Demand Response Using Linear Supply
Function Bidding
Na Li1, Lijun Chen2, Munther Dahleh3
1Harvard
University
2University of Colorado at Boulder
3Massachusetts Institute of Technology
PESGM, 07/28/2015
Demand Response: Demand
• Electricity demand: highly timevarying
• Provision for peak load
– Low load factor
• National load factor is about 55%
– Underutilized
• 10% of generation and 25% of
distribution facilities are used less than
5% of the time
• A way out: Shape the demand
– Reduce the peak
– Smooth the variation
Source: DoE, Smart Grid Intro, 2008
Demand Response: Generation
• Supply becomes highly time-varying
– steady rise of renewable energy resources
• Intermittent generation
– Large storage is not available
This work
• A way out: Match the supply
Demand Response
Use incentive mechanisms such as real-time pricing to induce
customers to shift usage or reduce (even increase) consumption
Special
Climate
Shape
Match
activitie
Price
the Else…
the
s
Weathe
(Market)
Demand
Supply
r
Overall structure
generation
customer
wholesale
market
retail
market
utility
company
Main issues
The role of utility as an intermediary
– Play in multiple wholesale markets to provision aggregate power
to meet demands
• day-ahead, real-time balancing, ancillary services
– Resell, with appropriate pricing, to the 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 the end users.
How to quantify these values and price them in the
form of appropriate contracts/pricing schemes?
Main issues
Utility/end users interaction
– Design objective
• Welfare-maximizing, profit-maximizing
– Price-taking (Competitive) vs Price-anticipating (Game)
– Distributed implementation
– Real-time demand response
The basics of supply and demand
• Supply function: quantity demanded at given prices
q = S ( p)
• Demand function: quantity supplied at given prices
q = D( p)
• Market equilibrium:
q* S=
( p* ) D ( p* )
(q* , p* ) such that=
– No surplus, no shortage, price clears the market
supply
p*
demand
q*
Problem setting
• Supply deficit (or surplus) on electricity: d
weather change, unexpected events, …
• Supply is inelastic
Problem: How to allocate the deficit among demandresponsive customers?
load (demand) as a resource to allocate
Supply function bidding
• Customer i load to shed: qi
• Customer i supply function (SF):
qi (bi , p ) = bi p
– the amount of load that the
customer is committed to shed
p
given price
• Market-clearing pricing:
∑ qi (bi , p) = d
i
p = p (b)  d / ∑ bi
i
utility company:
deficit d
p q1
customer 1:
q1 = b1 p
p
qn
customer n:
qn = bn p
Parameterized supply function
• Adapts better to changing market conditions than does a simple
commitment to a fixed price of 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
Competitive market: Optimal demand response
• Customer i cost (or disutility)
function: Ci (qi )
– continuous, increasing, and strictly
convex
• Competitive market and pricetaking customers
• Optimal demand response
max pqi (bi , p ) − Ci (qi (bi , p ))
bi
utility company:
deficit d
p b
1
p
bn
customer i:
max pqi (bi , p ) − Ci (qi (bi , p ))
bi
Competitive equilibrium
• Definition: A competitive equilibrium (CE) is defined as a
tuple {(bi )i∈N , p} such that
(Ci′(qi (bi , p )) − p )(bˆi − bi ) ≥ 0, ∀bˆi ≥ 0
∑ q (b , p) = d
i
i
i
• Theorem: There exist a unique CE. Moreover, the CE is
efficient, i.e., maximizes social welfare
max − Ci (qi ) s.t.
qi
∑q
i
=d
i
• Proof: The equilibrium condition is the optimality condition of
the social welfare problem.
Proof
Proof Idea: Compare the equilibrium condition with the optimality
condition (KKT) of the optimization problem.
Equilibrium
max pqi (bi , p ) − Ci (qi (bi , p ))
bi
∑ q (b , q) = d
i
i
i
Social Welfare optimization
max − Ci (qi ) s.t.
qi
∑q
i
i
=d
Competitive equilibrium
Theorem (A water-filling structure):
1
Corollary (Individual Rantionality):
2
3
4
5
q
Iterative supply function bidding
• Upon receiving the price information, each
customer i updates its supply function
(Ci′) ( p (k )) +
bi (k ) = [
]
p(k )
−1
• Upon gathering bids from the customers,
the utility company updates price
p (k + 1) = [ p (k ) − γ (∑ bi (k ) p (k ) − d )]+
i
• Requires
– timely two-way communication
– certain computational capability of the
customers
p (k + 1) = [ p (k ) − γ (∑ bi (k ) p (k ) − d )]+
i
utility company:
deficit d
p (k ) p (k + 1)
q1 (k )
q1 (k + 1)
customer 1:
(C1′) −1 ( p (k )) +
b1 (k ) = [
]
p(k )
Strategic demand response
• Price-anticipating customer
with
utility company:
deficit d
max ui (bi , b−i )
bi
ui (bi , b−i ) = p (b)qi (bi , p (b)) − Ci (qi (bi , p (b)))
• Definition: A supply function profile
is a Nash equilibrium (NE) if, for all
customers i and bi ≥ 0 ,
ui (bi* , b−*i ) ≥ ui (bi , b−*i )
p
p q1
qn
b*
customer i:
max ui (bi , b−i )
bi
Nash Equilibrium
• Price-anticipating customer
Supply
deficit d
max p (bi , b− i )qi (bi , p (bi , b− i )) − Ci (qi (bi , p (bi , b− i )))
bi
• Nash equilibrium exits and is unique when the
number of customers is larger than 2
• Each customer will shed a load of less than d/2
at the equilibrium
• Solving another global optimization problem
max − Di (qi ) s.t. ∑ qi = d
i
qi
qi
d
Ci(xi )dxi
Di(qi ) = (1 +
)Ci (qi ) − ∫
2
0
(d-2 xi )
d − 2qi
0 ≤ qi ≤ d / 2
p q1
p
qn
customer i:
max p (bi , b− i )qi (bi , p (bi , b− i ))
bi
− Ci (qi (bi , p (bi , b− i )))
Nash equilibrium
Theorem
Proof
Proof Idea: Compare the equilibrium condition with the optimality
condition (KKT) of the optimization problem.
NE Equilibrium
max pqi ( p (b), p ) − Ci (qi ( p (b), p )
bi
max
qi
= d bi / (Σ j b j ) − Ci (dbi / Σ j b j )
2
∑ q (b , q) = d
i
i
Optimization
i
2
− Di ( qi ) s.t.
∑q
i
=
d
i
(1 + qi / d − 2qi )Ci ( qi )
Di (qi ) =
qi
− ∫ d / (d - 2 xi )2 Ci (xi )dxi
0
Nash equilibrium
Theorem (A water-filling structure):
Corollary (Individual Rationality):
Iterative supply function bidding
• Each customer i updates its
supply function
( Di′) −1 ( p (k )) +
]
bi (k ) = [
p(k )
• The utility company updates
price
p(k + 1) = [ p(k ) − γ (∑ bi (k ) p (k ) − d )]+
i
p (k + 1) = [ p (k ) − γ (∑ bi (k ) p (k ) − d )]+
i
utility company:
deficit d
p (k ) p (k + 1)
q1 (k )
q1 (k + 1)
customer 1:
( Di′) −1 ( p (k )) +
]
bi ( k ) = [
p(k )
Numerical example
Optimal supply function bidding (upper panels) v.s. strategic bidding (lower panels)
Efficiency Loss of Nash Equilibrium
Theorem:
Homogeneous Customers
Corollary:
Numerical example
Price
Total disutility
A Special Case: Quadratic Disutility Function
Theorem:
Message:
Both the competitive equilibrium and game equilibrium
are independent of the supply deficit d!
Concluding remarks
• Studied one abstract models for demand response
– Characterized competitive as well as strategic equilibria
– Proposed distributed demand response algorithms
based on optimization problem characterizations
– Characterized the efficiency loss and price of the gametheoretic equilibrium
• Further work
– Bring in the detailed dynamics and realistic constraints
of demand response appliances
– Put on networks
Thank you!