Arman cdc11

Wholesale Electricity Market: Dynamic Modeling and
Stability

Arman Kiani and Anuradha Annaswamy

Institute of Automatic Control Engineering, Technische Universit¨t M¨nchen, Germany,
a u
Department of Mechanical Engineering, Massachussets Institute of Technology
arman.kiani@tum.de

January 23, 2012

50th IEEE Conference on Decision and Control 2011

Table of contents

1 Motivation
Next Generation Grid
Electricity Market

2 Dynamic Modeling
Dynamical Market
State Based Games
Market Model: Stability Analsys
Asymptotic 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 Sensors: Advanced metering infrastructure (Smart Meters)
Smart Market Design: optimize assets, operate eﬃciently → utilize dynamic
information

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 disequilibrium
process.”


process.”
In Smart Grid we have several active agents as self-interested decision
makers.
Game theory is beginning to emerge as a powerful tool for the design and
coordinate 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 objective
function for each decision maker.
2 Specifying a distributed learning algorithm that enables the agents to

reach a desirable operating point, e.g., a Nash equilibrium of the
designed game.
Learning Dynamics in Games = Dynamics of Disequilibrium
What Does Disequilibrium Mean? A situation where internal and/or external
forces (Uncertainties and Perturbations) prevent market equilibrium from being
reached or cause the market to fall out of balance.



process.”



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



process.”
Adaptive Learning Dynamics
Adjusting adaptively the expectations, myopic [Example: Gradient
Play, Best Response Dynamics, Fictitious Play, ... ]



process.”




process.”


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 to
the framework of strategic form games and introduces an underlying
state space to the game theoretic framework.
In the proposed state based games we focus on myopic players and
static equilibrium concepts similar to that of pure Nash equilibrium.
The state can take on a variety of interpretations ranging from
1 Dynamics for equilibrium selection
2 Dummy players in a strategic form game that are preprogrammed to
behave due to the speciﬁc strategy
3 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 N
Underlying finite coordination state space X
State invariant action set Ai
State dependent cost function Ji : X × A → R
Coordinator 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 , each
player i ∈ N myopically selects an action ai (t) ∈ Ai according to some
specified decision rule.


Electricity Market: Our proposed Model Set Up

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

Minimize f (x)
s.t. gi (x) = 0, ∀i = 1, . . . , N
N
Rji hi (x) ≤ cj , ∀j = 1, . . . L
i=1


Electricity Market: Our proposed Model Set Up

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

Minimize f (x)
s.t. gi (x) = 0, ∀i = 1, . . . , N
N
Rji hi (x) ≤ cj , ∀j = 1, . . . L
i=1

Lagrange function L(x, λ, µ)
L(x, λ, µ) is called Lagrange function of the above optimization problem with
Lagrange multipliers λ and µ as
N L
L(x, λ, µ) = f (x) + λi gi (x) + µj (Rji hi (x) − cj )
i=1 j=1


Dynamic Model of Wholesale Market

Gradient play can be viewed as progressively adjusting x, λ and µ as
follows:
x(t + ε) = x(t) − kx x L(x, λ, µ)ε
λ(t + ε) = λ(t) + kλ λ L(x, λ, µ)ε
+
µ(t + ε) = µ(t) + kµ [ µ L(x, λ, µ)]µ ε

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



Gradient play can be viewed as progressively adjusting x, λ and µ as
follows:
x(t + ε) = x(t) − kx x L(x, λ, µ)ε
λ(t + ε) = λ(t) + kλ λ L(x, λ, µ)ε
+
µ(t + ε) = µ(t) + kµ [ µ L(x, λ, µ)]µ ε

where kx , kλ and kµ are positive scaling parameters which control the
amount 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 the
nonnegative orthant in R+ m

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


Given the following DC Economic Dispatch with quadratic cost functions
Maximize SW = UDj (PDj ) − CGi (PGi )
j∈Dq i∈Gf

s.t. − PGi + PDj + Bnm [δn − δm ] = 0; ρn
i∈θn j∈ϑn m∈Ωn
max
Bnm [δn − δm ] ≤ Pnm ; γnm , ∀n ∈ N; ∀m ∈ Ω


j∈Dq i∈Gf

max
Use the gradient play, we will have:

τGi P˙ = ρn(i) − cGi PGi − bGi
Gi → Dynamics for GenCoi


j∈Dq i∈Gf

max

τD P˙ = cDj PDj + bDj − ρn(j)
j Dj → Dynamics for ConCo j


j∈Dq i∈Gf

max

τδn δ˙n = − Bnm [ρn − ρm + γnm − γmn ] → Phase angles at bus n
m∈Ωn


j∈Dq i∈Gf

max

m∈Ωn

τρn ρ˙n = − PGi + PDj + Bnm [δn − δm ] → Real-Time Price at bus n


j∈Dq i∈Gf

max

m∈Ωn

τρn ρ˙n = − PGi + PDj + Bnm [δn − δm ] → Real-Time Price at bus n
max +
τnm γnm = [Bnm [δn − δm ] − Pnm ]γnm
˙ → Congestion Price for line n − m



Gi

j Dj

τδn δ˙n = − Bnm [ρn − ρm + γnm − γmn ]
m∈Ωn (1)
τρn ρ˙n = − PGi + PDj + Bnm [δn − δm ]
max +
τnm γnm = [Bnm [δn − δm ] −
˙ Pnm ]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 the
market to arrive at the equilibrium


A New Market Model


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)
˙ A1 A2 x1 (t) b
= + (2)
x2 (t)
˙ 0 0 x2 (t) f2 (x1 , x2 )

where
x1 (t) = PG PD δ ρ T +N +2N−1)×1
(Ng d
T −1 T
A2 = 0 0 −Bline Ar τδ 0
x2 (t) = γ1 ... γNt T
Nt×1

−1 −1 T
−τg cg τg A T T −1 T −1
 
0 0 g b = bg τg bd τd 0
−1 −1
 0 τd cd 0 −τd AT d

A1 = 
 
−1
−τδ AT Bline A −1

 0 0 0 r f2 (x1 , x2 ) = τγ [Bline Ar Rx1 − P max ]+
x
−1 −1 −1 2
−τρ Ag τρ Ad τρ AT Bline Ar 0



Let x = [x1 x2 ]T , E = {(x1 , x2 )|A1 x1 + A2 x2 + b = 0 ∧ f2 (x1 , x2 ) = 0},
T T

and Ω(γ) := {x| ||x|| < γ}
Deﬁnition of Market Stability
∗ ∗
The equilibrium point (x1 , x2 ) ∈ E is stable if given > 0, ∃σ such that
x(t) ∈ Ω(σ) ∀x(0) ∈ Ω( )

There exist the feasible sequences of PGi , PDj , and δn such that solutions
starting ”close enough” to the equilibrium (x(0) ∈ Ω( )) remain ”close
enough” forever (x(t) ∈ Ω(σ)).



Let x = [x1 x2 ]T , E = {(x1 , x2 )|A1 x1 + A2 x2 + b = 0 ∧ f2 (x1 , x2 ) = 0},
T T

and Ω(γ) := {x| ||x|| < γ}
Deﬁnition of Market Stability
∗ ∗
The equilibrium point (x1 , x2 ) ∈ E is stable if given > 0, ∃σ such that
x(t) ∈ Ω(σ) ∀x(0) ∈ Ω( )

There exist the feasible sequences of PGi , PDj , and δn such that solutions
starting ”close enough” to the equilibrium (x(0) ∈ Ω( )) remain ”close
enough” forever (x(t) ∈ Ω(σ)).
Is the market stable?

Dynamic Modeling Asymptotic Stability


∗ ∗
Let y1 = x1 − x1 , y2 = x2 − x2 , a positive deﬁnite Lyapunov function
V (y1 , y2 ) = y1 P1 y1 + y2 P2 y2 , and d = 2λmin (P2 )ψmin2 min (Q)
T T
τγ β
λ
max



∗ ∗
T T
τγ β
λ
max

Theorem (Asymptotic Stability)
∗ ∗
Let A1 be Hurwitz. Then the equilibrium (x1 , x2 ) ∈ E is asymptotically
stable for all initial conditions in Ωcmax = {(y1 , y2 ) | V (y1 , y2 ) ≤ c} for a
cmax > 0 such that Ωcmax D = {(y1 , y2 ) | ||y2 || ≤ d}.



∗ ∗
T T
τγ β
λ
max

∗ ∗

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



∗ ∗
T T
τγ β
λ
max

∗ ∗

Remarks
The region of attraction Ωmax for which stability and asymptotic
stability hold places an implicit bound on the congestion price.
In particular, it implies that the congestion price needs to be smaller
than d, which is proportional to thermal limit of transmission lines.

Simulation Results

Simulation Results

Trajectories of the resulting dynamics
The corresponding region of attraction Ωcmax such that Ωcmax D. It was
found 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” result
Related to congestion rent
Advantages: Model allows us to
design a stable market
utilize uncertain renewable generation
incorporate elastic demands

Summary and Ongoing work

Ongoing work

Strong relation to state-based games
Dynamic model: A disequilibrium process
Kenneth Arrow: ”The attainment of equilibrium requires a
disequilibrium process.”
Eﬀect of renewable sources uncertainty on stability of electricity
market
Uncertainty analysis
Equality constraints and local stability

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