Paper Review A Control Theorist’s Perspective on Dynamic ...cis800/lectures/truong_review.pdf ·...

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

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.


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.


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.


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 (?).


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.


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


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))]


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


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〉.


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.


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.


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.


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


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


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)


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.


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!

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