This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 Unported License.!
See http://creativecommons.org/licenses/by-nc-nd/3.0/!
©  July 13, 2010 , P. R. Kumar !
©  July 13, 2010 , P. R. Kumar !
Variability of Wind!
Demand Response with Stochastic
Renewables and Inertial Thermal Loads:
A Study of Some Fundamental Issues in
Modeling and Optimization!
Goal: Use renewable energy – wind!
◆  Problem: Highly variable – “stochastic”!
◆
!
Gaurav Sharma, Le Xie and P. R. Kumar!
!
!
Dept. of Electrical and Computer Engineering!
Texas A&M University!
Email: prk.tamu@gmail.com
Web: http://cesg.tamu.edu/faculty/p-r-kumar/!
NINTH ANNUAL CARNEGIE MELLON CONFERENCE
ON THE ELECTRICITY INDUSTRY!
CMU!
February 4, 2014!
1/40!
2/40!
©  July 13, 2010 , P. R. Kumar !
Demand Response!
©  July 13, 2010 , P. R. Kumar !
Limited capability of demand response!
◆
Adjust demand to match supply!
◆
Inertial thermal loads – building air conditioners!
◆
Renewable energy may not be enough to satisfy
load requirements!
Need to turn on
air-conditioner!
–  Air conditioner can be switched off for a short while
without loss of comfort!
–  Traditionally under thermostatic control!
Tmax!
Tmax!
Tmin!
Tmin!
◆
So there are limitations to demand response!
3/40!
4/40!
©  July 13, 2010 , P. R. Kumar !
Variability of power demand not met
by renewables!
◆
Several Questions – 1!
Residual power demand not met by renewables!
To what extent can demand of inertial loads be
met by renewable sources?!
◆  How does flexibility of load requirements, such as
comfort level settings, influence how much
renewable power can be used?!
◆  How much flexibility can be extracted from
thermal inertial loads for maximum utilization of
variable generation such as wind?!
◆
Time!
◆
Prefer less variability so that operating reserve
requirements are less!
Time!
©  July 13, 2010 , P. R. Kumar !
5/40!
6/40!
©  July 13, 2010 , P. R. Kumar !
©  July 13, 2010 , P. R. Kumar !
Several Questions – 2!
Several Questions – 3!
To what extent can operating reserve required be
minimized?!
◆  How beneficial is “demand pooling”?!
◆  Can we come up with quantitative answers?!
◆  How can demand “pooling” be done?!
◆  What are the communication requirements?!
◆  How much information exchange is needed
between suppliers and consumers?!
◆
What are the privacy implications? !
Does it require intrusive sensing?!
◆  How distributed can the solution be?!
◆  How tractable (computational complexity) is the
solution?!
◆  How robust is the solution?!
◆  How implementable is it?!
◆
◆
Role of model features, cost functions, stochasticity
assumptions, convexity, asymptotics, etc!
7/40!
8/40!
©  July 13, 2010 , P. R. Kumar !
©  July 13, 2010 , P. R. Kumar !
Load Aggregator!
Load service entity
a.k.a. Load aggregator
Features of problem!
•  Acts as coordinator
•  Possibly monitors
temperature of A/Cs
•  Possibly controls power
to cool A/Cs
•  Appears as aggregated
load
9/40!
10/40!
©  July 13, 2010 , P. R. Kumar !
Wind model!
◆
©  July 13, 2010 , P. R. Kumar !
Temperature dynamics !
Model wind as a finite state Markov process!
xi (t)
Temperature of load i
Wind power!
! given to load i
Power
Time
Time!
!
◆
Even simpler model for illustration!
ON!
OFF!
◆
Inertial thermal load (A/C) dynamics!
–  On-Off process!
ẋi (t) = hi (t)
W
Wind power!
Time!
Pi (t)
»  Rate of change in temperature = Ambient heating – Power for
cooling.!
11/40!
!
12/40!
©  July 13, 2010 , P. R. Kumar !
Stochastic control problem:
Comfort violation probability!
User specified comfort range !
Range of comfortable temperature ![⇥min , ⇥max ]
◆  Either: Enforce hard constraint!
◆
◆
Minimize the probability of leaving a user
specified comfort range ! [⇥min , ⇥max ]
!
⇥max
⇥min
!
!
©  July 13, 2010 , P. R. Kumar !
Or: Penalize the violations !
◆
!
◆
Temperature dynamics ! ẋi (t) = hi
!
◆
⇥max
⇥min
13/40!
t
Piw (t) ⇠ Markov process
Wind process!
t
Allow but penalize the violation !
X
◆
Cost function!
!
!
1
T !1 T
lim
Z
T
0
X
Piw (t)
I(xi (t) > ⇥max )dt
i
14/40!
©  July 13, 2010 , P. R. Kumar !
Optimal policy in comfort violation
probability model!
!
!
! Temperature!
!
!
Stochastic control problem:
Variance minimization!
!  Stochastic
Load 2!
⇥max
⇥min
©  July 13, 2010 , P. R. Kumar !
control problem:!
Z T
– Cost function!
Load 1!
lim
T !1
1
T
Wind power!
!
!
!
0
X
[(xi (t)
⇥max )+ ]2 dt
i
Time!
!  Theorem:
Optimal policy “synchronizes” loads!
Theorem: Provide power to the coolest load that is above the
temperature range. !
!
◆
Temperature!
Issue: Unfair, temperatures of some loads will remain higher
than others!
⇥max
⇥min
Loads will remain synchronized !
after this time instant !
Load 2!
Load 1!
Wind power!
Time!
◆
Possible solution: Minimize the variance of comfort violation 15/40
! !
16/40!
!
!
©  July 13, 2010 , P. R. Kumar !
Requirement for reserves (of nonrenewable power) !
◆
◆
Cost function for reducing operating
reserves!
◆
Temperatures can go very high occasionally!
Temperature!
⇥max
⇥min
©  July 13, 2010 , P. R. Kumar !
Desire low operating reserve requirements!
Load 2!
More variability!
Less variability!
Load 1!
Hard constraints require reliable non-renewable
source !
Use non-renewable power to!
t!
Prefer this!
t!
Need to turn on air-conditioner,
maintain constraint !
but no wind available!
◆
⇥max
⇥min
17/40!
Impose a quadratic cost on non-renewable power usage !
Z X
(
Pin (t))2 dt
18/40!
©  July 13, 2010 , P. R. Kumar !
©  July 13, 2010 , P. R. Kumar !
Stochastic control problem: Reduction of
variability with temperature constraint!
Stochastic control Problem:!
X
Piw (t) ⇠ Markov process
◆  Wind process!
!
◆  Temperature dynamics ! ẋi (t) = hi
Piw (t)
n
Non-renewable power! Pi (t)
◆
Optimal solution: Reduction of
variability with temperature constraint!
◆
Loads will remain synchronized !
after this time instant !
⇥max
Pin (t)
!
!
!
1
T !1 T
lim
Z
T
[
0
Load 1!
⇥min
!
!
0
Quadratic cost to reduce variability!
◆
Load 2!
Temperature!
Temperature constraint! xi (t) 2 [⇥min , ⇥max ], 8i
◆
Theorem: Optimal policy still synchronizes loads!!
!
X
Pin (t)]2 dt
i
◆
Counter-intuitive??!
◆
Question: Is there some modification in the model or cost
function which leads to de-synchronization ?!
19/40!
20/40!
©  July 13, 2010 , P. R. Kumar !
How to induce desynchronization:
Markov model for changes in Θmax !
! Stochastic control problem: Stochastic
! variation of temperature constraints!
! ◆  Wind process:! X P w (t) ⇠ Markov process
i
!
Suppose users occasionally change ⇥max
setting at the same time !
◆
◆
–  E.g. Super Bowl Sundays @ game time.!
! ◆
!!
E.g. ⇥max is a two state Markov process!
◆
©  July 13, 2010 , P. R. Kumar !
1/⌧h
Comfort range!
⇥min
1/⌧l
!
t
Pin (t)
0
Stochastic comfort level! ⇥max (t) ⇠ Markov process , ⇥max (t) 2 {⇥1max , ⇥2max }
◆
Temperature constraint:! xi (t) 2 [⇥min , ⇥max ], 8i
!
⇥1max
⇥2max
n
Non-renewable power ! Pi (t)
Piw (t)
◆
!
⇥2max
⇥1max
Temperature dynamics:! ẋi (t) = hi
2
◆
Maximum cooling rate:! Pin (t) = M If xi (t) > ⇥max (t)
◆
Quadratic cost:!
21/40!
1
T !1 T
lim
Z
T
[
0
X
Pin (t)]2 dt
i
22/40!
!
©  July 13, 2010 , P. R. Kumar !
Optimal de-synchronization and resynchronization!
Vector field of optimal solution!
It is optimal to break symmetry at high
temperatures !
◆
Nature of optimal solution!
2
max
!
1
max
Temperature load 2
–  De-synchronization at high temperatures!
–  Re-synchronization at low temperatures!
–  Hedges against the future eventuality that the
thermostats are switched low!
Temperature
◆
©  July 13, 2010 , P. R. Kumar !
⇥min
Time
Temperature load 1
Vector field of temperature changes!
23/40!
24/40!
©  July 13, 2010 , P. R. Kumar !
Local concavity/convexity of optimal
cost-to-go resulting from HJB equation!
A heuristic!
◆
Keep the temperatures apart!
Locally concave!
©  July 13, 2010 , P. R. Kumar !
900
»  Provide power to load with minimum temperature amongst all loads with
temperature higher than !✓2
»  Bring the temperatures together for loads in[⇥
! min , ✓2 ]
Cost to go
800
700
600
500
Bring the temperatures!
together!
400
300
Approximate optimal policy !
–  De-synchronization above temperature!
1000
–  Power is assumed affine in [⇥min , ✓2 ] and [✓!2 , ⇥max ]
–  Policy is a function of a few parameters, optimize iteratively!
200
100
0
80
60
40
Load 2
Two loads optimal solution along!
x1 = x2
20
0
0
10
20
30
40
50
60
70
80
90
100
Load 1
Non−renewable power
100
⇥min
!
But optimal policy is difficult to compute!
25/40!
◆
✓2
⇥2max
Temperature
But requires intrusive sensing!
26/40!
©  July 13, 2010 , P. R. Kumar !
©  July 13, 2010 , P. R. Kumar !
Thermostatic control with set points!
Z1
A possibly implementable
architecture of a solution!
Load service
entity
- “Senses” wind ddpower
- “Sets” set
ddpoints Zi
Z2
ZN
27/40!
28/40!
©  July 13, 2010 , P. R. Kumar !
Control policy!
©  July 13, 2010 , P. R. Kumar !
Information flow in architecture!
◆
–  Information from LSE to consumer!
–  Minimal information needed to be responsive?!
Wind not blowing!
Zi!
Wind blowing or not = “Price signal”!
◆
LSE need not “set” thermostat set-points!
–  Only needs to set empirical distribution of set-points!
–  Not detailed actuation!
Ambient
temperature
rise!
◆
t!
0!
Cooling
using
“wind”!
◆
No flow of state information from home to LSE!
Information and communication requirements!
–  Price signal to consumers!
–  Infrequent distribution signaling to consumers!
Cooling using
“non-renewable”!
29/40!
◆
(LSE also monitors total power usage by consumers)!
30/40!
©  July 13, 2010 , P. R. Kumar !
©  July 13, 2010 , P. R. Kumar !
Overall optimization problem!
◆
N
X
Stochastic Wind process:!
1
◆
Overall optimization problem!
Piw ⇠ Wt
◆
1
Temperature dynamics:!
2
Z T
4 lim 1
Min
◆  Cost:!
T !1 T 0
N
X
Pig (t)
1
!2
+
N
X
[(xi (t)
1
!
Grid power variation cost!
N
X
Stochastic Wind process:!
3
+ 2 5
⇥M
i (t)) ] dt
◆
Temperature dynamics:!
◆
Comfort setting dynamics:!
2
Z T
4 lim 1
Min
◆  Cost:!
T !1 T 0
N
X
Piw ⇠ Wt
Pig (t)
1
!2
+
N
X
[(xi (t)
1
!
Grid power variation cost!
User discomfort cost!
3
+ 2 5
⇥M
i (t)) ] dt
User discomfort cost!
31/40!
32/40!
©  July 13, 2010 , P. R. Kumar !
©  July 13, 2010 , P. R. Kumar !
Overall optimization problem!
◆
Continuum limit of Z-policy!
How2to choose {Z1 , Z2 , . . . , ZN } so as to minimize:
!
3
Continuum of loads in [0,1]!
u(x)= fraction of loads with set-points less than x,
! = empirical distribution of set-points!
◆  Cost function!
◆
!2
Z
N
N
X
!
1 T X g
+ 2 5
Min 4 lim
Pi (t) +
[(xi (t) ⇥M
i (t)) ] dt
T !1 T 0
!
1
1
◆  Difficult:!
◆
–  Complex as N is large, high dimensional. !
–  Need to solve different problem for different N!
◆
C
[0,1]
(u) =
Solution:!
Z ⇥2
–  Study asymptotic limit as !N ! 1.
[0,1]
(u) =
(z)u0 (z)dz
–  Solution becomes explicit!! C
0
–  And asymptotic solution is also nearly optimal even
for
small N!
2
+ (h) (
⇥2
+
Z
Z
⇥2
0
⇥2
0
2
(z)u (z)dz + (h) (
⇥2
+
0
Z
⇥2
u2 (z)P({Xz = z} \ {Xz+dz < z + dz}))
–  Masses at set points!
33/40!
34/40!
©  July 13, 2010 , P. R. Kumar !
◆
Resulting
variational
problem:!
u0⇤ (z)
Solution to variational problem (Euler-Lagrange):!
@F
@u
◆
d @F
= 0 ) 2(h)2 u(x)D(x)
dx @u0
Not so fast, singularity:!
d
dx
(x) = 0 ) u(x) =
0
(x)
2(h)2 D(x)
@2F
=0
@u02
!
◆
©  July 13, 2010 , P. R. Kumar !
Optimal solution of continuum limit!
Continuum limit optimization problem!
◆
u2 (z)P({Xz
0
Solution is given by: ! u⇤ (z) =
(
⇣
0
(z)
min 1, 2(h)2 D(z)
1
⌘
⇥1
⇥2
Z!
Optimal desynchronization
of demand response!
If z < ⇥2
If z = ⇥2 .
35/40!
36/40!
©  July 13, 2010 , P. R. Kumar !
Z-policy: Finite population approximation
from continuum limit asymptotic!
◆
©  July 13, 2010 , P. R. Kumar !
Some simulation results!
This appears to work very well even when N
is small !
◆  Even N = 5!
◆
Generate {Zi }N
1 to approximate continuum
limit!
1
0.8
Approximation based finite distribution
Optimal finite distribution
0.6
Variation cost
0.4
0.2
0
0.4
0.2
0
0.1
10
20
30
40
50
60
70
80
90
0.5
1
1.5
2
2.5
Time
3
3.5
0
0.5
1
1.5
2
2.5
Time
3
3.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
4
4.5
5
4
4.5
Optimal temperature variation cost 4
Variation cost for generated Z x 10
0.04
0.02
100
Setpoint, Z
0
0.06
0
0.8
0
Optimal grid cost
Grid cost for generated Z
0.6
0.08
0.6
Total cost
Distribution
Nonrenewable cost
Optimal infinite case distribution
0.8
Optimal total cost
4
Total cost for generated x
Z 10
0.4
0.2
0
Time
37/40!
5
4
x 10
38/40!
©  July 13, 2010 , P. R. Kumar !
©  July 13, 2010 , P. R. Kumar !
Concluding remarks!
◆
Attempt to develop an architecture and
tractable solution for demand response!
◆
Many extensions needed and feasible!
–  Response to comfort variations!
–  Availability of wind power!
–  Generalize wind model, temperature dynamics,
etc.!
Thank you!
39/40!
40/40!