Load-side
Frequency Control
Changhong Zhao
Enrique Mallada
Steven Low
EE, Caltech
Ufuk Topcu
Lina Li
Elec & Sys Engr
U Penn
LIDS
MIT
January 2014
Outline
Motivation
Dynamic network model
Load-side frequency control
Simulations
Zhao, Topcu, Li, Low, TAC 2014
Mallada, Low, 2013
Motivation: frequency control
Synchronous network
! All buses synchronized to same nominal
frequency (US: 60 Hz)
! Supply-demand imbalance " frequency
fluctuation
Frequency regulation
! Generator based
! Frequency sensitive (motor-type) loads
Controllable loads
! Do not react to frequency deviation
! … but intelligent
! Need active control – how?
Frequency control
Frequency control is traditionally done on generation side
secondary
freq control
economic
dispatch
unit
commitment
primary
freq control
sec
min
dynamic model
e.g. swing eqtn
5 min
60 min
day
power flow model
e.g. DC/AC power flow
year
Advantages of load-side control
Distributed loads can supplement
generator-side control
!
!
!
!
secondary
freq control
primary
freq control
sec
min
dynamic model
e.g. swing eqtn
5 min
faster (no/low inertia!)
no waste or emission
more reliable (large #)
localize disturbance
60 min
day
year
It’s about supply-demand balance, but synchronous
frequency helps
Idea dates back to 1970s
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Homeostatic utility control :
• freq adaptive loads
• spot prices
FIGURE 1: The Energy Marketplace
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• IT infrastructure
acts as a broker for the electricity. The Marketing
computational
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rces
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der
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y generators.
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g frequency and duration of an FRfrequency setting schemes are
rations of the FR-GFA are then
results.
dlyTM appliances, load frequency
ency regulation, frequency response,
nagement, automated load control.
NTRODUCTION
(a)
ces suchUS:
as frequency regulation,
operating
reserve:
13%
nning reserves
were
provided
byof peak
total GFA
capacity:
18%
gency where
the system
frequency
hold, under-frequency relays are
tore the load-to-generator balance.
ms, the services provided may be
Lu can
& Hammerstrom
ad control
play a role (2006),
very PNNL
• Residential load accounts
for ~1/3 of peak demand
• 61% residential appliances
are Grid Friendly
(a)
Small demo: PNNL
PNNL Grid Friendly Appliance Demo Project
(early 2006 – March 2007)
•
•
•
•
150 clothes dryers, 50 water heaters
Under-frequency threshold: 59.95 Hz (0.08% dev)
358 under-freq events during project, lasting secs – 10 mins
All GFA detected events correctly and loads shedded as
designed, despite wide geographical distribution
• Survey reported no customer inconvenience
PNNL-17079
from the controlled appliance. Remaining output pins were assigned to facilitate testing and
troubleshooting, but these additional signals were not used for appliance control.
Figure 1.3. GFA Controller Board used in the Grid Friendly Appliance Project
Hammerstrom et al (2007), PNNL
Can household Grid Friendly
appliances follow its own PV
production?
•
•
•
•
60,000 AC
avg demand ~ 140 MW
wind var: +- 40MW
temp var: 0.15 degC
Dynamically adjust
Fig. 8. Hysteresis-based
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Fig. 7. Load control example for balancing variability from
intermittent renewable generators, where the end-use functionVin
this case, thermostat setpointVis used as the input signal.
availability of TCLs for control. Fo
drop in ambient temperature woul
Callaway, Hiskens (2011)
fewer air
conditioning loads. Sy
Callaway (2009)
need to take account of such tem
Outline
Motivation
Dynamic network model
Load-side frequency control
Simulations
Zhao, Topcu, Li, Low, TAC 2014
Mallada, Low, 2013
Network model
j
reactance
xij
generation
i
Pi m
di + d̂i
loads:
controllable + freq-sensitive
i : bus/control area/balancing authority
Network model
j
i
Pij
Pi m
di + d̂i
DC approximation
! Lossless network (r=0)
! Fixed voltage magnitudes
! Reactive power ignored
! Do not assume small angle difference
Dynamic model
j
Swing equation on bus i
frequency
mechanical
power
M iω i = Pi − Pi
m
electrical
power
e
i
Pij
Pi m
di + d̂i
! Newton’s 2nd law
! Variables: deviations from nominal values
Dynamic model
j
Swing equation on bus i
i
Pij
Pi m
M iω i = Pi − Pi
m
di + d̂i
e
e
Pi := di + Diωi +
∑P
ij
i~ j
controllable
loads
freq-sens
loads
branch
power flow
Dynamic model
j
Swing equation on bus i
i
Pij
Pi m
m
e

M iωi = Pi − Pi
di + d̂i
e
Pi := di + Diωi +

Pij = bij (ωi − ω j )
bij = 3
Vi V j
xij
0
0
cos θ i − θ j
(
∑P
ij
i~ j
)
linearization around nominal
Network model
Generator bus (may contain load):
'
1 $
m

&
ωi = − & di + Diωi − Pi + ∑ P ij −∑ Pji ))
Mi %
(
i→ j
j→i
Load bus (no generator):
0 = di + Diωi − Pi + ∑ P ij −∑ Pji
m
i→ j
j→i
Real branch power flow:
Pij = bij (ωi − ω j )
∀ i→ j
swing dynamics
Outline
Motivation
Dynamic network model
Load-side frequency control
Simulations
Zhao, Topcu, Li, Low, TAC 2014
Mallada, Low, 2013
Frequency control
'
1 $
m

&
ωi = − & di + Diωi − Pi + ∑ P ij −∑ Pji ))
Mi %
(
i→ j
j→i
0 = di + Diω i − Pi + ∑ P ij −∑ Pji
m
i→ j
Pij = bij (ω i − ω j )
j→i
∀ i→ j
Suppose the system is in steady state
ω i = 0 Pij = 0
and suddenly …
Frequency control
Given: disturbance in gens/loads
Current: adapt remaining generators Pi
m
! to re-balance power
! (and restore nominal freq, zero ACE)
Our goal: adapt controllable loads di
! to re-balance power
! while minimizing disutility of load control
Frequency control
'
1 $
m

&
ωi = − & di + Diωi − Pi + ∑ P ij −∑ Pji ))
Mi %
(
i→ j
j→i
0 = di + Diω i − Pi + ∑ P ij −∑ Pji
m
i→ j
Pij = bij (ω i − ω j )
proposed
approach
j→i
∀ i→ j
current
approach
this talk: ignores generator-side control
Load-side controller design
'
1 $
m

&
ωi = − & di + Diωi − Pi + ∑ P ij −∑ Pji ))
Mi %
(
i→ j
j→i
0 = di + Diω i − Pi + ∑ P ij −∑ Pji
m
i→ j
Pij = bij (ω i − ω j )
j→i
∀ i→ j
How to design feedback control law
di = Fi (ω (t), P(t))
Load-side controller design
'
1 $
m

&
ωi = − & di + Diωi − Pi + ∑ P ij −∑ Pji ))
Mi %
(
i→ j
j→i
0 = di + Diω i − Pi + ∑ P ij −∑ Pji
m
i→ j
Pij = bij (ω i − ω j )
j→i
∀ i→ j
Control goals
Zhao, Topcu, Li, Low
TAC 2014
Mallada, Low 2013
!
!
!
!
Rebalance power
Resynchronize/stabilize frequency
Restore nominal frequency
Restore scheduled inter-area flows
Load-side controller design
'
1 $
m

&
ωi = − & di + Diωi − Pi + ∑ P ij −∑ Pji ))
Mi %
(
i→ j
j→i
0 = di + Diω i − Pi + ∑ P ij −∑ Pji
m
i→ j
Pij = bij (ω i − ω j )
j→i
∀ i→ j
Desirable properties of
di = Fi (ω (t), P(t))
! simple, scalable
! decentralized/distributed
Load-side controller design
'
1 $
m

&
ωi = − & di + Diωi − Pi + ∑ P ij −∑ Pji ))
Mi %
(
i→ j
j→i
0 = di + Diω i − Pi + ∑ P ij −∑ Pji
m
i→ j
Pij = bij (ω i − ω j )
j→i
∀ i→ j
Proposed approach: forward engineering
! formalize control goals into OLC objective
! derive local control as distributed solution
Outline
Motivation
Dynamic network model
Load-side frequency control
! Primary control
! Secondary control
Simulations
Zhao, Topcu, Li, Low, TAC 2014
Optimal load control (OLC)
controllable
load
min
over
s. t.
uncontrollable
load
!
1 2$
∑#" ci (di ) + 2D d̂i &%
i
i
loads dl ∈ )*d l , dl +,, d̂i
∑(
i
)
di + d̂i = ∑ Pi m
i
demand = supply
across network
Optimal load control (OLC)
controllable
load
min
over
s. t.
uncontrollable
load
!
1 2$
∑#" ci (di ) + 2D d̂i &%
i
i
loads dl ∈ )*d l , dl +,, d̂i
∑(
i
)
di + d̂i = ∑ Pi m
i
demand = supply
across network
Optimal load control (OLC)
controllable
load
min
over
s. t.
uncontrollable
load
!
1 2$
∑#" ci (di ) + 2D d̂i &%
i
i
loads dl ∈ )*d l , dl +,, d̂i
∑(
i
)
di + d̂i = ∑ Pi m
i
disturbances
demand = supply
across network
Punchline
Theorem
swing dynamics
+ frequency-based load control
= primal-dual algorithm that solves OLC
!
!
!
!
Completely decentralized
Not need explicit communication
Not need detailed network data
Exploit free global control signal
… reverse engineering swing dynamics
Recall OLC
controllable
load
min
over
s. t.
freq-sens
load
!
1 2$
∑#" ci (di ) + 2D d̂i &%
i
i
loads dl ∈ )*d l , dl +,, d̂i
∑(
i
)
di + d̂i = ∑ Pi m
i
demand = supply
across network
Punchline
swing dynamics (recap)
'
1 $
ω i = − && di (t) + Diωi (t) − Pi m + ∑ Pij (t) − ∑ Pji (t)))
Mi %
(
i→ j
j→i
Pij = bij (ωi (t) − ω j (t))
implicit
load control
di (t) := "#c
'−1
i
di
(ωi (t))$%d
i
active control
Punchline
Theorem
(
)
system trajectory d(t), d̂(t), ω (t), P(t)
converges to
!
!
!
(
(d , d̂ , ω , P )
*
*
*
*
as t → ∞
)
d *, d̂ * is unique optimal load control
ω * is unique optimal for DOLC
*
P is optimal for dual of DOLC
Zhao, Topcu, Li, Low, TAC 2014
Punchline
Theorem
(
)
system trajectory d(t), d̂(t), ω (t), P(t)
converges to
!
!
!
(
(d , d̂ , ω , P )
*
*
*
*
as t → ∞
)
d *, d̂ * is unique optimal load control
ω * is unique optimal for DOLC
*
P is optimal for dual of DOLC
Load-side primary frequency control works !
Zhao, Topcu, Li, Low, TAC 2014
Implications
! Freq deviations contains right info on
global power imbalance for local decision
! Decentralized load participation in
primary freq control is stable
!
*
ω : Lagrange multiplier of OLC
info on power imbalance
!
*
P : Lagrange multiplier of DOLC
info on freq asynchronism
Recap: control goals
Yes
Yes
No
No
! Rebalance power
! Resynchronize/stabilize frequency
(
)
! Restore nominal frequency ω * ≠ 0
! Restore scheduled inter-area flows
Proposed approach: forward engineering
! formalize control goals into OLC objective
! derive local control as distributed solution
Outline
Motivation
Dynamic network model
Load-side frequency control
! Primary control
! Secondary control
Simulations
Mallada, Low, 2013
Freq preserving OLC
min
s. t.
!
1 2$
∑#" ci ( di ) + 2D d̂i &%
i
i
∑(
i
)
di + d̂i = ∑ Pi m
demand = supply
across network
i
min
!
1 2$
∑#" ci ( di ) + 2D d̂i &%
i
i
s. t.
di + d̂i = Pi m − ∑ Cie Pe
demand = supply
per bus
e∈E
di
= Pi m − ∑ Cie Re
e∈E
to restore nominal
frequency
Recall primary control for OLC
swing dynamics:
'
1 $
m
ω i = −
& di (t) + Diω i (t) − Pi + ∑ Cie Pe (t))
Mi %
(
e∈E
Pij = bij (ω i (t) − ω j (t))
load control:
di (t) := "#c
'−1
i
implicit
di
(ωi (t))$%d i
active
control
Recall primary control for OLC
swing dynamics:
'
1 $
m
ω i = −
& di (t) + Diω i (t) − Pi + ∑ Cie Pe (t))
Mi %
(
e∈E
Pij = bij (ω i (t) − ω j (t))
load control:
implicit
di (t) := "#c
'−1
i
di
(ωi (t) + λ (t)i )$%d
i
computation & communication:
$
'
m
λi = −γ i & di (t) − Pi + ∑ Cie Re (t)),
%
(
e∈E
Rij = aij ( λi (t) − λ j (t))
Punchline
Theorem
(
)
system trajectory d(t), d̂(t), ω (t), P(t)
converges to
!
(
!
ω* = 0
d *, d̂ *
(d , d̂ , ω , P )
*
*
*
*
as t → ∞
) is unique optimal load control
Load-side secondary frequency control works !
Mallada, Low 2014
Recap: control goals
Yes
Yes
Yes
No
No
! Rebalance power
! Resynchronize/stabilize frequency
(
)
! Restore nominal frequency ω * ≠ 0
! Restore scheduled inter-area flows
Secondary control restores nominal
frequency but requires communication
with neighbors
Outline
Motivation
Dynamic network model
Load-side frequency control
Simulations
Zhao, Topcu, Li, Low, TAC 2014
Mallada, Low, 2013
Simulations
Dynamic simulation of IEEE 68-bus system
• Power System Toolbox (RPI)
• Detailed generation model
• Exciter model, power system
stabilizer model
• Nonzero resistance lines
Simulations
59.964 Hz
ERCOT threshold
for freq control
Simulations
- 4.5%
- 7.0%
.r,3.
Frequency
evolution
by definition
of F
,P =
P (0) + BC ✓ andusing OLC controllers of (Z
n
et al.,
2013)
Simulations
0) + BC ✓)
=P
d
()
(37)
T
m
⇤
m
⇤
25
20
CBC ✓ = P
d
CP (0).
(38)
hat except for 60.05
✓ the terms of (38) are fixed. The
T
CBC is a Laplacian matrix with null space given
CBC T ) = span (1). Thus, given ✓1 6= ✓2 both
60
g (38) we must have
(✓1 ✓2 ) 2 span(1).
P 1 = P (0) + BC T ✓1 and P 2 = P (0) + BC T ✓2 .
59.95
ωi
P 2 = (P (0) + BC T ✓1 )
1
(P (0) + BC T ✓1 )
1
= BC T (✓59.9
✓2 ) = 0.
re, there is a unique vector P such that (x, ) 2
argument also shows that there is
(0) . A similar 59.85
nique R such that (x, ) 2 M \ FP (0) .
re the set M \FP59.8
(0) is a singleton to which (x, )(t)
0
5
10
es.
2
d i (ω i + λ i )
T
T
c i (d i )
P (0)
15
10
5
0.5
0
−0.5
0
−1
−0.5
0
0.5
−1
−10
1
−5
0
5
ωi + λi
di
Fig. 2. Disutility ci (di ) and load function di (!i +
in general will a↵ect the convergence rate. We
the OLC-system proposed in (Zhao et al., 2013) a
the FP-OLC-system (25)-(26), after introducing a
bation at bus 1 of P1m = .5p.u.. Figures 3 and
the evolution of the bus frequencies for the OLC
OLC systems. It can be seen that while the OL
controllers fail to recover the nominal frequency,
OLC controllers
can jointly
rebalance
the power a
15
20
25
30
recovering the nominal frequency.
t
60.05
60
. 4. Frequency evolution using FP-OLC controllers
ωi
59.95
59.9
59.85
ally, we evaluate the “social” cost that the loads m
59.8
59.75
0
5
10
15
t
20
25
30