Market Model and Algorithmic Design for Demand Response in...
Transcript of Market Model and Algorithmic Design for Demand Response in...
Market Model and Algorithmic Design for Demand Response in Power Networks
Lijun Chen, Na Li, and Steven Low
Computing and Mathematical Sciences
California Institute of Technology
April 18, 2011
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Demand response
Use incentive mechanisms such as real-time pricing to induce customers/appliances to shift usage or reduce (even increase) consumption
Smart appliances responding to price/event signals
Load shifting technologies such as storage
Peak-eliminating techniques such as distributed generation or simply turning off appliances
…
Demand response techniques
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Enabler
Smart grid
Timely two-way communications between customers and utility companies
Individual customers and appliances are empowered with certain computing capability
High speed WAN allows real-time and global monitoring at control centers
High performance computing allows faster control decisions
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Outline
Motivation for demand response
Main issues in demand response design
Demand response: Match the supply
Demand response: Shape the demand
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Time-varying demand
Electricity demand is highly time-varying
Provision for peak load
Low load factor
• US national load factor is about 55%
Underutilized
• 10% of generation and 25% of distribution facilities used less than 5% of the time
Source: DoE, Smart Grid Intro, 2008
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Shape the demand
Reduce peak load
Flatten load profile
Benefits
Lower generation cost
Larger safety margin
Reduce or slow down the need for new generation and distribution infrastructure
Uncertainty of renewables
Source: Rosa Yang
change at timescale of minutes
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Uncertainty of renewables
Source: Rosa Yang
change at timescale
of seconds
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Dealing with uncertainty
Reduce uncertainty by
Aggregating supply types
Aggregating over space
Aggregating over time (but large-scale storage is currently not available)
Accommodate uncertainty
Reliability as resource to trade off
• Optimize risk tolerance
Match time-varying supply (demand response)
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Outline
Motivation for demand response
Main issues in demand response design
Demand response: Match the supply
Demand response: Shape the demand
Main challenge
Matching supply and demand
Market challenge
• achieve efficient and economic generation, delivery, and consumption
Engineering challenge
electricity must be consumed at the moments it is generated
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Overall structure
generation customers
utility
company
wholesale
market
retail
market
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Bilateral contractsAuction market
day-aheadreal-time ancillary service
Main issues
The role of utility as an intermediary Play in multiple wholesale markets to provision aggregate power
to meet demands
Resell, with appropriate pricing, to end users
Provide two important values
• Aggregate demand at the wholesale level so that overall system is more efficient
• Absorb large uncertainty/complexity in wholesale markets and translate them into a smoother environment (both in prices and supply) for end users.
How to quantify these values and price them in the
form of appropriate contracts/pricing schemes?
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Main issues
Utility/end users interaction
Design objective• Welfare-maximizing, profit-maximizing, …
Distributed implementation
Real-time demand response
our focus
The impact of distribution network
i.e., put in physical network (Kirkoff Law, and other constraints)
How does it change the algorithm and optimality
Can we exploit radial structure of distribution network
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Retail market
Retail (utility-user) essentially uses fixed prices
Tiered, some time-of-day
Demand response will (likely) use real-time pricing to better manage load
How should utility company design real-time retail prices to optimize demand response?
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The basics of supply and demand
Supply function: quantity supplied at given price
Demand function: quantity demanded at given price
Market equilibrium: such that No surplus, no shortage, price clears the market
( )q D p
( )q S p
* * *( ) ( )q S p D p * *( , )q p
*p
*q
supply
demand
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Competitive vs oligopolistic markets
Competitive market: no market participant is large enough to have market power to set the price
Price-taking behavior
e.g., individual residential customers
Oligopolistic market: (a few) market players can influence and be influenced by the actions of others
Price-anticipating behavior
e.g., large commercial customers
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Utility function
Given the set of possible alternatives, a function
is a utility function representing preference relation among alternatives, if for all ,
To use utility function to characterize preferences is a fundamental assumption in economics
,x y X
X
:U X R
" is at least as good as " U( ) ( )x y x U y
x
U
x
Uinelastic elastic
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Outline
Motivation for demand response
Main issues in demand response design
Demand response: Match the supply
Demand response: Shape the demand
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Problem setting
Supply deficit (or surplus) on electricity:
weather change, unexpected events, …
Supply is inelastic
because of technical reasons such as supply friction
Problem: How to allocate the deficit/surplus among demand-responsive customers?
load (demand) as a resource to trade
d
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Supply function bidding
Customer load to shed:
Customer supply function (SF):
parameterized by ;
the amount of load that the customer is committed to shed given price
Market-clearing pricing:
i N iq
pbpbq iii ),(
dpbqi
ii ),(
p
i
customer 1:
p1q
1 1q b p
customer n:
n nq b p
pnq
utility company:
deficit d
( ) / i
i
p p b d b
i i Nb b
0ib
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Parameterized supply function
Adapts better to changing market conditions than does a simple commitment to a fixed price or quantity (Klemper & Meyer ’89)
widely used in the analysis of the wholesale electricity markets
Green & Newbery ‘92, Rudkevich et al ‘98, Baldick et al ‘02, ‘04, …
Parameterized SF
easy to implement
control information revelation
…
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Optimal demand response
Customer cost (or disutility) function:
continuous, increasing, and strictly convex
Competitive market and price-taking customers
Given price , each customer solves
i)( ii qC
)),((),( max pbqCpbpq iiiiibi
p
customer i:
p
utility company:
deficit d
)),((),( max pbqCpbpq iiiiibi
nb
1b
p i
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Competitive equilibrium
Definition: A competitive equilibrium (CE) is defined as a tuple such that
Theorem: There exist a unique CE. Moreover, the equilibrium is efficient, i.e., maximizes social welfare
* * * *
0
* *
arg max ( , ) ( ( , )),
( , )
i
i i i i i ib
i i
i
b p q b p C q b p i
q b p d
dqqCi
iiiqi
s.t. )( max
* *
*{( ) , }i Nb p
Social Welfare optimizationEquilibrium
Proof
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Show the equilibrium condition is the optimality condition (KKT) of the optimization problem
dqqCi
iiiqi
s.t. )( max
* * * *
0
* *
arg max ( , ) ( ( , )),
( , )
i
i i i i i ib
i i
i
b p q b p C q b p i
q b p d
* * * * *
* *
( ( ( , )))( ) 0, 0
( , )
i i i i i i
i i
i
p C q b p b b p b
q b p d
* * *
*
( ( ))( ) 0, 0i i i i i
i
i
p C q q q q
q d
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Iterative supply function bidding
Upon receiving the price information, each customer updates its supply function
Upon gathering bids from the customers, the utility company updates price
Requires
timely two-way communication
certain computing capability of the customers
])(
))(()([)(
1
kp
kpCkb i
i
i
]))()(()([)1(i
i dkpkbkpkp
( )p k
customer 1:
1( )b k
1
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( ) ( ( ))( ) [ ]
( )
C p kb k
p k
1( 1)b k
( 1)p k
]))()(()([)1(i
i dkpkbkpkp
utility company:
deficit d
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Strategic demand response
Oligopoly market and price-anticipating customer
Given others’ supply functions , each customer solves
with
pnb
p1b
)))(,(())(,()(),( bpbqCbpbqbpbbu iiiiiiii
),( max iiib
bbui
utility company:
deficit d
customer i:
),( max iiib
bbui
ibi
It is a game
( ) / i
i
p p b d b
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Definition: A supply function profile is a Nash equilibrium (NE) if, for all customers and ,
Theorem: There exists a unique NE when the number of customers is larger than 2. Moreover, the equilibrium solves
pnb
p1b
*b
* * *( , ) ( , ).i i i i i iu b b u b b
i 0ib
utility company:
deficit d
customer i:
),( max iiib
bbui
Game-theoretic equilibrium
iq
iii
i
ii
i
iii )dx(xC
)x(d-
dqC
qd
q(qD
0 22)()
21()
dqqDi
iiidqi
s.t. )( max2/0
OptimizationNash Equilibrium
Proof
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Show the equilibrium condition is the optimality condition (KKT) of the optimization problem.
2
0
max ( ) s.t.
) (1 / 2 ) ( )
/ 2
i
i
i i iq
i
i i i i i i
q
i i i i
D q q d
D (q q d q C q
d (d - x ) C (x )dx
2 2
max ( ( ), ) ( ( ( ), )
= / ( ) ( / )
( , )
i
i i ib
i j j i i j j
i i
i
pq p b p C q p b p
d b b C db b
q b q d
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Iterative supply function bidding
Each customer updates its supply function
The utility company updates price
])(
))(()([)(
1
kp
kpDkb i
i
i
]))()(()([)1(i
i dkpkbkpkp
])(
))(()([)(
1
kp
kpDkb i
i
( )p k
customer 1:
1( )b k1( 1)b k
( 1)p k
]))()(()([)1(i
i dkpkbkpkp
utility company:
deficit d
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Numerical example
Optimal supply function bidding (upper panels) v.s. strategic bidding (lower panels)
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Outline
Motivation for demand response
Main issues in demand response design
Demand response: Match the supply
Demand response: Shape the demand
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Problem setting
Load is deferrable and reducible
Subject to various constraints, depending on the types of appliances
minimal/maximal load over certain period of time
minimal/maximal load at each time
battery has finite capacity and usage-dependent cost
…
Problem: How to shape deferrable load over certain period of time, so as to reduce peak, flatten load profile and even conserve energy?
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Customer-side model (abstract)
Each customer , each of the appliances :
Load at time : ; define:
Load constraint:
Total load at time :
Utility:
The appliances divided into 4 categories
Energy Storage: one battery for each customer
Load at time : ; define
positive means charging
negative means discharging
Load constraints:
Cost function:
i
, ( )i aq t
, ,i a i aU q
t
ia
, ,i a i aq C
i
( )ir t
i iD r
t
i ir R
, , ( )i a i a t Tq q t
( )i i t Tr r t
t ,( ) ( ) ( )i i a i
a
Q t q t r t
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Utility-side model
The utility company incurs cost when the supply is
convex, with a positive, increasing marginal cost
Piecewise quadratic cost functions
with
( )C Q Q
2
1 1 1 1
2
2 2 2 1 2
2
1
; 0
; Q( )
; Qm m m m
c Q b Q a Q Q
c Q b Q a Q QC Q
c Q b Q a Q
1 1 0m mc c c Q
C(Q)
Q1 Q2
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Objective: induce customers’ consumption to maximize social welfare
, ,,
, ,
max
max ( ) ( )
s.t.
0 ( )
i
i a i a i i iq r
i a A t i
i a i a
i i
i i
U q D r C Q t
q C
r R
Q t Q
Utility-side model
proof of conception, to see how effective
real-time pricing can be
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Utility-customer interaction
Utility sets prices to induce customer behaviors
Customer maximizes his own net benefit
( ( ))t Tp p t
, ,,
, ,
max
max ( ) ( ) ( )
s.t.
0 ( )
i i
i a i a i i iq r
a t
i a i a
i i
i i
U q D r Q t p t
q C
r R
Q t Q
i
price-taking
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Definition: The prices and customer demands is in equilibrium if maximizes the social-welfare, and also maximizes customer net benefit for given price .
Theorem: There exists an equilibrium . Moreover, the equilibrium price .
follow from the welfare theorem and imply that setting the price to be the marginal cost of power is optimal
similar proof
* * *
,( , , )i a ip q r* *
,( , )i a iq r
* ' *( ) ( ( ))i
i
p t C Q t
Market equilibrium
i*p
* * *
,( , , )i a ip q r
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Customer-side model (appliances)
Air ConditionerRefrigeratorEtc
Ui,a qi,a Ui,a Ti,a(t),Ti,acomf
t
Ti,amin Ti,a (t) Ti,a
max
Ti,a (t) g(Ti,a (t 1),qi,a (t))
0 qi,a (t) qi,amax (t)
Utility function:
Constraints: temperature
PHEVWasherEtc
Ui,a qi,a Ui,a qi,a(t)t
0 qi,a (t) qi,amax (t)
Qi,amin qi,a (t)
t
Qi,amax
Utility function:
Constraints:
TTcomf
U
Q
U
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Lighting
Ui,a qi,a Ui,a qi,a(t),t t
0 qi,a(t) qi,amax(t)
Utility function:
Constraints:
0 qi,a (t) qi,amax (t)
Qi,amin qi,a (t)
t
Qi,amax
Utility function:
Constraints:
Entertainment
Ui,a qi,a Ui,a qi,a(t),t t
a crude model
Customer-side model (appliances)
q
U
q
U
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Customer-side model (Battery)
2 2
1 2 3( ) ( ) ( ) ( 1) (min{ ( ) ,0})i i i i i i i
t t t
D r r t r t r t B t B
Cost function:
Constraints:
t
r(t)
bettereven better
min max
0 ( )
( )
( )
i i
i i i
i i i
B t B
B T B
r r t r
charging& discharging
charging -discycles
deep discharging
Numerical example: no battery
4 households with people at home all the day; 4 with no person at home during day time
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Total
PHEV
AC
Entertainment
Washer
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Total
PHEVAC
Entertainment WasherLight
Battery
Numerical example: with battery
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Numerical example
Numerical experiments
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Load Factor
Peak Demand Per Household, kwh
Total Demand Per Household, kwh
Number of Households
load factor
average load=
peak load
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Concluding remarks
Demand response: Match the supply
iterative supply function bidding (competitive vs oligopolistic)
Demand response: Shape the demand
Real-time pricing based on marginal cost is “ideally” very effective
Future work: extend the models to study the afore-mentioned issues in demand response design
Current focus: real-time demand response; coordinated control with Volt/Var
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References
Two Market Models for Demand Response in Power Networks,L. Chen, N. Li, S. Low and J. Doyle, IEEE SmartGridComm, 2010.http:\\cds.caltech.edu\~chen\papers\DemandResponse.pdf
Optimal Demand Response Based on Utility Maximization inPower Networks, L. Chen, N. Li and S. Low, IEEE PESGM 2011.http:\\cds.caltech.edu\~chen\papers\ODemandResponse.pdf
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Thanks