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

2

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

3

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

4

Outline

Motivation for demand response

Main issues in demand response design

Demand response: Match the supply

Demand response: Shape the demand

5

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

6

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

7

Uncertainty of renewables

Source: Rosa Yang

change at timescale

of seconds

8

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)

9

10

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

11

Overall structure

generation customers

utility

company

wholesale

market

retail

market

12

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?

13

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

14

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?

15

16

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

17

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

19

Outline

Motivation for demand response

Main issues in demand response design

Demand response: Match the supply

Demand response: Shape the demand

20

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

21

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

22

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

25

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

11

( ) ( ( ))( ) [ ]

( )

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

28

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

29

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

33

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

35

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

36

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

37

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

38

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

39

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

41

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

43

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