Kyeong Soo Kim, "Atomic scheduling of appliance energy consumption in residential smart grid," Invited talk, CNU International Workshop on Industrial Mathematics, Chungnam National University (CNU), Daejeon, Korea, Oct. 7, 2016.

1. Atomic Scheduling of Appliance
Energy Consumption in Residential
Smart Grid
Kyeong Soo (Joseph) Kim
(With S. Lee, T. O. Ting@XJTLU and X.-S. Yang@Middlesex)
Department of Electrical and Electronic Engineering
Xi’an Jiaotong-Liverpool University
International workshop on Industrial Mathematics
Center for Industrial Mathematics Initiative
Chungnam National University
6-8 October 2016
1 / 66

3. Outline
Introduction
Review of Current Formulation of Appliance Energy
Consumption Scheduling
Atomic Scheduling of Appliance Energy Consumption
Starting-Time-Based Formulation
Optimal-Routing-Based Formulation
Successive Convex Relaxation
Examples
Numerical Results
Experimental Parameters
Comparison of Bounds with True Optimal Values
Cost Minimization
PAR Minimization
Next Steps: Taking into Account Local Energy Storage
and Generation
Conclusions
3 / 66

8. Next . . .
Introduction
Review of Current Formulation of Appliance Energy
Consumption Scheduling
Atomic Scheduling of Appliance Energy Consumption
Starting-Time-Based Formulation
Optimal-Routing-Based Formulation
Successive Convex Relaxation
Examples
Numerical Results
Experimental Parameters
Comparison of Bounds with True Optimal Values
Cost Minimization
PAR Minimization
Next Steps: Taking into Account Local Energy Storage
and Generation
Conclusions
8 / 66

9. Autonomous Demand-Side Management in
Smart Grid
Gateway
(Traditional)
Electricity
Grid
Greenfield
Power
Line
Bi-Directional
Communication
Links
Power Plant
9 / 66

10. Scheduling of Appliance Energy
Consumption
I A key to autonomous DSM in
optimizing energy production and
consumption.
I Based on two-way digital
communications between a utility
company and users through smart
meters at users’ premises.
I Typical objectives
I Peak-to-average ratio (PAR)
I Total energy cost
10 / 66

11. Scheduling of Appliance Energy
Consumption
I A key to autonomous DSM in
optimizing energy production and
consumption.
I Based on two-way digital
communications between a utility
company and users through smart
meters at users’ premises.
I Typical objectives
I Peak-to-average ratio (PAR)
I Total energy cost
10 / 66

12. Scheduling of Appliance Energy
Consumption
I A key to autonomous DSM in
optimizing energy production and
consumption.
I Based on two-way digital
communications between a utility
company and users through smart
meters at users’ premises.
I Typical objectives
I Peak-to-average ratio (PAR)
I Total energy cost
10 / 66

13. Scheduling of Appliance Energy
Consumption
I A key to autonomous DSM in
optimizing energy production and
consumption.
I Based on two-way digital
communications between a utility
company and users through smart
meters at users’ premises.
I Typical objectives
I Peak-to-average ratio (PAR)
I Total energy cost
10 / 66

14. A Question on Scheduled Energy
Consumption
I Can a washing machine
successfully complete its job with
the energy consumption scheduled
as follows?
11 / 66

15. A Question on Scheduled Energy
Consumption
I Can a washing machine
successfully complete its job with
the energy consumption scheduled
as follows?
9am 3pm
12am
11 / 66

16. A Question on Scheduled Energy
Consumption
I Can a washing machine
successfully complete its job with
the energy consumption scheduled
as follows?
9am 3pm
12am
or
11 / 66

17. A Question on Scheduled Energy
Consumption
I Can a washing machine
successfully complete its job with
the energy consumption scheduled
as follows?
9am 3pm
12am
or
9am 3pm
2pm
10am
11 / 66

18. Appliance Wattprint1
I Each appliance has its own unique energy consumption pattern.
I Some appliances require atomic — i.e., uninterruptible and
unthrottleable — patterns for their operations.
1
http://www.jmp.com/en_us/success/plotwatt.html
12 / 66

19. Next . . .
Introduction
Review of Current Formulation of Appliance Energy
Consumption Scheduling
Atomic Scheduling of Appliance Energy Consumption
Starting-Time-Based Formulation
Optimal-Routing-Based Formulation
Successive Convex Relaxation
Examples
Numerical Results
Experimental Parameters
Comparison of Bounds with True Optimal Values
Cost Minimization
PAR Minimization
Next Steps: Taking into Account Local Energy Storage
and Generation
Conclusions
13 / 66

20. Optimization Variables
I Based on energy consumption over equally-divided time
slots of a day (typically hourly time slots) as
optimization variables, i.e.,
xn ,
h
x0
n, . . . , xh
n, . . . , xH−1
n
i
for
I User n ∈ N , {1, . . ., N};
I Time slot h ∈ H , {0, . . ., H−1}.
I Because the optimization variables take continuous
values, we can easily apply
I Convex optimization [1];
I Distributed algorithms through concave n-person
games [2].
14 / 66

21. Optimization Variables
I Based on energy consumption over equally-divided time
slots of a day (typically hourly time slots) as
optimization variables, i.e.,
xn ,
h
x0
n, . . . , xh
n, . . . , xH−1
n
i
for
I User n ∈ N , {1, . . ., N};
I Time slot h ∈ H , {0, . . ., H−1}.
I Because the optimization variables take continuous
values, we can easily apply
I Convex optimization [1];
I Distributed algorithms through concave n-person
games [2].
14 / 66

22. Optimization Variables
I Based on energy consumption over equally-divided time
slots of a day (typically hourly time slots) as
optimization variables, i.e.,
xn ,
h
x0
n, . . . , xh
n, . . . , xH−1
n
i
for
I User n ∈ N , {1, . . ., N};
I Time slot h ∈ H , {0, . . ., H−1}.
I Because the optimization variables take continuous
values, we can easily apply
I Convex optimization [1];
I Distributed algorithms through concave n-person
games [2].
14 / 66

23. Optimization Variables
I Based on energy consumption over equally-divided time
slots of a day (typically hourly time slots) as
optimization variables, i.e.,
xn ,
h
x0
n, . . . , xh
n, . . . , xH−1
n
i
for
I User n ∈ N , {1, . . ., N};
I Time slot h ∈ H , {0, . . ., H−1}.
I Because the optimization variables take continuous
values, we can easily apply
I Convex optimization [1];
I Distributed algorithms through concave n-person
games [2].
14 / 66

24. Optimization Variables
I Based on energy consumption over equally-divided time
slots of a day (typically hourly time slots) as
optimization variables, i.e.,
xn ,
h
x0
n, . . . , xh
n, . . . , xH−1
n
i
for
I User n ∈ N , {1, . . ., N};
I Time slot h ∈ H , {0, . . ., H−1}.
I Because the optimization variables take continuous
values, we can easily apply
I Convex optimization [1];
I Distributed algorithms through concave n-person
games [2].
14 / 66

25. Optimization Variables
I Based on energy consumption over equally-divided time
slots of a day (typically hourly time slots) as
optimization variables, i.e.,
xn ,
h
x0
n, . . . , xh
n, . . . , xH−1
n
i
for
I User n ∈ N , {1, . . ., N};
I Time slot h ∈ H , {0, . . ., H−1}.
I Because the optimization variables take continuous
values, we can easily apply
I Convex optimization [1];
I Distributed algorithms through concave n-person
games [2].
14 / 66

26. Feasible Set
A feasible energy consumption scheduling set for user n
Xn=
(
xn

32. X
h∈Hn
xh
n=En, γmin
n ≤xh
n≤γmax
n , ∀h∈Hn, xh
n=0, ∀h∈HHn
)
where
I γmin
n : Minimum energy level;
I γmax
n : Maximum energy level;
I En: Total daily energy consumption;
I Hn: Scheduling interval defined as follows:
Hn ,
n
h

35. h = i mod H, ∀i∈
αn, βn
o
with αn∈[0, H−1], βn∈[1, 2H−2], and 1≤βn−αn≤H−1.
15 / 66

36. Feasible Set
A feasible energy consumption scheduling set for user n
Xn=
(
xn

42. X
h∈Hn
xh
n=En, γmin
n ≤xh
n≤γmax
n , ∀h∈Hn, xh
n=0, ∀h∈HHn
)
where
I γmin
n : Minimum energy level;
I γmax
n : Maximum energy level;
I En: Total daily energy consumption;
I Hn: Scheduling interval defined as follows:
Hn ,
n
h

45. h = i mod H, ∀i∈
αn, βn
o
with αn∈[0, H−1], βn∈[1, 2H−2], and 1≤βn−αn≤H−1.
15 / 66

46. Feasible Set
A feasible energy consumption scheduling set for user n
Xn=
(
xn

52. X
h∈Hn
xh
n=En, γmin
n ≤xh
n≤γmax
n , ∀h∈Hn, xh
n=0, ∀h∈HHn
)
where
I γmin
n : Minimum energy level;
I γmax
n : Maximum energy level;
I En: Total daily energy consumption;
I Hn: Scheduling interval defined as follows:
Hn ,
n
h

55. h = i mod H, ∀i∈
αn, βn
o
with αn∈[0, H−1], βn∈[1, 2H−2], and 1≤βn−αn≤H−1.
15 / 66

56. Feasible Set
A feasible energy consumption scheduling set for user n
Xn=
(
xn

62. X
h∈Hn
xh
n=En, γmin
n ≤xh
n≤γmax
n , ∀h∈Hn, xh
n=0, ∀h∈HHn
)
where
I γmin
n : Minimum energy level;
I γmax
n : Maximum energy level;
I En: Total daily energy consumption;
I Hn: Scheduling interval defined as follows:
Hn ,
n
h

65. h = i mod H, ∀i∈
αn, βn
o
with αn∈[0, H−1], βn∈[1, 2H−2], and 1≤βn−αn≤H−1.
15 / 66

66. Feasible Set
A feasible energy consumption scheduling set for user n
Xn=
(
xn

72. X
h∈Hn
xh
n=En, γmin
n ≤xh
n≤γmax
n , ∀h∈Hn, xh
n=0, ∀h∈HHn
)
where
I γmin
n : Minimum energy level;
I γmax
n : Maximum energy level;
I En: Total daily energy consumption;
I Hn: Scheduling interval defined as follows:
Hn ,
n
h

75. h = i mod H, ∀i∈
αn, βn
o
with αn∈[0, H−1], βn∈[1, 2H−2], and 1≤βn−αn≤H−1.
15 / 66

76. Feasible Set
A feasible energy consumption scheduling set for user n
Xn=
(
xn

82. X
h∈Hn
xh
n=En, γmin
n ≤xh
n≤γmax
n , ∀h∈Hn, xh
n=0, ∀h∈HHn
)
where
I γmin
n : Minimum energy level;
I γmax
n : Maximum energy level;
I En: Total daily energy consumption;
I Hn: Scheduling interval defined as follows:
Hn ,
n
h

85. h = i mod H, ∀i∈
αn, βn
o
with αn∈[0, H−1], βn∈[1, 2H−2], and 1≤βn−αn≤H−1.
15 / 66

86. Optimal Scheduling
The optimal scheduling is formulated as an optimization
problem for a given objective function φ(·) (e.g., total
energy cost or PAR) as follows:
minimize
xn∈Xn, ∀n∈N
φ (L(x))
where
I x , [x1, . . . , xN]: A vector of user energy consumption
vectors;
I L(x) , [L0 (x) , . . . , LH−1 (x)]: A vector of aggregate
loads across all users at each time slot, which are
defined as
Lh (x) ,
X
n∈N
xh
n.
16 / 66

87. Optimal Scheduling
The optimal scheduling is formulated as an optimization
problem for a given objective function φ(·) (e.g., total
energy cost or PAR) as follows:
minimize
xn∈Xn, ∀n∈N
φ (L(x))
where
I x , [x1, . . . , xN]: A vector of user energy consumption
vectors;
I L(x) , [L0 (x) , . . . , LH−1 (x)]: A vector of aggregate
loads across all users at each time slot, which are
defined as
Lh (x) ,
X
n∈N
xh
n.
16 / 66

88. Optimal Scheduling
The optimal scheduling is formulated as an optimization
problem for a given objective function φ(·) (e.g., total
energy cost or PAR) as follows:
minimize
xn∈Xn, ∀n∈N
φ (L(x))
where
I x , [x1, . . . , xN]: A vector of user energy consumption
vectors;
I L(x) , [L0 (x) , . . . , LH−1 (x)]: A vector of aggregate
loads across all users at each time slot, which are
defined as
Lh (x) ,
X
n∈N
xh
n.
16 / 66

89. Optimal Scheduling
The optimal scheduling is formulated as an optimization
problem for a given objective function φ(·) (e.g., total
energy cost or PAR) as follows:
minimize
xn∈Xn, ∀n∈N
φ (L(x))
where
I x , [x1, . . . , xN]: A vector of user energy consumption
vectors;
I L(x) , [L0 (x) , . . . , LH−1 (x)]: A vector of aggregate
loads across all users at each time slot, which are
defined as
Lh (x) ,
X
n∈N
xh
n.
16 / 66

90. Objective Functions
I For energy cost minimization:
φ (L(x)) =
X
h∈H
Ch (Lh(x))
where
I Ch(·): A cost function for generating or distributing
electricity energy at a time slot h.
I For PAR minimization:
φ (L(x)) =
H max
h∈H
Lh(x)
X
n∈N
En
17 / 66

91. Objective Functions
I For energy cost minimization:
φ (L(x)) =
X
h∈H
Ch (Lh(x))
where
I Ch(·): A cost function for generating or distributing
electricity energy at a time slot h.
I For PAR minimization:
φ (L(x)) =
H max
h∈H
Lh(x)
X
n∈N
En
17 / 66

92. Atomic vs. Non-Atomic Scheduling
αn βn
αn βn
Gap Gap
γmin
n
γmax
n
(a)
γop
n (·)
γmin
n
γmax
n
(b)
Examples of (a) non-atomic and (b) atomic scheduling.
I γmin
n : Minimum energy level
I γmax
n : Maximum energy level
I γ
op
n (·): Operating energy level
18 / 66

93. Scheduling Example
19 / 66

94. Scheduling Example: Case 1
20 / 66

95. Scheduling Example: Case 2
Appliance 1
Appliance 2
Scheduled
Consumption
21 / 66

96. Scheduling Example: Case 3
22 / 66

97. Next . . .
Introduction
Review of Current Formulation of Appliance Energy
Consumption Scheduling
Atomic Scheduling of Appliance Energy Consumption
Starting-Time-Based Formulation
Optimal-Routing-Based Formulation
Successive Convex Relaxation
Examples
Numerical Results
Experimental Parameters
Comparison of Bounds with True Optimal Values
Cost Minimization
PAR Minimization
Next Steps: Taking into Account Local Energy Storage
and Generation
Conclusions
23 / 66

98. Generalized Optimal Scheduling
We now generalize the optimal scheduling problem to
include atomic tasks:
minimize
xn∈Xn,yn∈Yn,∀n∈N
φ L(x) + L(y)
where
I xn: Non-atomic energy consumption scheduling
vector of user n;
I Xn: Feasible set for xn;
I yn: Atomic energy consumption scheduling vector of
user n;
I Yn: Feasible set for yn.
24 / 66

99. Overview of Atomic Scheduling Problem
Formulation Solution
Starting-Time-Based
Formulation
Optimal-Routing-Based
Formulation
Convex Relaxation
Successive Convex
Relaxation with
Fractional-Value
Dropping
Convex
Optimization
(Feasible Upper Bound)
Combinatorial
Optimization
Boolean-Convex
Optmization
Convex
Optmization
(Lower Bound)
25 / 66

100. Next . . .
Introduction
Review of Current Formulation of Appliance Energy
Consumption Scheduling
Atomic Scheduling of Appliance Energy Consumption
Starting-Time-Based Formulation
Optimal-Routing-Based Formulation
Successive Convex Relaxation
Examples
Numerical Results
Experimental Parameters
Comparison of Bounds with True Optimal Values
Cost Minimization
PAR Minimization
Next Steps: Taking into Account Local Energy Storage
and Generation
Conclusions
26 / 66

101. Starting-Time-Based Formulation
I Optimization variables:
s , [s1, . . . , sN]
I A feasible set for user n:
Sn ,
n
sn

104. sn = i mod H, ∀i∈[αn, βn−δn+1]
o
I Aggregate load across all users at each time slot h:
Lh(s) ,
X
n∈N
γ
op
n ((h − sn) mod H) IRn(sn)(h)
where
I IRn(sn)(h): An indicator function for a set Rn(sn).
I Rn(sn): A range of user n’s appliance operation for sn
defined as follows:
Rn(sn) ,
n
h

107. h = i mod H, ∀i∈ [sn, sn + δn − 1]
o
27 / 66

108. Starting-Time-Based Formulation
I Optimization variables:
s , [s1, . . . , sN]
I A feasible set for user n:
Sn ,
n
sn

111. sn = i mod H, ∀i∈[αn, βn−δn+1]
o
I Aggregate load across all users at each time slot h:
Lh(s) ,
X
n∈N
γ
op
n ((h − sn) mod H) IRn(sn)(h)
where
I IRn(sn)(h): An indicator function for a set Rn(sn).
I Rn(sn): A range of user n’s appliance operation for sn
defined as follows:
Rn(sn) ,
n
h

114. h = i mod H, ∀i∈ [sn, sn + δn − 1]
o
27 / 66

115. Starting-Time-Based Formulation
I Optimization variables:
s , [s1, . . . , sN]
I A feasible set for user n:
Sn ,
n
sn

118. sn = i mod H, ∀i∈[αn, βn−δn+1]
o
I Aggregate load across all users at each time slot h:
Lh(s) ,
X
n∈N
γ
op
n ((h − sn) mod H) IRn(sn)(h)
where
I IRn(sn)(h): An indicator function for a set Rn(sn).
I Rn(sn): A range of user n’s appliance operation for sn
defined as follows:
Rn(sn) ,
n
h

121. h = i mod H, ∀i∈ [sn, sn + δn − 1]
o
27 / 66

122. Starting-Time-Based Formulation
I Optimization variables:
s , [s1, . . . , sN]
I A feasible set for user n:
Sn ,
n
sn

125. sn = i mod H, ∀i∈[αn, βn−δn+1]
o
I Aggregate load across all users at each time slot h:
Lh(s) ,
X
n∈N
γ
op
n ((h − sn) mod H) IRn(sn)(h)
where
I IRn(sn)(h): An indicator function for a set Rn(sn).
I Rn(sn): A range of user n’s appliance operation for sn
defined as follows:
Rn(sn) ,
n
h

128. h = i mod H, ∀i∈ [sn, sn + δn − 1]
o
27 / 66

129. Starting-Time-Based Formulation
I Optimization variables:
s , [s1, . . . , sN]
I A feasible set for user n:
Sn ,
n
sn

132. sn = i mod H, ∀i∈[αn, βn−δn+1]
o
I Aggregate load across all users at each time slot h:
Lh(s) ,
X
n∈N
γ
op
n ((h − sn) mod H) IRn(sn)(h)
where
I IRn(sn)(h): An indicator function for a set Rn(sn).
I Rn(sn): A range of user n’s appliance operation for sn
defined as follows:
Rn(sn) ,
n
h

135. h = i mod H, ∀i∈ [sn, sn + δn − 1]
o
27 / 66

136. Starting-Time-Based Formulation
I Optimization variables:
s , [s1, . . . , sN]
I A feasible set for user n:
Sn ,
n
sn

139. sn = i mod H, ∀i∈[αn, βn−δn+1]
o
I Aggregate load across all users at each time slot h:
Lh(s) ,
X
n∈N
γ
op
n ((h − sn) mod H) IRn(sn)(h)
where
I IRn(sn)(h): An indicator function for a set Rn(sn).
I Rn(sn): A range of user n’s appliance operation for sn
defined as follows:
Rn(sn) ,
n
h

142. h = i mod H, ∀i∈ [sn, sn + δn − 1]
o
27 / 66

143. Issues with Starting-Time-Based Formulation
I Because the feasible set is now discrete, we have to
evaluate the objective function for all the elements in the
feasible set.
I The optimization by direct enumeration becomes
impractical for large N and H.
I When N=100 and H=24 with the worst case scenario
of αn=0, βn=23, and δn=1 for all n∈N, we need to
evaluate the objective function 24100 times, which is
on the order of 10138 times!
28 / 66

144. Next . . .
Introduction
Review of Current Formulation of Appliance Energy
Consumption Scheduling
Atomic Scheduling of Appliance Energy Consumption
Starting-Time-Based Formulation
Optimal-Routing-Based Formulation
Successive Convex Relaxation
Examples
Numerical Results
Experimental Parameters
Comparison of Bounds with True Optimal Values
Cost Minimization
PAR Minimization
Next Steps: Taking into Account Local Energy Storage
and Generation
Conclusions
29 / 66

145. Why London Eye?
Does the London Eye
have something to
do with the atomic
scheduling?
30 / 66

146. A Network, A Path, and Links
S
D
0 1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
l9,10
l10,11
l11,12
p9,3
A network connecting
the source (S) and the
destination (D)
through 24
intermediate nodes
with a path (p9,3
) and
its constituent links
(l9,10
, l10,11
, and l11,12
).
31 / 66

147. Mapping of Atomic Operations to Flows
0 1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
f0
1
f1
1
f2
1
f3
1
f4
1
f9
2
f10
2
f11
2
f12
2
S
D
Mapping of all
possible atomic
operations of two
appliances into two
groups of flows
(f0
1
, . . . , f4
1
and
f9
2
, . . . , f12
2
) over
multiple paths on the
network.
32 / 66

148. Optimization Variables and Feasible Set
I Optimization variables: Flow configurations of all users
defined as
f ,
h
f1, . . . , fn, . . . , fN
i
where
fn ,
h
f0
n , . . . , fH−1
n
i
.
I A feasible atomic energy consumption scheduling set
for user n:
Fn =
(
fn

154. X
s∈Sn
fs
n=1, fs
n∈ {0, 1} , ∀s∈Sn, fs
n=0, ∀s∈HSn
)
where Sn is the feasible set of starting times for user n
that is already defined in starting-time-based
formulation.
33 / 66

155. Optimization Variables and Feasible Set
I Optimization variables: Flow configurations of all users
defined as
f ,
h
f1, . . . , fn, . . . , fN
i
where
fn ,
h
f0
n , . . . , fH−1
n
i
.
I A feasible atomic energy consumption scheduling set
for user n:
Fn =
(
fn

161. X
s∈Sn
fs
n=1, fs
n∈ {0, 1} , ∀s∈Sn, fs
n=0, ∀s∈HSn
)
where Sn is the feasible set of starting times for user n
that is already defined in starting-time-based
formulation.
33 / 66

162. Optimization Variables and Feasible Set
I Optimization variables: Flow configurations of all users
defined as
f ,
h
f1, . . . , fn, . . . , fN
i
where
fn ,
h
f0
n , . . . , fH−1
n
i
.
I A feasible atomic energy consumption scheduling set
for user n:
Fn =
(
fn

168. X
s∈Sn
fs
n=1, fs
n∈ {0, 1} , ∀s∈Sn, fs
n=0, ∀s∈HSn
)
where Sn is the feasible set of starting times for user n
that is already defined in starting-time-based
formulation.
33 / 66

169. Optimization Variables and Feasible Set
I Optimization variables: Flow configurations of all users
defined as
f ,
h
f1, . . . , fn, . . . , fN
i
where
fn ,
h
f0
n , . . . , fH−1
n
i
.
I A feasible atomic energy consumption scheduling set
for user n:
Fn =
(
fn

175. X
s∈Sn
fs
n=1, fs
n∈ {0, 1} , ∀s∈Sn, fs
n=0, ∀s∈HSn
)
where Sn is the feasible set of starting times for user n
that is already defined in starting-time-based
formulation.
33 / 66

176. Atomic Optimal Scheduling
I Atomic optimal scheduling for an objective function
of φ(·) is formulated as follows:
minimize
fn∈Fn,∀n∈N
φ (L (f)) .
where
I L (f) , [L0 (f) , . . . , LH−1 (f)]: A vector of aggregate loads
across all flows at each time slot, which are defined as
Lh(f) ,
X
n∈N
γ
op
n ((h−s) modH)








X
s∈Sn
fs
nIRn(s)(h)








.
I Note that for a convex objective function, this
problem becomes a Boolean-convex problem, since the
optimization variable fs
n is restricted to only 0 or 1.
34 / 66

177. Atomic Optimal Scheduling
I Atomic optimal scheduling for an objective function
of φ(·) is formulated as follows:
minimize
fn∈Fn,∀n∈N
φ (L (f)) .
where
I L (f) , [L0 (f) , . . . , LH−1 (f)]: A vector of aggregate loads
across all flows at each time slot, which are defined as
Lh(f) ,
X
n∈N
γ
op
n ((h−s) modH)








X
s∈Sn
fs
nIRn(s)(h)








.
I Note that for a convex objective function, this
problem becomes a Boolean-convex problem, since the
optimization variable fs
n is restricted to only 0 or 1.
34 / 66

178. Atomic Optimal Scheduling
I Atomic optimal scheduling for an objective function
of φ(·) is formulated as follows:
minimize
fn∈Fn,∀n∈N
φ (L (f)) .
where
I L (f) , [L0 (f) , . . . , LH−1 (f)]: A vector of aggregate loads
across all flows at each time slot, which are defined as
Lh(f) ,
X
n∈N
γ
op
n ((h−s) modH)








X
s∈Sn
fs
nIRn(s)(h)








.
I Note that for a convex objective function, this
problem becomes a Boolean-convex problem, since the
optimization variable fs
n is restricted to only 0 or 1.
34 / 66

179. Atomic Optimal Scheduling
I Atomic optimal scheduling for an objective function
of φ(·) is formulated as follows:
minimize
fn∈Fn,∀n∈N
φ (L (f)) .
where
I L (f) , [L0 (f) , . . . , LH−1 (f)]: A vector of aggregate loads
across all flows at each time slot, which are defined as
Lh(f) ,
X
n∈N
γ
op
n ((h−s) modH)








X
s∈Sn
fs
nIRn(s)(h)








.
I Note that for a convex objective function, this
problem becomes a Boolean-convex problem, since the
optimization variable fs
n is restricted to only 0 or 1.
34 / 66

180. Next . . .
Introduction
Review of Current Formulation of Appliance Energy
Consumption Scheduling
Atomic Scheduling of Appliance Energy Consumption
Starting-Time-Based Formulation
Optimal-Routing-Based Formulation
Successive Convex Relaxation
Examples
Numerical Results
Experimental Parameters
Comparison of Bounds with True Optimal Values
Cost Minimization
PAR Minimization
Next Steps: Taking into Account Local Energy Storage
and Generation
Conclusions
35 / 66

181. Relaxed Atomic Optimal Scheduling
I We can relax the atomic optimal scheduling problem
by replacing fs
n∈ {0, 1} with 0≤fs
n≤1 in constraints as
follows:
minimize
fn∈F̂n,∀n∈N
φ (L (f))
where
F̂n =
(
fn

187. X
s∈Sn
fs
n=1, 0 ≤ fs
n ≤ 1, ∀s∈Sn, fs
n=0, ∀s∈HSn
)
.
I For a convex objective function, this problem
becomes convex because F̂n is now a convex set. It
can be solved efficiently, for instance, using the
well-known interior-point method [1].
36 / 66

188. Relaxed Atomic Optimal Scheduling
I We can relax the atomic optimal scheduling problem
by replacing fs
n∈ {0, 1} with 0≤fs
n≤1 in constraints as
follows:
minimize
fn∈F̂n,∀n∈N
φ (L (f))
where
F̂n =
(
fn

194. X
s∈Sn
fs
n=1, 0 ≤ fs
n ≤ 1, ∀s∈Sn, fs
n=0, ∀s∈HSn
)
.
I For a convex objective function, this problem
becomes convex because F̂n is now a convex set. It
can be solved efficiently, for instance, using the
well-known interior-point method [1].
36 / 66

195. Relaxed Atomic Optimal Scheduling
I We can relax the atomic optimal scheduling problem
by replacing fs
n∈ {0, 1} with 0≤fs
n≤1 in constraints as
follows:
minimize
fn∈F̂n,∀n∈N
φ (L (f))
where
F̂n =
(
fn

201. X
s∈Sn
fs
n=1, 0 ≤ fs
n ≤ 1, ∀s∈Sn, fs
n=0, ∀s∈HSn
)
.
I For a convex objective function, this problem
becomes convex because F̂n is now a convex set. It
can be solved efficiently, for instance, using the
well-known interior-point method [1].
36 / 66

202. Relaxed vs. Original Scheduling Problems
I The relaxed atomic optimal scheduling problem is
not equivalent to the original problem.
I The elements of the optimal solution from the relaxed
problem can take fractional values (e.g., 0.75).
I The optimal solution of the relaxed problem,
however, provides a lower bound on the optimal
solution of the original problem.
I The feasible set for the relaxed problem contains the
feasible set for the original problem.
37 / 66

203. Relaxed vs. Original Scheduling Problems
I The relaxed atomic optimal scheduling problem is
not equivalent to the original problem.
I The elements of the optimal solution from the relaxed
problem can take fractional values (e.g., 0.75).
I The optimal solution of the relaxed problem,
however, provides a lower bound on the optimal
solution of the original problem.
I The feasible set for the relaxed problem contains the
feasible set for the original problem.
37 / 66

204. Relaxed vs. Original Scheduling Problems
I The relaxed atomic optimal scheduling problem is
not equivalent to the original problem.
I The elements of the optimal solution from the relaxed
problem can take fractional values (e.g., 0.75).
I The optimal solution of the relaxed problem,
however, provides a lower bound on the optimal
solution of the original problem.
I The feasible set for the relaxed problem contains the
feasible set for the original problem.
37 / 66

205. Relaxed vs. Original Scheduling Problems
I The relaxed atomic optimal scheduling problem is
not equivalent to the original problem.
I The elements of the optimal solution from the relaxed
problem can take fractional values (e.g., 0.75).
I The optimal solution of the relaxed problem,
however, provides a lower bound on the optimal
solution of the original problem.
I The feasible set for the relaxed problem contains the
feasible set for the original problem.
37 / 66

206. Successive Convex Relaxation
First, we solve the relaxed convex optimization problem.
Then, carry out the following procedures:
1. Identify the maximum element of each user flow
configuration vector (i.e., corresponding to fn) and
exclude them in the following procedures.
2. Arrange in ascending order the remaining elements
that are less than 1.
3. Drop the smallest element and add a zero constraint
for it.
38 / 66

207. Successive Convex Relaxation
First, we solve the relaxed convex optimization problem.
Then, carry out the following procedures:
1. Identify the maximum element of each user flow
configuration vector (i.e., corresponding to fn) and
exclude them in the following procedures.
2. Arrange in ascending order the remaining elements
that are less than 1.
3. Drop the smallest element and add a zero constraint
for it.
38 / 66

208. Successive Convex Relaxation
First, we solve the relaxed convex optimization problem.
Then, carry out the following procedures:
1. Identify the maximum element of each user flow
configuration vector (i.e., corresponding to fn) and
exclude them in the following procedures.
2. Arrange in ascending order the remaining elements
that are less than 1.
3. Drop the smallest element and add a zero constraint
for it.
38 / 66

209. Successive Convex Relaxation
First, we solve the relaxed convex optimization problem.
Then, carry out the following procedures:
1. Identify the maximum element of each user flow
configuration vector (i.e., corresponding to fn) and
exclude them in the following procedures.
2. Arrange in ascending order the remaining elements
that are less than 1.
3. Drop the smallest element and add a zero constraint
for it.
38 / 66

210. Successive Convex Relaxation (Cont.)
4. For the rest of the elements, drop up to ND elements
— i.e., starting from the next smallest element and
including the one in step 3 — as far as the element is
less than a dropping threshold (θD) and add zero
constraints for them; otherwise, stop dropping and
go to the next step.
5. If there remains only one nonzero element per user
flow configuration vector, stop here (a solution
found); otherwise, solve a new relaxed convex
optimization problem with augmented constraints
and repeat the whole procedure from step 1.
39 / 66

211. Successive Convex Relaxation (Cont.)
4. For the rest of the elements, drop up to ND elements
— i.e., starting from the next smallest element and
including the one in step 3 — as far as the element is
less than a dropping threshold (θD) and add zero
constraints for them; otherwise, stop dropping and
go to the next step.
5. If there remains only one nonzero element per user
flow configuration vector, stop here (a solution
found); otherwise, solve a new relaxed convex
optimization problem with augmented constraints
and repeat the whole procedure from step 1.
39 / 66

212. Successive Convex Relaxation: An Example
Consider a simple case of N=2, H=4, and ND = 1.
I Initial condition:
f = [ 0.25 0.25 0.25 0.25 | 0.3 0.2 0.25 0.25 ]
I After 1st step:
f = [ 0.0 0.2 0.3 0.5 | 0.7 0.0 0.2 0.1 ]
I After 2nd step:
f = [ 0.0 0.0 0.2 0.8 | 0.9 0.0 0.1 0.0 ]
Stop here. Solution found!
40 / 66

213. Successive Convex Relaxation: An Example
Consider a simple case of N=2, H=4, and ND = 1.
I Initial condition:
f = [ 0.25 0.25 0.25 0.25 | 0.3 0.2 0.25 0.25 ]
I After 1st step:
f = [ 0.0 0.2 0.3 0.5 | 0.7 0.0 0.2 0.1 ]
I After 2nd step:
f = [ 0.0 0.0 0.2 0.8 | 0.9 0.0 0.1 0.0 ]
Stop here. Solution found!
40 / 66

214. Successive Convex Relaxation: An Example
Consider a simple case of N=2, H=4, and ND = 1.
I Initial condition:
f = [ 0.25 0.25 0.25 0.25 | 0.3 0.2 0.25 0.25 ]
I After 1st step:
f = [ 0.0 0.2 0.3 0.5 | 0.7 0.0 0.2 0.1 ]
I After 2nd step:
f = [ 0.0 0.0 0.2 0.8 | 0.9 0.0 0.1 0.0 ]
Stop here. Solution found!
40 / 66

215. Successive Convex Relaxation: An Example
Consider a simple case of N=2, H=4, and ND = 1.
I Initial condition:
f = [ 0.25 0.25 0.25 0.25 | 0.3 0.2 0.25 0.25 ]
I After 1st step:
f = [ 0.0 0.2 0.3 0.5 | 0.7 0.0 0.2 0.1 ]
I After 2nd step:
f = [ 0.0 0.0 0.2 0.8 | 0.9 0.0 0.1 0.0 ]
Stop here. Solution found!
40 / 66

216. Next . . .
Introduction
Review of Current Formulation of Appliance Energy
Consumption Scheduling
Atomic Scheduling of Appliance Energy Consumption
Starting-Time-Based Formulation
Optimal-Routing-Based Formulation
Successive Convex Relaxation
Examples
Numerical Results
Experimental Parameters
Comparison of Bounds with True Optimal Values
Cost Minimization
PAR Minimization
Next Steps: Taking into Account Local Energy Storage
and Generation
Conclusions
41 / 66

217. Energy Cost and PAR Minimization
I Energy cost minization
minimize
fn∈F̂n,∀n∈N
X
h∈H
Ch (Lh(f)) .
I PAR minimization
minimize
Γ,fn∈F̂n,∀n∈N
Γ
subject to Γ ≥ Lh(f), ∀h ∈ H.
I Note that PAR minimization is formulated as a relaxed
linear program by introducing a new auxiliary variable
Γ.
42 / 66

218. Energy Cost and PAR Minimization
I Energy cost minization
minimize
fn∈F̂n,∀n∈N
X
h∈H
Ch (Lh(f)) .
I PAR minimization
minimize
Γ,fn∈F̂n,∀n∈N
Γ
subject to Γ ≥ Lh(f), ∀h ∈ H.
I Note that PAR minimization is formulated as a relaxed
linear program by introducing a new auxiliary variable
Γ.
42 / 66

219. Energy Cost and PAR Minimization
I Energy cost minization
minimize
fn∈F̂n,∀n∈N
X
h∈H
Ch (Lh(f)) .
I PAR minimization
minimize
Γ,fn∈F̂n,∀n∈N
Γ
subject to Γ ≥ Lh(f), ∀h ∈ H.
I Note that PAR minimization is formulated as a relaxed
linear program by introducing a new auxiliary variable
Γ.
42 / 66

220. Next . . .
Introduction
Review of Current Formulation of Appliance Energy
Consumption Scheduling
Atomic Scheduling of Appliance Energy Consumption
Starting-Time-Based Formulation
Optimal-Routing-Based Formulation
Successive Convex Relaxation
Examples
Numerical Results
Experimental Parameters
Comparison of Bounds with True Optimal Values
Cost Minimization
PAR Minimization
Next Steps: Taking into Account Local Energy Storage
and Generation
Conclusions
43 / 66

221. Next . . .
Introduction
Review of Current Formulation of Appliance Energy
Consumption Scheduling
Atomic Scheduling of Appliance Energy Consumption
Starting-Time-Based Formulation
Optimal-Routing-Based Formulation
Successive Convex Relaxation
Examples
Numerical Results
Experimental Parameters
Comparison of Bounds with True Optimal Values
Cost Minimization
PAR Minimization
Next Steps: Taking into Account Local Energy Storage
and Generation
Conclusions
44 / 66

222. Appliance Energy Consumption
Requirements
Appliance
Parameters
α [h] β [h] γop
[kWh] δ [h]
Dish Washer 0 23 0.7200 2
Washing Machine
0 23 0.4967 3
(Energy Star)
Washing Machine
0 23 0.6467 3
(Regular)
Clothes Dryer 0 23 0.6250 4
PHEV1
222
292
3.3000 3
1
Plug-in hybrid electric vehicle.
2
Scheduling interval of 10 PM–5 AM.
I An appliance is randomly selected for each user.
I Constant operating energy levels (γop
) are assumed. 45 / 66

223. Hourly Cost Function
We assume a simple quadratic hourly cost function as in
[3], i.e.,
Ch (Lh) = ahL2
h [cent]
where
ah =
(
0.2 if h ∈ [0, 7],
0.3 if h ∈ [8, 23].
46 / 66

224. Hourly Cost Function
We assume a simple quadratic hourly cost function as in
[3], i.e.,
Ch (Lh) = ahL2
h [cent]
where
ah =
(
0.2 if h ∈ [0, 7],
0.3 if h ∈ [8, 23].
46 / 66

225. Performance Measures and Parameter Values
I Performance measures
I Lower bound: LB
I Upper bound with ND: UB (ND)
I Gap: G , UB(D) − LB
I Number of iterations
I Parameter values for successive convex relaxation
I Dropping threshold (θD): 0.1
I Maximum number of fractional-valued elements that
can be dropped per iteration (ND): 1, 2, 5, 10
47 / 66

226. Performance Measures and Parameter Values
I Performance measures
I Lower bound: LB
I Upper bound with ND: UB (ND)
I Gap: G , UB(D) − LB
I Number of iterations
I Parameter values for successive convex relaxation
I Dropping threshold (θD): 0.1
I Maximum number of fractional-valued elements that
can be dropped per iteration (ND): 1, 2, 5, 10
47 / 66

227. Performance Measures and Parameter Values
I Performance measures
I Lower bound: LB
I Upper bound with ND: UB (ND)
I Gap: G , UB(D) − LB
I Number of iterations
I Parameter values for successive convex relaxation
I Dropping threshold (θD): 0.1
I Maximum number of fractional-valued elements that
can be dropped per iteration (ND): 1, 2, 5, 10
47 / 66

228. Performance Measures and Parameter Values
I Performance measures
I Lower bound: LB
I Upper bound with ND: UB (ND)
I Gap: G , UB(D) − LB
I Number of iterations
I Parameter values for successive convex relaxation
I Dropping threshold (θD): 0.1
I Maximum number of fractional-valued elements that
can be dropped per iteration (ND): 1, 2, 5, 10
47 / 66

229. Performance Measures and Parameter Values
I Performance measures
I Lower bound: LB
I Upper bound with ND: UB (ND)
I Gap: G , UB(D) − LB
I Number of iterations
I Parameter values for successive convex relaxation
I Dropping threshold (θD): 0.1
I Maximum number of fractional-valued elements that
can be dropped per iteration (ND): 1, 2, 5, 10
47 / 66

230. Performance Measures and Parameter Values
I Performance measures
I Lower bound: LB
I Upper bound with ND: UB (ND)
I Gap: G , UB(D) − LB
I Number of iterations
I Parameter values for successive convex relaxation
I Dropping threshold (θD): 0.1
I Maximum number of fractional-valued elements that
can be dropped per iteration (ND): 1, 2, 5, 10
47 / 66

231. Performance Measures and Parameter Values
I Performance measures
I Lower bound: LB
I Upper bound with ND: UB (ND)
I Gap: G , UB(D) − LB
I Number of iterations
I Parameter values for successive convex relaxation
I Dropping threshold (θD): 0.1
I Maximum number of fractional-valued elements that
can be dropped per iteration (ND): 1, 2, 5, 10
47 / 66

232. Performance Measures and Parameter Values
I Performance measures
I Lower bound: LB
I Upper bound with ND: UB (ND)
I Gap: G , UB(D) − LB
I Number of iterations
I Parameter values for successive convex relaxation
I Dropping threshold (θD): 0.1
I Maximum number of fractional-valued elements that
can be dropped per iteration (ND): 1, 2, 5, 10
47 / 66

233. Next . . .
Introduction
Review of Current Formulation of Appliance Energy
Consumption Scheduling
Atomic Scheduling of Appliance Energy Consumption
Starting-Time-Based Formulation
Optimal-Routing-Based Formulation
Successive Convex Relaxation
Examples
Numerical Results
Experimental Parameters
Comparison of Bounds with True Optimal Values
Cost Minimization
PAR Minimization
Next Steps: Taking into Account Local Energy Storage
and Generation
Conclusions
48 / 66

234. Upper/Lower Bounds vs. True Optimal
Values: Cost Minimization
2 3 4 5 6 7 8 9 10
0
5 · 10−2
0.1
0.15
0.2
0.25
0.3
0.35
N
Energy
Cost
[USD] LB
GO
UB(1)
UB(2)
UB(5)
UB(10)
49 / 66

235. Upper/Lower Bounds vs. True Optimal
Values: PAR Minimization
2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10
N
PAR
in
Aggregated
Load LB
GO
UB(1)
UB(2)
UB(5)
UB(10)
50 / 66

236. Next . . .
Introduction
Review of Current Formulation of Appliance Energy
Consumption Scheduling
Atomic Scheduling of Appliance Energy Consumption
Starting-Time-Based Formulation
Optimal-Routing-Based Formulation
Successive Convex Relaxation
Examples
Numerical Results
Experimental Parameters
Comparison of Bounds with True Optimal Values
Cost Minimization
PAR Minimization
Next Steps: Taking into Account Local Energy Storage
and Generation
Conclusions
51 / 66

237. Cost Minimization: Upper/Lower Bounds
2 10 20 30 40 50
0
1
2
3
4
5
N
Energy
Cost
[USD]
LB
UB(1)
UB(2)
UB(5)
UB(10)
52 / 66

238. Cost Minimization: Gaps
2 10 20 30 40 50
0
1
2
3
4
5
6
7
×10−2
N
Gap
[USD]
UB(1)−LB
UB(2)−LB
UB(5)−LB
UB(10)−LB
53 / 66

239. Cost Minimization: Number of Iterations
2 10 20 30 40 50
0
200
400
600
800
1,000
N
Number
of
Iterations
UB(1)
UB(2)
UB(5)
UB(10)
54 / 66

240. Next . . .
Introduction
Review of Current Formulation of Appliance Energy
Consumption Scheduling
Atomic Scheduling of Appliance Energy Consumption
Starting-Time-Based Formulation
Optimal-Routing-Based Formulation
Successive Convex Relaxation
Examples
Numerical Results
Experimental Parameters
Comparison of Bounds with True Optimal Values
Cost Minimization
PAR Minimization
Next Steps: Taking into Account Local Energy Storage
and Generation
Conclusions
55 / 66

241. Par Minimization: Upper/Lower Bounds
2 10 20 30 40 50
0
5
10
15
20
25
N
PAR
in
Aggregated
Load
LB
UB(1)
UB(2)
UB(5)
UB(10)
56 / 66

242. Par Minimization: Gaps
2 10 20 30 40 50
0
1
2
3
N
Gap
UB(1)−LB
UB(2)−LB
UB(5)−LB
UB(10)−LB
57 / 66

243. Par Minimization: Number of Iterations
2 10 20 30 40 50
0
200
400
600
800
1,000
N
Number
of
Iterations
UB(1)
UB(2)
UB(5)
UB(10)
58 / 66

244. Next . . .
Introduction
Review of Current Formulation of Appliance Energy
Consumption Scheduling
Atomic Scheduling of Appliance Energy Consumption
Starting-Time-Based Formulation
Optimal-Routing-Based Formulation
Successive Convex Relaxation
Examples
Numerical Results
Experimental Parameters
Comparison of Bounds with True Optimal Values
Cost Minimization
PAR Minimization
Next Steps: Taking into Account Local Energy Storage
and Generation
Conclusions
59 / 66

245. Impact of Energy Storage on DSM
t
t
(a)
(b)
(a) An original pattern and (b) a smoothened pattern (i.e., solid line) by
energy storage where up and down arrows denote charging and discharging.
60 / 66

246. Electric Battery vs. Packet Buffer
61 / 66

247. Challenges and Opportunities
I The energy storage is expected to be a key component
in smart homes in the future smart grid [4].
I Joint optimal scheduling of appliance energy
consumption and energy storage device
charging/discharging (optionally with local energy
generation) is for further study.
I Note that the impact of energy storage devices in
energy scheduling is similar to that of packet buffers in
packet scheduling.
I It is interesting to investigate this similarity in the
study of the joint optimal scheduling.
62 / 66

248. Challenges and Opportunities
I The energy storage is expected to be a key component
in smart homes in the future smart grid [4].
I Joint optimal scheduling of appliance energy
consumption and energy storage device
charging/discharging (optionally with local energy
generation) is for further study.
I Note that the impact of energy storage devices in
energy scheduling is similar to that of packet buffers in
packet scheduling.
I It is interesting to investigate this similarity in the
study of the joint optimal scheduling.
62 / 66

249. Challenges and Opportunities
I The energy storage is expected to be a key component
in smart homes in the future smart grid [4].
I Joint optimal scheduling of appliance energy
consumption and energy storage device
charging/discharging (optionally with local energy
generation) is for further study.
I Note that the impact of energy storage devices in
energy scheduling is similar to that of packet buffers in
packet scheduling.
I It is interesting to investigate this similarity in the
study of the joint optimal scheduling.
62 / 66

250. Challenges and Opportunities
I The energy storage is expected to be a key component
in smart homes in the future smart grid [4].
I Joint optimal scheduling of appliance energy
consumption and energy storage device
charging/discharging (optionally with local energy
generation) is for further study.
I Note that the impact of energy storage devices in
energy scheduling is similar to that of packet buffers in
packet scheduling.
I It is interesting to investigate this similarity in the
study of the joint optimal scheduling.
62 / 66

251. Next . . .
Introduction
Review of Current Formulation of Appliance Energy
Consumption Scheduling
Atomic Scheduling of Appliance Energy Consumption
Starting-Time-Based Formulation
Optimal-Routing-Based Formulation
Successive Convex Relaxation
Examples
Numerical Results
Experimental Parameters
Comparison of Bounds with True Optimal Values
Cost Minimization
PAR Minimization
Next Steps: Taking into Account Local Energy Storage
and Generation
Conclusions
63 / 66

252. Conclusions
I We have provided a new formulation of appliance energy
consumption scheduling based on the optimal routing
framework.
I It guarantees the atomicity of resulting scheduled
energy consumption.
I We have also provided an efficient solution technique
based on successive convex relaxation for a
Boolean-convex problem resulting from a convex
objective function.
I It enables us to carry out systematic analysis of the
original problem with both upper and lower bounds.
I Possible extensions to the current work include
I Distributed atomic energy consumption scheduling;
I Joint optimal scheduling of appliance energy
consumption and energy storage device
charging/discharging (optionally with local energy
generation).
64 / 66

253. Conclusions
I We have provided a new formulation of appliance energy
consumption scheduling based on the optimal routing
framework.
I It guarantees the atomicity of resulting scheduled
energy consumption.
I We have also provided an efficient solution technique
based on successive convex relaxation for a
Boolean-convex problem resulting from a convex
objective function.
I It enables us to carry out systematic analysis of the
original problem with both upper and lower bounds.
I Possible extensions to the current work include
I Distributed atomic energy consumption scheduling;
I Joint optimal scheduling of appliance energy
consumption and energy storage device
charging/discharging (optionally with local energy
generation).
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254. Conclusions
I We have provided a new formulation of appliance energy
consumption scheduling based on the optimal routing
framework.
I It guarantees the atomicity of resulting scheduled
energy consumption.
I We have also provided an efficient solution technique
based on successive convex relaxation for a
Boolean-convex problem resulting from a convex
objective function.
I It enables us to carry out systematic analysis of the
original problem with both upper and lower bounds.
I Possible extensions to the current work include
I Distributed atomic energy consumption scheduling;
I Joint optimal scheduling of appliance energy
consumption and energy storage device
charging/discharging (optionally with local energy
generation).
64 / 66

255. Conclusions
I We have provided a new formulation of appliance energy
consumption scheduling based on the optimal routing
framework.
I It guarantees the atomicity of resulting scheduled
energy consumption.
I We have also provided an efficient solution technique
based on successive convex relaxation for a
Boolean-convex problem resulting from a convex
objective function.
I It enables us to carry out systematic analysis of the
original problem with both upper and lower bounds.
I Possible extensions to the current work include
I Distributed atomic energy consumption scheduling;
I Joint optimal scheduling of appliance energy
consumption and energy storage device
charging/discharging (optionally with local energy
generation).
64 / 66

256. Conclusions
I We have provided a new formulation of appliance energy
consumption scheduling based on the optimal routing
framework.
I It guarantees the atomicity of resulting scheduled
energy consumption.
I We have also provided an efficient solution technique
based on successive convex relaxation for a
Boolean-convex problem resulting from a convex
objective function.
I It enables us to carry out systematic analysis of the
original problem with both upper and lower bounds.
I Possible extensions to the current work include
I Distributed atomic energy consumption scheduling;
I Joint optimal scheduling of appliance energy
consumption and energy storage device
charging/discharging (optionally with local energy
generation).
64 / 66

257. Conclusions
I We have provided a new formulation of appliance energy
consumption scheduling based on the optimal routing
framework.
I It guarantees the atomicity of resulting scheduled
energy consumption.
I We have also provided an efficient solution technique
based on successive convex relaxation for a
Boolean-convex problem resulting from a convex
objective function.
I It enables us to carry out systematic analysis of the
original problem with both upper and lower bounds.
I Possible extensions to the current work include
I Distributed atomic energy consumption scheduling;
I Joint optimal scheduling of appliance energy
consumption and energy storage device
charging/discharging (optionally with local energy
generation).
64 / 66

258. Conclusions
I We have provided a new formulation of appliance energy
consumption scheduling based on the optimal routing
framework.
I It guarantees the atomicity of resulting scheduled
energy consumption.
I We have also provided an efficient solution technique
based on successive convex relaxation for a
Boolean-convex problem resulting from a convex
objective function.
I It enables us to carry out systematic analysis of the
original problem with both upper and lower bounds.
I Possible extensions to the current work include
I Distributed atomic energy consumption scheduling;
I Joint optimal scheduling of appliance energy
consumption and energy storage device
charging/discharging (optionally with local energy
generation).
64 / 66

259. References I
S. Boyd and L. Vandenberghe, Convex optimization.
Cambridge, U.K.: Cambridge Universtiy Press, 2004.
J. B. Rosen, “Existence and uniqueness of equilibrium
for concave N-person games,” Econometrica, vol. 33,
no. 3, pp. 520–534, Jul. 1965.
A.-H. Mohsenian-Rad, V. W. S. Wong, J. Jatskevich,
R. Schober, and A. Leon-Garcia, “Autonomous
demand-side management based on game-theoretic
energy consumption scheduling for the future smart
grid,” IEEE Trans. Smart Grid, vol. 1, no. 3, pp.
320–331, Dec. 2010.
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260. References II
W. Saad, Z. Han, H. V. Poor, and T. Başar,
“Game-theoretic methods for the smart grid: An
overview of microgrid systems, demand-side
management, and smart grid communications,” IEEE
Signal Process. Mag., vol. 29, no. 5, pp. 86–105, Sep.
2012.
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