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Wholesale Electricity Market: Dynamic Modeling andStability

Arman Kiani and Anuradha Annaswamy

Institute of Automatic Control Engineering, Technische Universitat Munchen, Germany,Department of Mechanical Engineering, Massachussets Institute of Technology

[email protected]

January 23, 2012

50th IEEE Conference on Decision and Control 2011

Table of contents

1 MotivationNext Generation GridElectricity Market

2 Dynamic ModelingDynamical MarketState Based GamesMarket Model: Stability AnalsysAsymptotic Stability

3 Simulation Results

4 Summary and Ongoing work

Motivation Next Generation Grid

Next Generation Grid: What makes a grid smart?

Smart Resources: Renewable Energy Resources

Smart Participation: Real-Time Pricing + Demand Response

Smart Sensors: Advanced metering infrastructure (Smart Meters)

Smart Market Design: optimize assets, operate efficiently → utilize dynamicinformation

Motivation Electricity Market

Auction Process in Electricity Market

Power generation scheduling is conducted through a market mechanism:

Use of an auction market - bids from Generating Companies (GenCo)and Consumer Companies (ConCo).

Any uncertainties are managed through a contingency analysis.

Dynamic Modeling

Electricity Market: Current practice

Dynamic Modeling Dynamical Market

Learning Dynamics in Games

Kenneth Arrow: ”The attainment of equilibrium requires a disequilibriumprocess.”

In Smart Grid we have several active agents as self-interested decisionmakers.

Game theory is beginning to emerge as a powerful tool for the design andcoordinate of multiagent systems.

Utilizing Game theory for this purpose requires two steps.1 Modeling the agent as self-interested decision makers in a game

theoretic environment. Defining a set of choices and a local objectivefunction for each decision maker.

2 Specifying a distributed learning algorithm that enables the agents toreach a desirable operating point, e.g., a Nash equilibrium of thedesigned game.

Learning Dynamics in Games = Dynamics of Disequilibrium

What Does Disequilibrium Mean? A situation where internal and/or externalforces (Uncertainties and Perturbations) prevent market equilibrium from beingreached or cause the market to fall out of balance.

In Smart Grid we have several active agents as self-interested decisionmakers.

Game theory is beginning to emerge as a powerful tool for the design andcoordinate of multiagent systems.

Utilizing Game theory for this purpose requires two steps.1 Modeling the agent as self-interested decision makers in a game

theoretic environment. Defining a set of choices and a local objectivefunction for each decision maker.

2 Specifying a distributed learning algorithm that enables the agents toreach a desirable operating point, e.g., a Nash equilibrium of thedesigned game.

Bayesian Learning Dynamics

Update beliefs (about an underlying state or opponent strategies)based on new information optimally (i.e., in a Bayesian manner)

Adaptive Learning Dynamics

Adjusting adaptively the expectations, myopic [Example: GradientPlay, Best Response Dynamics, Fictitious Play, ... ]

We will use the terms dynamics and learning dynamics in games interchangeably.

Dynamic Modeling State Based Games

State Based Games

Using the notion of state based game, we consider an extension tothe framework of strategic form games and introduces an underlyingstate space to the game theoretic framework.

In the proposed state based games we focus on myopic players andstatic equilibrium concepts similar to that of pure Nash equilibrium.

The state can take on a variety of interpretations ranging from1 Dynamics for equilibrium selection2 Dummy players in a strategic form game that are preprogrammed to

behave due to the specific strategy3 Disequilibrium process to attain the equilibrium

State Based Games

Definition: A State Based game

A State Based game G characterized by the tuple G =〈N,X , (Ai )i∈N , (Ji )i∈N , f 〉, which consists of

Player set NUnderlying finite coordination state space X

State invariant action set Ai

State dependent cost function Ji : X × A→ RCoordinator mechanism function f : X × A→ X

The sequence of actions a(0), a(1), .. and coordination states x(0), x(1), ...is generated according to the disequilibrium process. At any time t0, eachplayer i ∈ N myopically selects an action ai (t) ∈ Ai according to somespecified decision rule.

Electricity Market: Our proposed Model Set Up

A dynamic model based on sub-gradients in a nonlinear optimizationproblem stated below:

Minimize f (x)

s.t. gi (x) = 0, ∀i = 1, . . . ,N

N∑i=1

Rjihi (x) ≤ cj , ∀j = 1, . . . L

Lagrange function L(x , λ, µ)

L(x , λ, µ) is called Lagrange function of the above optimization problem withLagrange multipliers λ and µ as

L(x , λ, µ) = f (x) +N∑i=1

λigi (x) +L∑

µj(Rjihi (x)− cj)

Electricity Market: Our proposed Model Set Up

A dynamic model based on sub-gradients in a nonlinear optimizationproblem stated below:

Minimize f (x)

s.t. gi (x) = 0, ∀i = 1, . . . ,N

N∑i=1

Rjihi (x) ≤ cj , ∀j = 1, . . . L

Lagrange function L(x , λ, µ)

L(x , λ, µ) is called Lagrange function of the above optimization problem withLagrange multipliers λ and µ as

L(x , λ, µ) = f (x) +N∑i=1

λigi (x) +L∑

µj(Rjihi (x)− cj)

Dynamic Model of Wholesale Market

Gradient play can be viewed as progressively adjusting x , λ and µ asfollows:

x(t + ε) = x(t)− kx∇xL(x , λ, µ)ε

λ(t + ε) = λ(t) + kλ∇λL(x , λ, µ)ε

µ(t + ε) = µ(t) + kµ [∇µL(x , λ, µ)]+µ ε

where kx , kλ and kµ are positive scaling parameters which control theamount of change in the direction of the gradient.

Nonnegative projection of congestion cost

[h(x , y)]+y

denotes the projection of h(x , y) on euclidean projection on thenonnegative orthant in Rm

[h(x , y)

h(x , y) if y > 0,

max(0, h(x , y)) if y = 0.

Gradient play can be viewed as progressively adjusting x , λ and µ asfollows:

x(t + ε) = x(t)− kx∇xL(x , λ, µ)ε

λ(t + ε) = λ(t) + kλ∇λL(x , λ, µ)ε

µ(t + ε) = µ(t) + kµ [∇µL(x , λ, µ)]+µ ε

where kx , kλ and kµ are positive scaling parameters which control theamount of change in the direction of the gradient.

Nonnegative projection of congestion cost

[h(x , y)]+y

denotes the projection of h(x , y) on euclidean projection on thenonnegative orthant in Rm

[h(x , y)

h(x , y) if y > 0,

max(0, h(x , y)) if y = 0.

Given the following DC Economic Dispatch with quadratic cost functions

Maximize SW =∑j∈Dq

UDj(PDj)−∑i∈Gf

CGi (PGi )

s.t. −∑i∈θn

PGi +∑j∈ϑn

PDj +∑m∈Ωn

Bnm [δn − δm] = 0; ρn

Bnm [δn − δm] ≤ Pmaxnm ; γnm,∀n ∈ N;∀m ∈ Ω

Use the gradient play, we will have:

τGi˙PGi = ρn(i) − cGiPGi − bGi → Dynamics for GenCoi

τDj˙PDj = cDjPDj + bDj − ρn(j) → Dynamics for ConCo j

τδn δn = −∑m∈Ωn

Bnm [ρn − ρm + γnm − γmn] → Phase angles at bus n

τρn ρn = −∑i∈θn

PGi +∑j∈ϑn

PDj +∑m∈Ωn

Bnm [δn − δm] → Real-Time Price at bus n

τnmγnm = [Bnm [δn − δm]− Pmaxnm ]+

γnm→ Congestion Price for line n −m

CGi (PGi )

s.t. −∑i∈θn

PGi +∑j∈ϑn

PDj +∑m∈Ωn

PGi +∑j∈ϑn

PDj +∑m∈Ωn

CGi (PGi )

s.t. −∑i∈θn

PGi +∑j∈ϑn

PDj +∑m∈Ωn

PGi +∑j∈ϑn

PDj +∑m∈Ωn

CGi (PGi )

s.t. −∑i∈θn

PGi +∑j∈ϑn

PDj +∑m∈Ωn

PGi +∑j∈ϑn

PDj +∑m∈Ωn

CGi (PGi )

s.t. −∑i∈θn

PGi +∑j∈ϑn

PDj +∑m∈Ωn

PGi +∑j∈ϑn

PDj +∑m∈Ωn

CGi (PGi )

s.t. −∑i∈θn

PGi +∑j∈ϑn

PDj +∑m∈Ωn

PGi +∑j∈ϑn

PDj +∑m∈Ωn

τGi˙PGi = ρn(i) − cGiPGi − bGi

τDj˙PDj = cDjPDj + bDj − ρn(j)

Bnm [ρn − ρm + γnm − γmn]

PGi +∑j∈ϑn

PDj +∑m∈Ωn

Bnm [δn − δm]

τnmγnm = [Bnm [δn − δm]− Pmaxnm ]+γnm

Trajectory of (1):

Distinct from the equilibrium (solutions of KKT conditions)

Converges to the equilibrium if stable

Represents desired exchange of information between key players in themarket to arrive at the equilibrium

A New Market Model

Use of feedback in converging to the equilibrium.

GenCos and ConCos adjust their power level using a recursive process.

Price is a Public Signal that guides all entities to adjust efficiently.

A New Market Model

Use of feedback in converging to the equilibrium.

GenCos and ConCos adjust their power level using a recursive process.

Price is a Public Signal that guides all entities to adjust efficiently.

Dynamic Modeling Market Model: Stability Analsys

Market Model: Stability Analysis

A compact representation of the model:[x1(t)x2(t)

[A1 A2

] [x1(t)x2(t)

f2(x1, x2)

x1(t) =[PG PD δ ρ

]T(Ng+Nd +2N−1)×1

x2(t) =[γ1 . . . γNt

]TNt×1

−τ−1

g cg 0 0 τ−1g AT

0 τ−1d

cd 0 −τ−1d

0 0 0 −τ−1δ

ATr BlineA

−τ−1ρ Ag τ−1

ρ Ad τ−1ρ ATBlineAr 0

0 0 −BTlineArτ

−1δ

b =[bTg τ

−1g bTd τ

f2(x1, x2) =[τ−1γ [BlineArRx1 − Pmax ]+x2

Let x = [xT1 xT2 ]T , E = (x1, x2)|A1x1 + A2x2 + b = 0 ∧ f2(x1, x2) = 0,and Ω(γ) := x | ||x || < γ

Definition of Market Stability

The equilibrium point (x∗1 , x∗2 ) ∈ E is stable if given ε > 0, ∃σ such that

x(t) ∈ Ω(σ) ∀x(0) ∈ Ω(ε)

There exist the feasible sequences of PGi, PDj

, and δn such that solutionsstarting ”close enough” to the equilibrium (x(0) ∈ Ω(ε)) remain ”closeenough” forever (x(t) ∈ Ω(σ)).

Is the market stable?

Let x = [xT1 xT2 ]T , E = (x1, x2)|A1x1 + A2x2 + b = 0 ∧ f2(x1, x2) = 0,and Ω(γ) := x | ||x || < γ

Definition of Market Stability

The equilibrium point (x∗1 , x∗2 ) ∈ E is stable if given ε > 0, ∃σ such that

x(t) ∈ Ω(σ) ∀x(0) ∈ Ω(ε)

There exist the feasible sequences of PGi, PDj

, and δn such that solutionsstarting ”close enough” to the equilibrium (x(0) ∈ Ω(ε)) remain ”closeenough” forever (x(t) ∈ Ω(σ)).Is the market stable?

Dynamic Modeling Asymptotic Stability

Let y1 = x1 − x∗1 , y2 = x2 − x∗2 , a positive definite Lyapunov function

V (y1, y2) = yT1 P1y1 + yT2 P2y2, and d = 2λmin(P2)ψminλmin(Q)τγmax β

Theorem (Asymptotic Stability)

Let A1 be Hurwitz. Then the equilibrium (x∗1 , x∗2 ) ∈ E is asymptotically

stable for all initial conditions in Ωcmax = (y1, y2) | V (y1, y2) ≤ c for acmax > 0 such that Ωcmax ( D = (y1, y2) | ||y2|| ≤ d.

Remarks

The region of attraction Ωmax for which stability and asymptoticstability hold places an implicit bound on the congestion price.

In particular, it implies that the congestion price needs to be smallerthan d , which is proportional to thermal limit of transmission lines.

Remarks

Simulation Results

Trajectories of the resulting dynamics

The corresponding region of attraction Ωcmax such that Ωcmax ( D. It wasfound that cmax = 38.4.

The matrix A1 is Hurwitz.

Summary and Ongoing work

Summary

A New Market Model was proposed

Recursive, dynamic convergence to equilibrium

Enables stability analysis

Not globally stable

”Domain of attraction” resultRelated to congestion rent

Advantages: Model allows us to

design a stable marketutilize uncertain renewable generationincorporate elastic demands

Summary and Ongoing work

Ongoing work

Strong relation to state-based games

Dynamic model: A disequilibrium processKenneth Arrow: ”The attainment of equilibrium requires adisequilibrium process.”

Effect of renewable sources uncertainty on stability of electricitymarket

Uncertainty analysis

Equality constraints and local stability

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