Incentives for Demand-Response Programs with
Nonlinear, Piece-Wise Continuous Electricity Cost
Functions
Carlos Barreto and Alvaro Cárdenas
University of Texas at Dallas
2015 American Control Conference
C. Barreto and A. Cárdenas
Incentives for Demand-Response Programs
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Problem
Generation cost functions are usually assumed to be quadratic1 .
Consequently, the unitary price is linear, e.g., p(z) = az + b.
This does not reflect reality because does not capture incentives to reduce
peaks, e.g., supply of peak demand requires2
fast but expensive generators to supply demand.
more capacity in transmission and generation.
Short run unitary cost:
AB: Base load plants (nuclear)
BC: Coal fired plants
CD: Peaking plants (turbines
that use use natural gas, diesel,
or jet fuel)
1
Allen J Wood/Bruce F Wollenberg: Power generation, operation, and control, 2012.
2
W Kip Viscusi/Joseph E Harrington/John M Vernon: Economics of regulation and antitrust, 2005.
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Problem
The Independent System Operator (ISO) designs regulation to encourage
efficiency3
Peak reductions save money in facility/generation costs.
Saves money by slowing the need for costly facilities updates to
supply growing demand.
Models with linear unitary price might give a wrong idea about the
effectiveness of the DR scheme and achievement of goals.
3
John Malinowski/Keith Kaderl: Peak Shaving - A Method to ReduceUtility Costs, tech. rep. FM2404, Baldor Electric
Company, 2004, url: http://www.sustainableplant.com/assets/PeakSavingsFM2404Whitepaper.pdf.
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Contributions
1
2
We show that cost functions from the literature do not achieve peak
reduction.
We design an incentives scheme that has into account nonlinear
prices.
I
I
We show that there is no incentive mechanism that can achieve Pareto
optimality with budget balance.
we propose two incentive rules that which either require subsidies or
impose taxes on a population.
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1
Market Model
Efficiency Criteria
Non-flat Demand
Inefficiency in Strategic Environments
2
Design of Incentives
Budget Balance Impossibility
Budget Deficit
Budget Surplus
3
Conclusions and future work
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Market Model: Efficiency Criteria
The surplus of the i th user is defined by:
Ui (q) =
t=1
Cost
z }| { !
z }| {
vit (qit ) − p kq t k1 qit .
Satisfaction
XT
Ideal Solution
maximize
q
XN
i=1
Ui (q)
subject to qi ≥ 0.
P
Ui (q) is the social welfare minus the total cost, which is called
aggregate surplus.
We assume that Ui (·) is concave and derivable.
i∈P
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The Optimal Demand Might not be Flat
If the valuation functions belong to the family of functions
v : R≥0 × R≥0 → R that satisfy:
If αik1 > αjk2 , then v (q, αik1 ) > v (q, αjk2 ), for all i, j ∈ P and
αik1 , αjk2 , q ∈ R≥0 .
limq→0 v (q, α) = 0, for α ∈ R≥0 .
Proposition 1
If agents have different valuations at each time period, then the aggregated
demand is greater in the time period with greater valuation.
That
is,if
there are t1 and t2 , such that for all i αit1 > αit2 , then q t1 1 > q t2 1 .
Hence, in an efficient outcome the demand at the peak hour will be
higher. However it is not clear if the Peak to Average Ratio(PAR)
improves in the optimal outcome.
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Inefficiency in Strategic Environments
There is a loss of efficiency when users are strategic4 .
Individual optimization problem
Aggregated Profit at the Equilibrium
20
qi
Ui (q i , q −i )
18
Profit
maximize
Efficient outcome
Inefficient outcome
subject to qi ≥ 0.
16
14
12
10
0
Peak-to-Average Ratio (PAR) of
both solutions:
10
15
Time of day
Aggregated Demand at the Equilibrium
20
Efficient outcome
Inefficient outcome
4.5
Power
Inefficient (Nash
equilibrium): 1.1139
5
5
4
3.5
3
2.5
Efficient (Pareto solution):
1.1228
0
5
10
Time of day
15
20
Note that the efficient outcome has a slight major PAR.
4
Ramesh Johari/John N. Tsitsiklis: Efficiency of Scalar-Parameterized Mechanisms, in: Operations Research 57.4 (July
2009), pp. 823–839.
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Nonlinear Prices
Let us introduce the following price
per unit of electricity consumption:
Aggregated Demand at the Equilibrium
γ=
γ=
γ=
γ=
γ=
3.7
p̂(q) = p(q) + pk (q)
3.6
3.5
3.4
Power
pk (q) is a peak tax or incentive
designed to reduce demand peaks.
0,
if z ≤ k,
pk (z) =
γ(z − k)2 , if z > k,
0
0.2
1
2
10
3.3
3.2
3.1
3
2.9
2.8
0
5
10
Time of day
15
20
k is a demand threshold for the tax.
p(·) is nonlinear piece-wise continuous differentiable.
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1
Market Model
Efficiency Criteria
Non-flat Demand
Inefficiency in Strategic Environments
2
Design of Incentives
Budget Balance Impossibility
Budget Deficit
Budget Surplus
3
Conclusions and future work
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Incentives for Demand-Response Programs
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Design of Decentralized Incentives
Decentralized scheme based on economic incentives to achieve optimal
outcomes in strategic environments5 .
Individual optimization problem
Consumers
Utility
maximize
qi
q=
Ui (q i , q −i ) + Ii (q)
subject to qi ≥ 0.
N
P
i=1
q∗i
q
q∗1 ∈ arg max U1 (x1 , q−1 ) + I1 (q−1 )
x1
UN (xN , q−N ) + IN (q−N )
q∗N ∈ arg max
x
N
The incentives are of the form
Ii (q) = q −i 1 hi (q −i ) − p q 1 .
hi (·) : estimation of the externality imposed by the i th customer. Burden
on the system with the participation of the i th customer.
Dynamic Prices
Does not require full information.
Might need external subsidies.
5
C. Barreto/E. Mojica-Nava/N. Quijano: Design of Mechanisms for Demand Response Programs, in: 2013 IEEE 52nd
Annual Conference on Decision and Control (CDC), Dec. 2013.
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Budget Balance Property
Ideally net payments in the system should be equal to zero. That is,
X
Ii (q) = 0.
i∈V
A budget balanced system neither requires additional taxes nor requires
external subsidies to operate. This can be rewritten as
P
q −i hi (q −i )
i∈V
1
1 P
p q 1 =
i∈V q −i 1
Proposition 2
An incentive that is budget balanced must satisfy
p q̂ 1 = hi (q̂ −i 1 ),
given a uniform demand profile q̂ for all i ∈ V.
If p(·) is concave, then hi (·) must be concave as well.
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Incentives for Demand-Response Programs
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Budget Balance Impossibility
Theorem 3
There is no function h(·) that satisfies the budget balance property.
The implementation of this scheme might be limited because the
incentives scheme does not guarantee that net payments are equal to zero
for all cases.
The incentives might
Impose taxes over customers.
Require external subsidies.
Next we analyze the conditions in which the previous cases take place.
C. Barreto and A. Cárdenas
Incentives for Demand-Response Programs
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Budget Deficit: Requires External Subsidies
X
i∈V
Ii (q) =
X
i∈V
q −i q −i ) − p q h
(
≥ 0.
i
1
1
hi (q −i 1 ) = p
!
q −i 1
+ q −i 1
(N − 1)
The burden on the system imposed by the i th customer is estimated using
alternative system where the i th customer consumes q −i 1/(N − 1).
Properties:
1
Incentives for the i th and j th agents are equivalent if their
consumption is the same, i.e., if qi = qj , then Ii (q) = Ij (q).
2
If qi = qj for all i, j ∈ P, then Ii (q) = Ij (q) = 0 (special case of
budget balance).
3
A higher power consumption deserves a lower incentive, i.e., if
qj > qi , then Ij (q) < Ii (q).
C. Barreto and A. Cárdenas
Incentives for Demand-Response Programs
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Budget Surplus: Imposes Taxes on the Population
X
i∈V
Ii (q) =
X
i∈V
q −i q −i ) − p q h
(
≤ 0.
i
1
1
hi (q −i 1 ) = p
!
q −i 1
(N − 1)
The burden on the system imposed by the i th customer is estimated using
an alternative system where the i th customer has null consumption.
Properties:
1
Incentives for the i th and j th agents are equivalent if their
consumption is the same, i.e., if qi = qj , then Ii (q) = Ij (q).
2
If qi = qj for all i, j ∈ P, then Ii (q) = Ij (q) ≤ 0 (special case of
budget balance).
3
A higher power consumption deserves a lower incentive, i.e., if
qj > qi , then Ij (q) < Ii (q).
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Conclusions and future work
Popular electricity cost functions in the literature do not achieve the
peak minimization goal of practical DR programs.
We then introduced a new nonlinear pricing function that achieves
this objective.
We prove that there is no incentive mechanism that, in a distributed
way, can achieve Pareto efficiency with a balanced budget.
We introduce two incentive rules, which either require external
subsidies or impose taxes on the population to operate.
Future work:
Include in the modeling random components, such as renewable
generation.
Consider time interdependencies of the electricity usage. and
additional operation constraints.
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Thank You
Questions?
Contact:
Carlos Barreto, [email protected]
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