
2
=
d
k
2
2
+
r
k+1
n
− r
k
n
2
2
+ 2hd
k
, r
k+1
n
− r
k
n
i
≤
d
k
2
2
−
r
k+1
n
− r
k
n
2
2
5801
for all n ∈ {1, . . . , N} and all k ≥ 1.
Lemma 2: If feasible charging profiles exist, then for all
n ∈ {1, . . . , N} and all k ≥ 1, r
k+1
n
= r
k
n
if and only if
hd
k
, r
n
− r
k
n
i ≥ 0 (8)
for all r
n
∈ D
n
.
Proof: Follows from (7) and the strict convexity of (6).
Theorem 2: Let r
k
:= (r
k
1
, . . . , r
k
N
) be the charging profile
in iteration k of Algorithm ODC. If feasible charging profiles
exist, then r
k
→ S as k → ∞.
Proof: The proof is based on a Lyapunov-type argument.
Define L(r) :=
P
T
t=1
(D(t) + R
r
(t))
2
, then
L(r
k+1
) =
T
X
t=1
D(t) + R
k+1
(t)
2
= N
2
T
X
t=1
d
k
(t) +
1
N
N
X
n=1
r
k+1
n
(t) − r
k
n
(t)
!
2
≤ N
2
T
X
t=1
1
N
N
X
n=1
d
k
(t) + r
k+1
n
(t) − r
k
n
(t)
2
!
= N
N
X
n=1
d
k
+ r
k+1
n
− r
k
n
2
2
≤ N
N
X
n=1
d
k
2
2
−
r
k+1
n
− r
k
n
2
2
(9)
≤ N
2
d
k
2
2
= L(r
k
). (10)
The first inequality is due to Jessen’s inequality, and the
second inequality is due to Lemma 1. It is easy to show that
L(r
k+1
) = L(r
k
) if and only if r
k+1
n
= r
k
n
for all n, i.e.,
r
k+1
= r
k
.
If r
k+1
= r
k
, it follows from Lemma 2 that for all n,
hd
k
, r
n
− r
k
n
i ≥ 0 for all r
n
∈ D
n
. Hence,
hD + R
k
, R
r
0
− R
k
i = N
N
X
n=1
hd
k
, r
0
n
− r
k
n
i ≥ 0 (11)
for all r
0
= (r
0
1
, . . . , r
0
N
) ∈ D. This is the first order
optimality condition for (1), hence r
k
∈ S. On the other
hand, if r
k
∈ S, L(r
k
) ≤ L(r
k+1
), hence L(r
k+1
) = L(r
k
).
It follows that L(r
k+1
) = L(r
k
) ⇔ r
k+1
= r
k
⇔ r
k
∈ S.
Now we have D is compact; S minimizes L(D); L(r
k+1
) <
L(r
k
) if r
k
/∈ S. It follows that r
k
→ S as k → ∞.
Corollary 2: A charging profile r is an equilibrium point
of Algorithm ODC if and only if r ∈ S, the set of optimal
charging profiles.
Proof: Since r
k+1
= r
k
⇔ r
k
∈ S (showed in the proof of
Theorem 2), the set of equilibrium points E := {r
k
|r
k+1
=
r
k
} = S.
Theorem 3: Let r
∗
be an optimal charging profile, p
k
be
the price profile in iteration k of Algorithm ODC. If feasible
charging profiles exist, then
• aggregated charging profile converges to that of r
∗
, i.e.,
lim
k→∞
R
k
= R
r
∗
;
• price profile converges to that of r
∗
, i.e.,
lim
k→∞
p
k
= D + R
r
∗
;
• for each EV, the change in charging profile between
consecutive iterations of Algorithm ODC converges to
0, i.e.,
lim
k→∞
r
k+1
n
− r
k
n
2
= 0
for all n ∈ {1, . . . , N}.
Proof: Since the set S of optimal charging profiles is an
equivalence class, for all r
0
∈ S, R
r
0
= R
r
. Let r
k
be the
charging profile in iteration k. Theorem 2 implies that we
can find a sequence in S, such that the distance between
r
k
and the k
th
element of the sequence converges to 0 as
k → ∞. It follows that R
k
→ R
r
, as k → ∞. The price
profile converges as a direct consequence of (3). Inequality
(9) implies
L(r
k
) − L(r
k+1
) ≥ N
N
X
n=1
r
k+1
n
− r
k
n
2
2
.
Since L(r
k
) converges, L(r
k
) − L(r
k+1
) → 0. Hence
||r
k+1
n
− r
k
n
||
2
→ 0, for all n, as k → ∞. .
Theorem 3 implies that in Algorithm ODC, the aggregated
charging profile and price profile converge to that of an
optimal charging profile. Furthermore, change in the charging
profile for each EV between consecutive iterations vanishes
as k → ∞. Hence, the objective function (4) reduces to the
electricity cost after certain number of iterations.
Theorem 4: Algorithm ODC is a gradient projection
algorithm for minimizing L(r). That is,
r
k+1
n
= argmin
r
n
∈D
n
r
n
−
r
k
n
− γ
∂L
∂r
n
2
2
for all n ∈ {1, . . . , N} and all k ≥ 0, where γ = 1/(2N ).
Proof: Since
γ
∂L
∂r
n
(r
k
) = 2γ
D + R
k
= d
k
,
Theorem 4 directly follows from (6).
IV. SIMULATIONS
We compare the convergence properties and optimality of
Algorithm ODC with the decentralized scheduling algorithm
proposed in [14], in homogeneous and non-homogeneous
cases. For notational brevity, we call the algorithm in [14]
DAP, standing for “Deviation from Average Penalty”. Re-
call that the optimality of DAP is only guaranteed in the
homogeneous case. We choose the average residential load
profile in the service area of South California Edison from
20:00 on Feb. 13th 2011 to 19:00 on Feb. 14th 2011 as the
normalized base demand profile. According to the typical
charging characteristics of EVs in [15], we set r
n
(t) =
3.3 kW if EV n can be charged at time t, and 0 kW
otherwise. We assume charging rate can change continuously
in [0, r
n
(t)], hence choose r
n
(t) = 0 for all t. We set
R
n
= 10, 25, 40 for three different types (sedan, compact,
5802
![Fig. 1. Base demand curve is the average residential load in the service area of SCE from 20:00 on Feb. 13th to 19:00 on Feb 14th, 2011 [16]. Valleyfilling curve corresponds to the outcome of Algorithm ODC, and can be flat or non-flat. A hypothetical non-valley-filling curve is shown with the marker o.](/figures/fig-1-base-demand-curve-is-the-average-residential-load-in-a5rwklwh.webp)
