Pricing and regulations for a wholesale electricity...

Outline Introduction and motivation Modeling Market Market Power Efficient regulations and mechanism design

Pricing and regulations for a wholesaleelectricity market

Alejandro Jofre1

Center for Mathematical Modeling & DIMUniversidad de Chile

Terry Fest, Limoges, May 2015

1In collaboration with J. Escobar and B. Heymann1 / 41


Terry Fest

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Terry Fest

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Outline

• Introduction and motivation• Modeling market and Equilibrium.• Market Power• Efficient regulations and Extended Mechanism Design• Conclusions

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1 Introduction and motivation

2 Modeling MarketEquilibrium: Nash

3 Market Power

4 Efficient regulations and mechanism designThe benchmark gameComparing Benchmark with Optimal Mechanism

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ISO

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Transmission US

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ISOs USA

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3 Market Power


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A generation short term market: day-aheadmandatory pool

Today: generators taking into account an estimation of thedemand bid increasing piece-wise linear cost functions orequivalently piece-wise constant ”price”. Even generalconvex cost functions.Tomorrow: the (ISO) using this information and knowing arealization of the demand, minimizes the sum of the coststo satisfy demands at each node considering all thetransmission constraints: ”dispatch problem”.Tomorrow: the (ISO) sends back to generators the optimalquantities and ”prices” (multipliers associated to supply =demand balance equation at each node)

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ISO problem or dispatch DP (c, d)

The (ISO) knows a realization of the demand d ∈ IRV , receivesthe costs functions bid (ci)i∈G and compute: (qi)i∈G, (λi)i∈G

min(h,q)

∑i∈G

ci(qi). (1)

∑e∈Ki

re2h2e + di ≤ qi +

∑e∈Ki

hesgn(e, i), i ∈ G (2)

qi ∈ [0, qi], i ∈ G, (3)

0 ≤ he ≤ he (4)

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We denote Q(c, d) ⊂ IRG the generation component of theoptimal solution set associated to each cost vector submittedc = (ci) and demand d.We denote Λ(c, d) ⊂ IRG the set of multipliers associated to thesupply=demand in the ISO problem.

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Modeling Generators

1 At each node i ∈ G we have a generator with payoff

ui(λ, q) = λq − ci(q)

ci is the real cost.2 The strategic set for each player i denoted Si:

{ci : IR→ IR+| convex, nondecreasing, bounded subgradients }

∂ci ⊂ [0, p∗] , p∗ is a price cap.

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Equilibrium: Nash

Equilibrium

An equilibrium is (q, λ,m) such that q is a selection of Q(·, ·)and λ is a selection of Λ(·, ·) and m = (mi)i∈G is amixed-strategy equilibrium of the generator game in which eachgenerator submits costs ci ∈ Si with a payoff

Eui(λi(c, ·), qi(c, ·)) =

∫D

[λi(c, d)qi(c, d)− ci(qi(c, d))]dP(d),

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Equilibrium: Nash

Literature

In some cases, for example, using a supply functionequilibria approach there are previous works by Anderson,Philpott, or using variational inequality approach by Pang,Ralph or also using game theory by Smeers, Wilson,Joskow, Tirole, Oren, Borestein, Bushnel, Wolak...Limited network representation or strategic behavior orstrategy space.

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Equilibrium: Nash

Generation costs

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Equilibrium: Nash

Prices NY ISO5/26/13 7:08 PM20120529NYISO.png 569×231 pixels

Page 1 of 1http://www.eia.gov/todayinenergy/images/2012.06.07/20120529NYISO.png

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3 Market Power


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Two-node case

Two nodes case

Symmetric Nash equilibrium

$

q

d d

Nash = c/(1‐2rd)

r

Profit = mul@plier × quan@ty − cost × quan@ty

d < 1/2r

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the ISO Problem: two-node case

Given that each generator reveals a cost ci, the (ISO) solves:

minq,h

2∑i=1

ciqi

s.t. qi − hi + h−i ≥ r2 [h21 + h22] + d for i = 1, 2

qi, hi ≥ 0 for i = 1, 2

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Result

Escobar and J. (ET (2010)) equilibrium exists butproducers charge a price above marginal cost:

Nash = c/(1− 2rd)

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Sensitivity formula

PropositionLet c ∈

∏i∈G Si and ci − ci a Lipschitz function with constant κ.

Then,|Qi(c, d)−Qi(ci, c−i, d)| ≤ κη,

where η = 2 (1+rihi)2

mini∈G ric+i (0)∈]0,+∞[ and

c+i (0) = limy→0+ci(y)−ci(0)

y .

Why? losses => the second-order growth

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Market Power formula

Dangerous incentive: If the number of generators is small orthe topology of the network isolates some demand nodes thenthe generators will play strategically with the ISO exercisingmarket power.

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PropositionThe equilibrium prices pi satisfy

E|pi − γ| ≥E[Qi(pi, p−i, d)]

η

where ηi = 2|Ki|2

(1+max{rehe : e∈Ki}

)2p∗mine∈K re

γ(p−i, d) is a measurable selection of ∂ci(Qi(pi, p−i, d)).

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PropositionLinear case: ci(q) = ciq, then

pi − ci ≥E[Qi(pi, p−i, d)]

η.

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3 Market Power


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The Questions

In an electric network with transmission costs and privateinformation:

Does the usual (price equal Lagrange multiplier) regulationmechanism minimize costs for the society?If not, what is the mechanism that achieves this objective?How does the performance of both systems compare?

Methodology:Bayesian Game TheoryMechanism Design

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Framework

Two-node network with demand d at each node.One producer at each node, with marginal cost ofproduction ci ∼ Fi[ci, ci].Transmission costs rh2, with h the amount sent from onenode to another.

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The ISO Problem

Given that each generator reveals a cost ci, the ISO solves:

minq,h

2∑i=1

ciqi

s.t. qi − hi + h−i ≥ r2 [h21 + h22] + d for i = 1, 2

qi, hi ≥ 0 for i = 1, 2

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The Solution for ISO problem

If we define

H(x, y) = d+1

2r

(x− yx+ y

)2

− 1

r

(x− yx+ y

)and

q = 2

[1−√

1− 2dr

r

]then the solution to this problem can be written as

qi(ci, c−i) =

H(ci, c−i) if H(ci, c−i) ≥ 0 and H(c−i, ci) ≥ 0q if H(c−i, ci) < 00 if H(ci, c−i) < 0

λi(ci, c−i) ≡ pi(ci, c−i) = ci if H(ci, c−i) ≥ 0

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The benchmark game

The Bayesian Game

The game:2 players. Strategies ci ∈ Ci = [ci, ci], i=1,2.Payoff ui(ci, c−i) = (λi(ci, c−i)− ci)qi(ci, c−i),

where ci is the real cost. The Equilibrium:A strategy bi : [ci, ci] −→ IR+ (convex at equilibrium!)In a Nash equilibrium

b(c) ∈ argmaxx

∫C−i

[λi(x, b(c−i))− c]qi(x, b(c−i))f−i(c−i)dc−i

(5)

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The benchmark game

Numerical Approximation

For simplicity Ci = [1, 2].Let k ∈ {0, ..., n− 1}, and b(c) = bk for c ∈ [ kn ,

k+1n ].

The weight of each interval is given bywk = F (k+1

n )− F ( kn).The approximate equilibrium is characterized by:

bk ∈ argmaxx

n−1∑l=0

[λi(x, bl)−rk]qi(x, bl)wl for all k ∈ {0, ..., n−1}

(6)

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The benchmark game

Optimal Mechanism. Principal Agent Model (Myerson)

A direct revelation mechanism M = (q, h, x) consists of anassignment rule (q1, q2, h1, h2) : C −→ R4 and a paymentrule x : C −→ R2.The ex-ante expected profit of a generator of type ci whenparticipates and declares c′i is

Ui(ci, c′i; (q, h, x)) = Ec−i [xi(c

′i, c−i)− ciqi(c′i, c−i)]

A mechanism (q, h, x) is feasible iff:

Ui(ci, ci; (q, h, x)) ≥ Ui(ci, c′i; (q, h, x)) for all ci, c

′i ∈ Ci

Ui(ci, ci; (q, h, x)) ≥ 0 for all ci ∈ Ci

qi(c)− hi(c) + h−i(c) ≥r

2[h21(c) + h22(c)] + d for all c ∈ C

qi(c), hi(c) ≥ 0 for all c ∈ C

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The benchmark game

The Regulator’s Problem

Using the revelation principle, the regulator’s problem can bewritten as:

min

∫C

2∑i=1

xi(c)f(c)dc (7)

subject to (q, h, x) being “feasible”

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The benchmark game

The Regulator’s Problem (II)

It can be rewritten as

min∫C

2∑i=1

qi(c)[ci + Fi(ci)fi(ci)

]f(c)dc

s.t∫

C−i

qi(ci, c−i)f−i(c−i)dc−i is non-increasing in ci

qi(c)− hi(c) + h−i(c) ≥ r2 [h21(c) + h22(c)] + d for all c ∈ C

qi(c), hi(c) ≥ 0 for all c ∈ C

We denote by Ji(ci) = ci + Fi(ci)fi(ci)

the virtual cost of agent i. Weassume it is increasing (Monotone likelihood ratio property:true for any log concave distribution)

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The benchmark game

Solution

An optimal mechanism is given by

qi(ci, c−i) =

H(Ji(ci), J−i(c−i)) if H(Ji(ci), J−i(c−i)) ≥ 0 and H(J−i(c−i), Ji(ci)) ≥ 0q if H(J−i(c−i), Ji(ci)) < 00 if H(Ji(ci), J−i(c−i)) < 0

xi(ci, c−i) = ciqi(ci, c−i) +

ci∫ci

qi(s, c−i)ds

Such a mechanism is dominant strategy incentive compatible.

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Comparing Benchmark with Optimal Mechanism


We consider the family of distributions with densities

fa(x) =

{a(x− 1) + (1− a

4 ) if x ≤ 1.5−a(x− 1) + (1 + 3a

4 ) if x ≥ 1.5

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Asymmetric information

1 1.2 1.4 1.6 1.8 20.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Marginal Cost

Density Functions ga

1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Cumulatives Functions Ga

Marginal Cost

a =1

a =0

a=2

a =3

a =0

a =1

a =2

a =3

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Informational rent

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

F(c)

/f(c)

c

F(x,0)/f(x,0)F(x,1)/f(x,1)F(x,2)/f(x,2)F(x,3)/f(x,3)F(x,4)/f(x,4)

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Social costs for different mechanisms

1

2

3

4

5

6

7

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Socia

l Cos

t

r

Optimal 0 Standard 0

Optimal 2Standard 2

Optimal 4Standard 4

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Robustness and Practical Implementation

The optimal mechanism is detail free. If the designer iswrong about common beliefs, then the mechanism is stillnot bad:

||Xf −Xf || ≤ ||x||1||f − f ||∞ ≤ cq||f − f ||∞

The assignment rule is computationally simple toimplement. It requires solving once the dispatcherproblem, with modified costs.However, the payments are computationally difficult

ciqi(ci, c−i) +

ci∫ci

qi(s, c−i)ds

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Bagh, Adib; Jofre, Alejandro. Weak reciprocallyupper-semicontinuity and better reply secure games: Acomment. Econometrica, Vol. 74 Issue 6 (2006),1715-1721.

Bosh, Paul; Jofre, Alejandro; Schultz, Ruediger. Two-StageStochastic Programs with Mixed Probabilities. SIAMOptimization (2007) 778-788.

Escobar, Juan and Jofre, Alejandro. MonopolisticCompetition in Electricity Markets. Economics Theory, 44,Number 1, 101-121 (2010)

Escobar, Juan and Jofre, Alejandro. Equilibrium analysis ofelectricity auctions.

Escobar, Juan F.; Jofre, Alejandro. Equilibrium analysis fora network market model. Robust optimization-directed

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design, 63–72, Nonconvex Optim. Appl., 81, Springer, NewYork, (2006).

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