The document discusses atomic scheduling for appliance energy consumption within a residential smart grid framework, focusing on optimizing energy production and consumption using smart meter data for two-way communication between utility companies and users. It reviews current formulations and presents various optimization strategies, including start-time and optimal-routing-based methods, alongside numerical results demonstrating cost and peak-to-average ratio minimization. The work emphasizes the importance of feasible energy consumption scheduling and the impacts of distributed algorithms on achieving optimal outcomes.

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
CeSGIC 1st International Workshop on
Smart Grid Technology and Data Processing
19 June 2015
1 / 55

2. 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
Conclusions
2 / 55

4. 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
Conclusions
4 / 55

5. Autonomous Demand-Side Management in
Smart Grid
Gateway
(Traditional)
Electricity
Grid
Greenfield
Power
Line
Bi-Directional
Communication
Links
Power Plant
5 / 55

6. Scheduling of Appliance Energy
Consumption
A key to autonomous DSM in
optimizing energy production and
consumption.
Based on two-way digital
communications between a utility
company and users through smart
meters at users’ premises.
Typical objectives
Peak-to-average ratio (PAR)
Total energy cost
6 / 55

7. Scheduling of Appliance Energy
Consumption
A key to autonomous DSM in
optimizing energy production and
consumption.
Based on two-way digital
communications between a utility
company and users through smart
meters at users’ premises.
Typical objectives
Peak-to-average ratio (PAR)
Total energy cost
6 / 55

8. Scheduling of Appliance Energy
Consumption
A key to autonomous DSM in
optimizing energy production and
consumption.
Based on two-way digital
communications between a utility
company and users through smart
meters at users’ premises.
Typical objectives
Peak-to-average ratio (PAR)
Total energy cost
6 / 55

9. Scheduling of Appliance Energy
Consumption
A key to autonomous DSM in
optimizing energy production and
consumption.
Based on two-way digital
communications between a utility
company and users through smart
meters at users’ premises.
Typical objectives
Peak-to-average ratio (PAR)
Total energy cost
6 / 55

10. A Question on Scheduled Energy
Consumption
Can a washing machine
successfully complete its job with
the energy consumption scheduled
as follows?
7 / 55

11. A Question on Scheduled Energy
Consumption
Can a washing machine
successfully complete its job with
the energy consumption scheduled
as follows?
9am 3pm12am
7 / 55

12. A Question on Scheduled Energy
Consumption
Can a washing machine
successfully complete its job with
the energy consumption scheduled
as follows?
9am 3pm12am
or
7 / 55

13. A Question on Scheduled Energy
Consumption
Can a washing machine
successfully complete its job with
the energy consumption scheduled
as follows?
9am 3pm12am
or
9am 3pm2pm10am
7 / 55

14. 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
Conclusions
8 / 55

15. Optimization Variables
Based on energy consumption over equally-divided time
slots of a day (typically hourly time slots) as
optimization variables, i.e.,
xn x0
n, . . . , xh
n, . . . , xH−1
n
for
User n ∈ N {1, . . ., N};
Time slot h ∈ H {0, . . ., H−1}.
Because the optimization variables take continuous
values, we can easily apply
Convex optimization [1];
Distributed algorithms through concave n-person
games [2].
9 / 55

16. Optimization Variables
Based on energy consumption over equally-divided time
slots of a day (typically hourly time slots) as
optimization variables, i.e.,
xn x0
n, . . . , xh
n, . . . , xH−1
n
for
User n ∈ N {1, . . ., N};
Time slot h ∈ H {0, . . ., H−1}.
Because the optimization variables take continuous
values, we can easily apply
Convex optimization [1];
Distributed algorithms through concave n-person
games [2].
9 / 55

17. Optimization Variables
Based on energy consumption over equally-divided time
slots of a day (typically hourly time slots) as
optimization variables, i.e.,
xn x0
n, . . . , xh
n, . . . , xH−1
n
for
User n ∈ N {1, . . ., N};
Time slot h ∈ H {0, . . ., H−1}.
Because the optimization variables take continuous
values, we can easily apply
Convex optimization [1];
Distributed algorithms through concave n-person
games [2].
9 / 55

18. Optimization Variables
Based on energy consumption over equally-divided time
slots of a day (typically hourly time slots) as
optimization variables, i.e.,
xn x0
n, . . . , xh
n, . . . , xH−1
n
for
User n ∈ N {1, . . ., N};
Time slot h ∈ H {0, . . ., H−1}.
Because the optimization variables take continuous
values, we can easily apply
Convex optimization [1];
Distributed algorithms through concave n-person
games [2].
9 / 55

19. Optimization Variables
Based on energy consumption over equally-divided time
slots of a day (typically hourly time slots) as
optimization variables, i.e.,
xn x0
n, . . . , xh
n, . . . , xH−1
n
for
User n ∈ N {1, . . ., N};
Time slot h ∈ H {0, . . ., H−1}.
Because the optimization variables take continuous
values, we can easily apply
Convex optimization [1];
Distributed algorithms through concave n-person
games [2].
9 / 55

20. Optimization Variables
Based on energy consumption over equally-divided time
slots of a day (typically hourly time slots) as
optimization variables, i.e.,
xn x0
n, . . . , xh
n, . . . , xH−1
n
for
User n ∈ N {1, . . ., N};
Time slot h ∈ H {0, . . ., H−1}.
Because the optimization variables take continuous
values, we can easily apply
Convex optimization [1];
Distributed algorithms through concave n-person
games [2].
9 / 55

21. Feasible Set
A feasible energy consumption scheduling set for user n
Xn= xn
h∈Hn
xh
n=En, γmin
n ≤xh
n≤γmax
n , ∀h∈Hn, xh
n=0, ∀h∈HHn
where
γmin
n : Minimum energy level;
γmax
n : Maximum energy level;
En: Total daily energy consumption;
Hn: Scheduling interval defined as follows:
Hn h h = i mod H, ∀i∈ αn, βn
with αn∈[0, H−1], βn∈[1, 2H−2], and 1≤βn−αn≤H−1.
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22. Feasible Set
A feasible energy consumption scheduling set for user n
Xn= xn
h∈Hn
xh
n=En, γmin
n ≤xh
n≤γmax
n , ∀h∈Hn, xh
n=0, ∀h∈HHn
where
γmin
n : Minimum energy level;
γmax
n : Maximum energy level;
En: Total daily energy consumption;
Hn: Scheduling interval defined as follows:
Hn h h = i mod H, ∀i∈ αn, βn
with αn∈[0, H−1], βn∈[1, 2H−2], and 1≤βn−αn≤H−1.
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23. Feasible Set
A feasible energy consumption scheduling set for user n
Xn= xn
h∈Hn
xh
n=En, γmin
n ≤xh
n≤γmax
n , ∀h∈Hn, xh
n=0, ∀h∈HHn
where
γmin
n : Minimum energy level;
γmax
n : Maximum energy level;
En: Total daily energy consumption;
Hn: Scheduling interval defined as follows:
Hn h h = i mod H, ∀i∈ αn, βn
with αn∈[0, H−1], βn∈[1, 2H−2], and 1≤βn−αn≤H−1.
10 / 55

24. Feasible Set
A feasible energy consumption scheduling set for user n
Xn= xn
h∈Hn
xh
n=En, γmin
n ≤xh
n≤γmax
n , ∀h∈Hn, xh
n=0, ∀h∈HHn
where
γmin
n : Minimum energy level;
γmax
n : Maximum energy level;
En: Total daily energy consumption;
Hn: Scheduling interval defined as follows:
Hn h h = i mod H, ∀i∈ αn, βn
with αn∈[0, H−1], βn∈[1, 2H−2], and 1≤βn−αn≤H−1.
10 / 55

25. Feasible Set
A feasible energy consumption scheduling set for user n
Xn= xn
h∈Hn
xh
n=En, γmin
n ≤xh
n≤γmax
n , ∀h∈Hn, xh
n=0, ∀h∈HHn
where
γmin
n : Minimum energy level;
γmax
n : Maximum energy level;
En: Total daily energy consumption;
Hn: Scheduling interval defined as follows:
Hn h h = i mod H, ∀i∈ αn, βn
with αn∈[0, H−1], βn∈[1, 2H−2], and 1≤βn−αn≤H−1.
10 / 55

26. Feasible Set
A feasible energy consumption scheduling set for user n
Xn= xn
h∈Hn
xh
n=En, γmin
n ≤xh
n≤γmax
n , ∀h∈Hn, xh
n=0, ∀h∈HHn
where
γmin
n : Minimum energy level;
γmax
n : Maximum energy level;
En: Total daily energy consumption;
Hn: Scheduling interval defined as follows:
Hn h h = i mod H, ∀i∈ αn, βn
with αn∈[0, H−1], βn∈[1, 2H−2], and 1≤βn−αn≤H−1.
10 / 55

27. 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
x [x1, . . . , xN]: A vector of user energy consumption
vectors;
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)
n∈N
xh
n.
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28. 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
x [x1, . . . , xN]: A vector of user energy consumption
vectors;
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)
n∈N
xh
n.
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29. 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
x [x1, . . . , xN]: A vector of user energy consumption
vectors;
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)
n∈N
xh
n.
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30. 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
x [x1, . . . , xN]: A vector of user energy consumption
vectors;
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)
n∈N
xh
n.
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31. Objective Functions
For energy cost minimization:
φ (L(x)) =
h∈H
Ch (Lh(x))
where
Ch(·): A cost function for generating or distributing
electricity energy at a time slot h.
For PAR minimization:
φ (L(x)) =
H max
h∈H
Lh(x)
n∈N
En
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32. Objective Functions
For energy cost minimization:
φ (L(x)) =
h∈H
Ch (Lh(x))
where
Ch(·): A cost function for generating or distributing
electricity energy at a time slot h.
For PAR minimization:
φ (L(x)) =
H max
h∈H
Lh(x)
n∈N
En
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33. Atomic vs. Non-Atomic Scheduling
αn,a βn,a
αn,a βn,a
Gap Gap
γmin
n
γmax
n
(a)
γop
n (·)
γmin
n
γmax
n
(b)
Examples of (a) non-atomic and (b) atomic scheduling.
γmin
n : Minimum energy level
γmax
n : Maximum energy level
γ
op
n (·): Operating energy level
13 / 55

34. Non-Atomic Scheduling Example
Appliance 1
Appliance 2
Scheduled
Consumption
?
14 / 55

35. Non-Atomic Scheduling Example: Case 1
Appliance 1
Appliance 2
Scheduled
Consumption
15 / 55

36. Non-Atomic Scheduling Example: Case 2
Appliance 1
Appliance 2
Scheduled
Consumption
16 / 55

37. Non-Atomic Scheduling Example: Case 3
Appliance 1
Appliance 2
Scheduled
Consumption
17 / 55

38. 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
Conclusions
18 / 55

39. 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)
19 / 55

40. 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
Conclusions
20 / 55

41. Starting-Time-Based Formulation
Optimization variables:
s [s1, . . . , sN]
A feasible set for user n:
Sn sn sn = i mod H, ∀i∈[αn, βn−δn+1]
Aggregate load across all users at each time slot h:
Lh(s)
n∈N
γ
op
n ((h − sn) mod H) IRn(sn)(h)
where
IRn(sn)(h): An indicator function for a set Rn(sn).
Rn(sn): A range of user n’s appliance operation for sn
defined as follows:
Rn(sn) h h = i mod H, ∀i∈ [sn, sn + δn − 1]
21 / 55

42. Starting-Time-Based Formulation
Optimization variables:
s [s1, . . . , sN]
A feasible set for user n:
Sn sn sn = i mod H, ∀i∈[αn, βn−δn+1]
Aggregate load across all users at each time slot h:
Lh(s)
n∈N
γ
op
n ((h − sn) mod H) IRn(sn)(h)
where
IRn(sn)(h): An indicator function for a set Rn(sn).
Rn(sn): A range of user n’s appliance operation for sn
defined as follows:
Rn(sn) h h = i mod H, ∀i∈ [sn, sn + δn − 1]
21 / 55

43. Starting-Time-Based Formulation
Optimization variables:
s [s1, . . . , sN]
A feasible set for user n:
Sn sn sn = i mod H, ∀i∈[αn, βn−δn+1]
Aggregate load across all users at each time slot h:
Lh(s)
n∈N
γ
op
n ((h − sn) mod H) IRn(sn)(h)
where
IRn(sn)(h): An indicator function for a set Rn(sn).
Rn(sn): A range of user n’s appliance operation for sn
defined as follows:
Rn(sn) h h = i mod H, ∀i∈ [sn, sn + δn − 1]
21 / 55

44. Starting-Time-Based Formulation
Optimization variables:
s [s1, . . . , sN]
A feasible set for user n:
Sn sn sn = i mod H, ∀i∈[αn, βn−δn+1]
Aggregate load across all users at each time slot h:
Lh(s)
n∈N
γ
op
n ((h − sn) mod H) IRn(sn)(h)
where
IRn(sn)(h): An indicator function for a set Rn(sn).
Rn(sn): A range of user n’s appliance operation for sn
defined as follows:
Rn(sn) h h = i mod H, ∀i∈ [sn, sn + δn − 1]
21 / 55

45. Starting-Time-Based Formulation
Optimization variables:
s [s1, . . . , sN]
A feasible set for user n:
Sn sn sn = i mod H, ∀i∈[αn, βn−δn+1]
Aggregate load across all users at each time slot h:
Lh(s)
n∈N
γ
op
n ((h − sn) mod H) IRn(sn)(h)
where
IRn(sn)(h): An indicator function for a set Rn(sn).
Rn(sn): A range of user n’s appliance operation for sn
defined as follows:
Rn(sn) h h = i mod H, ∀i∈ [sn, sn + δn − 1]
21 / 55

46. Starting-Time-Based Formulation
Optimization variables:
s [s1, . . . , sN]
A feasible set for user n:
Sn sn sn = i mod H, ∀i∈[αn, βn−δn+1]
Aggregate load across all users at each time slot h:
Lh(s)
n∈N
γ
op
n ((h − sn) mod H) IRn(sn)(h)
where
IRn(sn)(h): An indicator function for a set Rn(sn).
Rn(sn): A range of user n’s appliance operation for sn
defined as follows:
Rn(sn) h h = i mod H, ∀i∈ [sn, sn + δn − 1]
21 / 55

47. Issues with Starting-Time-Based Formulation
Because the feasible set is now discrete, we have to
evaluate the objective function for all the elements in the
feasible set.
The optimization by direct enumeration becomes
impractical for large N and H.
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!
22 / 55

48. 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
Conclusions
23 / 55

49. Why London Eye?
Does the London Eye
have something to
do with the atomic
scheduling?
24 / 55

50. A Network, A Path, and Links
S
D
0 1
2
3
4
5
6
7
8
9
10
111213
14
15
16
17
18
19
20
21
22
23
l9,10
l10,11l11,12
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
).
25 / 55

51. Mapping of Atomic Operations to Flows
0 1
2
3
4
5
6
7
8
9
10
111213
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
2f12
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.
26 / 55

52. Optimization Variables and Feasible Set
Optimization variables: Flow configurations of all users
defined as
f f1, . . . , fn, . . . , fN
where
fn f0
n , . . . , fH−1
n .
A feasible atomic energy consumption scheduling set
for user n:
Fn = fn
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.
27 / 55

53. Optimization Variables and Feasible Set
Optimization variables: Flow configurations of all users
defined as
f f1, . . . , fn, . . . , fN
where
fn f0
n , . . . , fH−1
n .
A feasible atomic energy consumption scheduling set
for user n:
Fn = fn
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.
27 / 55

54. Optimization Variables and Feasible Set
Optimization variables: Flow configurations of all users
defined as
f f1, . . . , fn, . . . , fN
where
fn f0
n , . . . , fH−1
n .
A feasible atomic energy consumption scheduling set
for user n:
Fn = fn
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.
27 / 55

55. Optimization Variables and Feasible Set
Optimization variables: Flow configurations of all users
defined as
f f1, . . . , fn, . . . , fN
where
fn f0
n , . . . , fH−1
n .
A feasible atomic energy consumption scheduling set
for user n:
Fn = fn
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.
27 / 55

56. Atomic Optimal Scheduling
Atomic optimal scheduling for an objective function
of φ(·) is formulated as follows:
minimize
fn∈Fn,∀n∈N
φ (L (f)) .
where
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)
n∈N
γ
op
n ((h−s) modH)


s∈Sn
fs
nIRn(s)(h)


.
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.
28 / 55

57. Atomic Optimal Scheduling
Atomic optimal scheduling for an objective function
of φ(·) is formulated as follows:
minimize
fn∈Fn,∀n∈N
φ (L (f)) .
where
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)
n∈N
γ
op
n ((h−s) modH)


s∈Sn
fs
nIRn(s)(h)


.
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.
28 / 55

58. Atomic Optimal Scheduling
Atomic optimal scheduling for an objective function
of φ(·) is formulated as follows:
minimize
fn∈Fn,∀n∈N
φ (L (f)) .
where
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)
n∈N
γ
op
n ((h−s) modH)


s∈Sn
fs
nIRn(s)(h)


.
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.
28 / 55

59. Atomic Optimal Scheduling
Atomic optimal scheduling for an objective function
of φ(·) is formulated as follows:
minimize
fn∈Fn,∀n∈N
φ (L (f)) .
where
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)
n∈N
γ
op
n ((h−s) modH)


s∈Sn
fs
nIRn(s)(h)


.
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.
28 / 55

60. 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
Conclusions
29 / 55

61. Relaxed Atomic Optimal Scheduling
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∈ ˆFn,∀n∈N
φ (L (f))
where
ˆFn = fn
s∈Sn
fs
n=1, 0 ≤ fs
n ≤ 1, ∀s∈Sn, fs
n=0, ∀s∈HSn .
For a convex objective function, this problem
becomes convex because ˆFn is now a convex set. It
can be solved efficiently, for instance, using the
well-known interior-point method [1].
30 / 55

62. Relaxed Atomic Optimal Scheduling
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∈ ˆFn,∀n∈N
φ (L (f))
where
ˆFn = fn
s∈Sn
fs
n=1, 0 ≤ fs
n ≤ 1, ∀s∈Sn, fs
n=0, ∀s∈HSn .
For a convex objective function, this problem
becomes convex because ˆFn is now a convex set. It
can be solved efficiently, for instance, using the
well-known interior-point method [1].
30 / 55

63. Relaxed Atomic Optimal Scheduling
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∈ ˆFn,∀n∈N
φ (L (f))
where
ˆFn = fn
s∈Sn
fs
n=1, 0 ≤ fs
n ≤ 1, ∀s∈Sn, fs
n=0, ∀s∈HSn .
For a convex objective function, this problem
becomes convex because ˆFn is now a convex set. It
can be solved efficiently, for instance, using the
well-known interior-point method [1].
30 / 55

64. Relaxed vs. Original Scheduling Problems
The relaxed atomic optimal scheduling problem is
not equivalent to the original problem.
The elements of the optimal solution from the relaxed
problem can take fractional values (e.g., 0.75).
The optimal solution of the relaxed problem,
however, provides a lower bound on the optimal
solution of the original problem.
The feasible set for the relaxed problem contains the
feasible set for the original problem.
31 / 55

65. Relaxed vs. Original Scheduling Problems
The relaxed atomic optimal scheduling problem is
not equivalent to the original problem.
The elements of the optimal solution from the relaxed
problem can take fractional values (e.g., 0.75).
The optimal solution of the relaxed problem,
however, provides a lower bound on the optimal
solution of the original problem.
The feasible set for the relaxed problem contains the
feasible set for the original problem.
31 / 55

66. Relaxed vs. Original Scheduling Problems
The relaxed atomic optimal scheduling problem is
not equivalent to the original problem.
The elements of the optimal solution from the relaxed
problem can take fractional values (e.g., 0.75).
The optimal solution of the relaxed problem,
however, provides a lower bound on the optimal
solution of the original problem.
The feasible set for the relaxed problem contains the
feasible set for the original problem.
31 / 55

67. Relaxed vs. Original Scheduling Problems
The relaxed atomic optimal scheduling problem is
not equivalent to the original problem.
The elements of the optimal solution from the relaxed
problem can take fractional values (e.g., 0.75).
The optimal solution of the relaxed problem,
however, provides a lower bound on the optimal
solution of the original problem.
The feasible set for the relaxed problem contains the
feasible set for the original problem.
31 / 55

68. 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.
32 / 55

69. 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.
32 / 55

70. 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.
32 / 55

71. 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.
32 / 55

72. Successive Convex Relaxation (Cont.)
4. For the rest of the elements, drop them and add zero
constraints from the smallest element up to ND
elements in total (including the one in step 3) as far as
the element is less than a dropping threshold (θD);
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.
33 / 55

73. Successive Convex Relaxation (Cont.)
4. For the rest of the elements, drop them and add zero
constraints from the smallest element up to ND
elements in total (including the one in step 3) as far as
the element is less than a dropping threshold (θD);
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.
33 / 55

74. Successive Convex Relaxation: An Example
Consider a simple case of N=2, H=4, and ND = 1.
Initial condition:
f = [ 0.25 0.25 0.25 0.25 | 0.3 0.2 0.25 0.25 ]
After 1st step:
f = [ 0.0 0.2 0.3 0.5 | 0.7 0.0 0.2 0.1 ]
After 2nd step:
f = [ 0.0 0.0 0.2 0.8 | 0.9 0.0 0.1 0.0 ]
Stop here. Solution found!
34 / 55

75. Successive Convex Relaxation: An Example
Consider a simple case of N=2, H=4, and ND = 1.
Initial condition:
f = [ 0.25 0.25 0.25 0.25 | 0.3 0.2 0.25 0.25 ]
After 1st step:
f = [ 0.0 0.2 0.3 0.5 | 0.7 0.0 0.2 0.1 ]
After 2nd step:
f = [ 0.0 0.0 0.2 0.8 | 0.9 0.0 0.1 0.0 ]
Stop here. Solution found!
34 / 55

76. Successive Convex Relaxation: An Example
Consider a simple case of N=2, H=4, and ND = 1.
Initial condition:
f = [ 0.25 0.25 0.25 0.25 | 0.3 0.2 0.25 0.25 ]
After 1st step:
f = [ 0.0 0.2 0.3 0.5 | 0.7 0.0 0.2 0.1 ]
After 2nd step:
f = [ 0.0 0.0 0.2 0.8 | 0.9 0.0 0.1 0.0 ]
Stop here. Solution found!
34 / 55

77. Successive Convex Relaxation: An Example
Consider a simple case of N=2, H=4, and ND = 1.
Initial condition:
f = [ 0.25 0.25 0.25 0.25 | 0.3 0.2 0.25 0.25 ]
After 1st step:
f = [ 0.0 0.2 0.3 0.5 | 0.7 0.0 0.2 0.1 ]
After 2nd step:
f = [ 0.0 0.0 0.2 0.8 | 0.9 0.0 0.1 0.0 ]
Stop here. Solution found!
34 / 55

78. 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
Conclusions
35 / 55

79. Energy Cost and PAR Minimization
Energy cost minization
minimize
fn∈ ˆFn,∀n∈N
h∈H
Ch (Lh(f)) .
PAR minimization
minimize
Γ,fn∈ ˆFn,∀n∈N
Γ
subject to Γ ≥ Lh(f), ∀h ∈ H.
Note that PAR minimization is formulated as a relaxed
linear program by introducing a new auxiliary variable
Γ.
36 / 55

80. Energy Cost and PAR Minimization
Energy cost minization
minimize
fn∈ ˆFn,∀n∈N
h∈H
Ch (Lh(f)) .
PAR minimization
minimize
Γ,fn∈ ˆFn,∀n∈N
Γ
subject to Γ ≥ Lh(f), ∀h ∈ H.
Note that PAR minimization is formulated as a relaxed
linear program by introducing a new auxiliary variable
Γ.
36 / 55

81. Energy Cost and PAR Minimization
Energy cost minization
minimize
fn∈ ˆFn,∀n∈N
h∈H
Ch (Lh(f)) .
PAR minimization
minimize
Γ,fn∈ ˆFn,∀n∈N
Γ
subject to Γ ≥ Lh(f), ∀h ∈ H.
Note that PAR minimization is formulated as a relaxed
linear program by introducing a new auxiliary variable
Γ.
36 / 55

82. 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
Conclusions
37 / 55

83. 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
Conclusions
38 / 55

84. 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.
An appliance is randomly selected for each user.
Constant operating energy levels (γop
) are assumed. 39 / 55

85. 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].
40 / 55

86. 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].
40 / 55

87. Performance Measures and Parameter Values
Performance measures
Lower bound: LB
Upper bound with ND: UB (ND)
Gap: G UB(D) − LB
Number of iterations
Parameter values for successive convex relaxation
Dropping threshold (θD): 0.1
Maximum number of fractional-valued elements that
can be dropped per iteration (ND): 1, 2, 5, 10
41 / 55

88. Performance Measures and Parameter Values
Performance measures
Lower bound: LB
Upper bound with ND: UB (ND)
Gap: G UB(D) − LB
Number of iterations
Parameter values for successive convex relaxation
Dropping threshold (θD): 0.1
Maximum number of fractional-valued elements that
can be dropped per iteration (ND): 1, 2, 5, 10
41 / 55

89. Performance Measures and Parameter Values
Performance measures
Lower bound: LB
Upper bound with ND: UB (ND)
Gap: G UB(D) − LB
Number of iterations
Parameter values for successive convex relaxation
Dropping threshold (θD): 0.1
Maximum number of fractional-valued elements that
can be dropped per iteration (ND): 1, 2, 5, 10
41 / 55

90. Performance Measures and Parameter Values
Performance measures
Lower bound: LB
Upper bound with ND: UB (ND)
Gap: G UB(D) − LB
Number of iterations
Parameter values for successive convex relaxation
Dropping threshold (θD): 0.1
Maximum number of fractional-valued elements that
can be dropped per iteration (ND): 1, 2, 5, 10
41 / 55

91. Performance Measures and Parameter Values
Performance measures
Lower bound: LB
Upper bound with ND: UB (ND)
Gap: G UB(D) − LB
Number of iterations
Parameter values for successive convex relaxation
Dropping threshold (θD): 0.1
Maximum number of fractional-valued elements that
can be dropped per iteration (ND): 1, 2, 5, 10
41 / 55

92. Performance Measures and Parameter Values
Performance measures
Lower bound: LB
Upper bound with ND: UB (ND)
Gap: G UB(D) − LB
Number of iterations
Parameter values for successive convex relaxation
Dropping threshold (θD): 0.1
Maximum number of fractional-valued elements that
can be dropped per iteration (ND): 1, 2, 5, 10
41 / 55

93. Performance Measures and Parameter Values
Performance measures
Lower bound: LB
Upper bound with ND: UB (ND)
Gap: G UB(D) − LB
Number of iterations
Parameter values for successive convex relaxation
Dropping threshold (θD): 0.1
Maximum number of fractional-valued elements that
can be dropped per iteration (ND): 1, 2, 5, 10
41 / 55

94. Performance Measures and Parameter Values
Performance measures
Lower bound: LB
Upper bound with ND: UB (ND)
Gap: G UB(D) − LB
Number of iterations
Parameter values for successive convex relaxation
Dropping threshold (θD): 0.1
Maximum number of fractional-valued elements that
can be dropped per iteration (ND): 1, 2, 5, 10
41 / 55

95. 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
Conclusions
42 / 55

96. 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
EnergyCost[USD] LB
GO
UB(1)
UB(2)
UB(5)
UB(10)
43 / 55

97. 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
PARinAggregatedLoad LB
GO
UB(1)
UB(2)
UB(5)
UB(10)
44 / 55

98. 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
Conclusions
45 / 55

99. Cost Minimization: Upper/Lower Bounds
2 10 20 30 40 50
0
1
2
3
4
5
N
EnergyCost[USD]
LB
UB(1)
UB(2)
UB(5)
UB(10)
46 / 55

100. 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
47 / 55

101. Cost Minimization: Number of Iterations
2 10 20 30 40 50
0
200
400
600
800
1,000
N
NumberofIterations
UB(1)
UB(2)
UB(5)
UB(10)
48 / 55

102. 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
Conclusions
49 / 55

103. Par Minimization: Upper/Lower Bounds
2 10 20 30 40 50
0
5
10
15
20
25
N
PARinAggregatedLoad
LB
UB(1)
UB(2)
UB(5)
UB(10)
50 / 55

104. 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
51 / 55

105. Par Minimization: Number of Iterations
2 10 20 30 40 50
0
200
400
600
800
1,000
N
NumberofIterations
UB(1)
UB(2)
UB(5)
UB(10)
52 / 55

106. 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
Conclusions
53 / 55

107. Conclusions
We have provided a new formulation of appliance energy
consumption scheduling based on the optimal routing
framework.
It guarantees the atomicity of resulting scheduled
energy consumption.
We have also provided an efficient solution technique
based on successive convex relaxation for a
Boolean-convex problem resulting from a convex
objective function.
It enables us to carry out systematic analysis of the
original problem with both upper and lower bounds.
Possible extensions to the current work include
Distributed atomic energy consumption scheduling;
Advanced techniques refining the feasible region at
each iterative step to reduce the total number of
iterations.
54 / 55

108. Conclusions
We have provided a new formulation of appliance energy
consumption scheduling based on the optimal routing
framework.
It guarantees the atomicity of resulting scheduled
energy consumption.
We have also provided an efficient solution technique
based on successive convex relaxation for a
Boolean-convex problem resulting from a convex
objective function.
It enables us to carry out systematic analysis of the
original problem with both upper and lower bounds.
Possible extensions to the current work include
Distributed atomic energy consumption scheduling;
Advanced techniques refining the feasible region at
each iterative step to reduce the total number of
iterations.
54 / 55

109. Conclusions
We have provided a new formulation of appliance energy
consumption scheduling based on the optimal routing
framework.
It guarantees the atomicity of resulting scheduled
energy consumption.
We have also provided an efficient solution technique
based on successive convex relaxation for a
Boolean-convex problem resulting from a convex
objective function.
It enables us to carry out systematic analysis of the
original problem with both upper and lower bounds.
Possible extensions to the current work include
Distributed atomic energy consumption scheduling;
Advanced techniques refining the feasible region at
each iterative step to reduce the total number of
iterations.
54 / 55

110. Conclusions
We have provided a new formulation of appliance energy
consumption scheduling based on the optimal routing
framework.
It guarantees the atomicity of resulting scheduled
energy consumption.
We have also provided an efficient solution technique
based on successive convex relaxation for a
Boolean-convex problem resulting from a convex
objective function.
It enables us to carry out systematic analysis of the
original problem with both upper and lower bounds.
Possible extensions to the current work include
Distributed atomic energy consumption scheduling;
Advanced techniques refining the feasible region at
each iterative step to reduce the total number of
iterations.
54 / 55

111. Conclusions
We have provided a new formulation of appliance energy
consumption scheduling based on the optimal routing
framework.
It guarantees the atomicity of resulting scheduled
energy consumption.
We have also provided an efficient solution technique
based on successive convex relaxation for a
Boolean-convex problem resulting from a convex
objective function.
It enables us to carry out systematic analysis of the
original problem with both upper and lower bounds.
Possible extensions to the current work include
Distributed atomic energy consumption scheduling;
Advanced techniques refining the feasible region at
each iterative step to reduce the total number of
iterations.
54 / 55

112. Conclusions
We have provided a new formulation of appliance energy
consumption scheduling based on the optimal routing
framework.
It guarantees the atomicity of resulting scheduled
energy consumption.
We have also provided an efficient solution technique
based on successive convex relaxation for a
Boolean-convex problem resulting from a convex
objective function.
It enables us to carry out systematic analysis of the
original problem with both upper and lower bounds.
Possible extensions to the current work include
Distributed atomic energy consumption scheduling;
Advanced techniques refining the feasible region at
each iterative step to reduce the total number of
iterations.
54 / 55

113. Conclusions
We have provided a new formulation of appliance energy
consumption scheduling based on the optimal routing
framework.
It guarantees the atomicity of resulting scheduled
energy consumption.
We have also provided an efficient solution technique
based on successive convex relaxation for a
Boolean-convex problem resulting from a convex
objective function.
It enables us to carry out systematic analysis of the
original problem with both upper and lower bounds.
Possible extensions to the current work include
Distributed atomic energy consumption scheduling;
Advanced techniques refining the feasible region at
each iterative step to reduce the total number of
iterations.
54 / 55

114. 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.
55 / 55