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Paper Review
A Control Theorist’s Perspective on DynamicCompetitive Equilibria in Electricity Markets
G. Wang, A. Kowli, M. Negrete-Pincetic, E. Shafieepoorfard andS. Meyn
Presenter: Truong X. NghiemESE, University of Pennsylvania
April 18, 2011
Summary Introduction Market Model Equilibria Conclusion
Outline
1 Summary2 Introduction3 Market Model4 Equilibria5 Conclusion
Summary Introduction Market Model Equilibria Conclusion
Summary
Economic model of electricity market that includes dynamics,uncertainties, and operational constraints (using control theorytechniques).
Conditions for the existence and optimality of competitive equilibriain the models.
Economic model and equilibrium result are very generic.
Does not provide a direct solution to market design problem, but aframework for modeling and equilibium characterization.
Summary Introduction Market Model Equilibria Conclusion
Features of Electricity Market
Electricity market (of the Smart Grid) couples:
Physical system: highly complex network with
power flowing through transmission lines;modulated by distributed generation units;governed by Kirchhoff’s laws;subject to operational and security constraints;uncertainty in loads and generations.
Economic system: consists of coupled markets, e.g., day-ahead andreal-time auctions, which are dynamic and subject to uncertainty(players/agents).
Many coupled complicating factors of two complex systems makeelectricity market design is a challenging task.
Summary Introduction Market Model Equilibria Conclusion
Features of Electricity Market
Electricity market (of the Smart Grid) couples:
Physical system: highly complex network with
power flowing through transmission lines;modulated by distributed generation units;governed by Kirchhoff’s laws;subject to operational and security constraints;uncertainty in loads and generations.
Economic system: consists of coupled markets, e.g., day-ahead andreal-time auctions, which are dynamic and subject to uncertainty(players/agents).
Many coupled complicating factors of two complex systems makeelectricity market design is a challenging task.
Summary Introduction Market Model Equilibria Conclusion
Features of Electricity Market
Electricity market (of the Smart Grid) couples:
Physical system: highly complex network with
power flowing through transmission lines;modulated by distributed generation units;governed by Kirchhoff’s laws;subject to operational and security constraints;uncertainty in loads and generations.
Economic system: consists of coupled markets, e.g., day-ahead andreal-time auctions, which are dynamic and subject to uncertainty(players/agents).
Many coupled complicating factors of two complex systems makeelectricity market design is a challenging task.
Summary Introduction Market Model Equilibria Conclusion
Control Theorist’s Perspective
Electricity market analysis is guided by idealized models of players’behavior and idealized models of the underlying physical system.
The underlying physical reality has deep impact on the marketoutcome, but is modeled simple ⇒ market behavior is often sodifferent from prediction.
Paper’s solution: incorporate dynamics, uncertainty in supply anddemand, operational constraints in electricity market models usingtechniques in control theory.
Summary Introduction Market Model Equilibria Conclusion
Electricity Market Model
Price-taking/Perfect-competition assumption
No players are large enough to have the market power to set the price.
Model structure
Single “consumer” represents all price-taking consumers.
Single “supplier” represents all price-taking suppliers.
N buses c.t. nodes in network.
L lossless transmission lines c.t. links in network.
Network is connected.
Price
Pn(t): energy price traded at bus n ∈ {1, 2, . . . ,N} at time t ≥ 0.Price-taking assumption: Pn(t) cannot be influenced by actions ofconsumers & suppliers (?).
Summary Introduction Market Model Equilibria Conclusion
Consumer
Consumer model
Dn(t): demand at time t at bus n;
EDn(t): energy consumed at time t at bus n; EDn(t) ≤ Dn(t);
EDn(t) = Dn(t): sufficient generation;EDn(t) < Dn(t): insufficient generation (blackout);
vn(EDn(t)): utility of consumption (possibly nonlinear);
cbon (Dn(t)− EDn(t)): disutility of blackout (possibly nonlinear);
Welfare of consumer at time t = sum of benefits and costs
WD(t) =∑n
[vn(EDn(t))− cbo
n (Dn(t)− EDn(t))− Pn(t)EDn(t)]
Notations
D(t), ED(t), P(t): vector of values;
D, ED , P: entire trajectory ∀t ≥ 0.
Summary Introduction Market Model Equilibria Conclusion
Supplier
Supplier model
ESn(t): energy produced at bus n; RSn(t): energy reserve at bus n;ESn(t) + RSn(t): generation capacity at bus n;
cEn (ESn(t)): production cost; cR
n (RSn(t)): reserve cost;
Welfare of supplier at time t = sum of revenue and costs
WS(t) =∑n
[Pn(t)ESn(t)− cE
n (ESn(t))− cRn (RSn(t))
]Constraints
Generic constraint: (ES ,RS) ∈ XS .
Rate constraint: for all t1 > t0 ≥ 0,
ζ−n ≤ESn(t1)− ESn(t0)
t1 − t0+
RSn(t1)− RSn(t0)
t1 − t0≤ ζ+n
Summary Introduction Market Model Equilibria Conclusion
Network
The network as a third player with network structure and constraints;
Lossless network: supply-demand balance constraint
1TES(t) = 1TED(t), t ≥ 0 (1)
Flow constraints
H ∈ [−1, 1]N×L: injection shift factor matrix based on a referencebus (e.g., bus 1); Hnl : power distributed on line l when 1MW isinjected into bus n and withdrawn at ref. bus.
Flow constraint: f maxl – capacity of line l
− f maxl ≤ (ES − ED)THl ≤ f max
l (2)
All constraints (1) and (2) summarized as (ES ,ED) ∈ XT .
Welfare of network at time t: WT (t) =∑
n [Pn(t) (EDn(t)− ESn(t))]
Summary Introduction Market Model Equilibria Conclusion
Example (Texas Model)
Figure: Source: Real-time Prices in an Entropic Grid.
Three nodes (buses), three links (lines).
If bus 1 as ref. bus and impedances are identical
H =1
3
0 0 0−2 −1 −1−1 −2 1
Summary Introduction Market Model Equilibria Conclusion
Uncertainty
All processes are stochastic.
Long-run discounted expected profit with rate γ:
KD = E[∫
e−γtWD(t)dt
]Denote
〈F,G〉 = E[∫
e−γtF (t)G (t)dt
]for two stochastic processes F and G.
KD = 〈WD , 1〉; KS = 〈WS , 1〉; KT = 〈WT , 1〉.
Summary Introduction Market Model Equilibria Conclusion
Competitive Equilibrium
When the objective functions of both the supplier and the consumer aremaximized.
Definition
A competitive equilibrium is a quadruple {ED ,ES ,RS ,P} such that:
1 (ES ,RS) ∈ arg maxES ,RS〈WS , 1〉 subject to (ES ,RS) ∈ XS ;
2 ED ∈ arg maxED〈WD , 1〉;
3 (ED ,ES) ∈ arg maxED ,ES〈WT , 1〉 subject to (ES ,ED) ∈ XT .
This paper: existence and optimality of a competitive equilibrium.
Summary Introduction Market Model Equilibria Conclusion
Social Planner
A social planner is an analytical device to maximize the economicwell-being of the entire system, as the total welfare:
Wtot(t) =WS(t) +WD(t) +WT (t)
=∑n
[vn(EDn(t))− cbo
n (Dn(t)− EDn(t))− cEn (ESn(t))− cR
n (RSn(t))]
independent of prices.
Social Planner’s Problem (SPP)
maxED ,ES ,RS
〈Wtot, 1〉
subject to all constraints.Its solution is called an efficient allocation.
Summary Introduction Market Model Equilibria Conclusion
Fundamental Theorems
If {ED ,ES ,RS ,P} is a competitive equilibrium and {ED ,ES ,RS} isan efficient allocation: the equilibrium is efficient.
If {ED ,ES ,RS} is an efficient allocation and P such that{ED ,ES ,RS ,P} is a competitive equilibrium: the allocation issupported by price P.
First Fundamental Theorem
Any competitive equilibrium, if it exists, is efficient.
Second Fundamental Theorem
If the market admits a competitive equilibrium, then for any efficientallocation {ED ,ES ,RS}, there exists a supporting price process P suchthat {ED ,ES ,RS ,P} is a competitive equilibrium.
Summary Introduction Market Model Equilibria Conclusion
Proof Sketch (1)
Lagrangian of the SPP
L =− 〈Wtot, 1〉+ 〈λ, 1TED − 1TES〉
+∑l
〈µ+l , (ES − ED)THl − f max
l 〉
+∑l
〈µ−l ,−(ES − ED)THl − f maxl 〉
where µ+l (t) ≥ 0 and µ−l (t) ≥ 0 for all t and l .
Candidate Price
Candidate price process P:
Pn(t) = λ(t) +∑l
(µ−l (t)− µ+
l (t))
Hln
Summary Introduction Market Model Equilibria Conclusion
Proof Sketch (2)
Lagrangian of the SPP with Candidate Price
L =−∑n
{〈vn(EDn)− cbo
n (Dn − EDn), 1〉 − 〈Pn,EDn〉}
−∑n
{〈Pn,ESn〉 − 〈cE
n (ESn) + cRn (RSn), 1〉
}−∑l
〈µ+l (t) + µ−l (t), f max
l 〉
Dual Function
h(λ,µ+,µ−) = min{ED ,ES ,RS}
L
Summary Introduction Market Model Equilibria Conclusion
Proof Sketch (3)
Using KKT conditions, complementary-slackness, and the constraints,can prove that
Existence of Competitive Equilibrium
The market admits a competitive equilibrium if and only if the SPPsatisfies strong duality:
−〈Wtot, 1〉 = h(λ,µ+,µ−)
{ED ,ES ,RS} are obtained through the solution of the SPP.
Price P is obtained as a solution to its dual.Corollary: If {E1
D ,E1S ,R
1S ,P
1} and {E2D ,E
2S ,R
2S ,P
2} are twocompetitive equilibria then {E2
D ,E2S ,R
2S ,P
1} is also a competitiveequilibrium. (price-taking assumption)
Summary Introduction Market Model Equilibria Conclusion
Illustration
In Real-time Prices in an Entropic Grid (submitted), authors providedsome examples for 6 S-market:
Supplier’s constraints are relaxed;
Welfare function of the supplier, WD , is identically zero.
Effectively, the supplier is removed from the model.
6 S-market for the Texas model with stationary demands, it is possible tohave negative prices or exceedingly high prices.
Summary Introduction Market Model Equilibria Conclusion
Conclusions
Generic economic model of electricity market that includesdynamics, uncertainties, and operational constraints.
Conditions for the existence and optimality of competitive equilibriain the models.
Dynamics are captured in the processes ED , ES , RS ; however, thispaper assumes no structure of dynamics.
Assumptions seem strict.
Focus on competitive equilibrium analysis, which may not be whatactually happens in real markets.
Cannot explain dramatic price dynamics, such as in Texas onFebruary 2nd, 2011 (price rose from $30/MWh to $3000/MWh).
Did not address stability of the equilibrium, or the impact ofstrategic behavior of players.
Thank You!