Arman cdc11
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Transcript of Arman cdc11
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
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 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 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 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 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 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.
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.
Dynamic Modeling Dynamical Market
Learning Dynamics in Games
Kenneth Arrow: ”The attainment of equilibrium requires a disequilibriumprocess.”
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, ... ]
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.
We will use the terms dynamics and learning dynamics in games interchangeably.
Dynamic Modeling Dynamical Market
Learning Dynamics in Games
Kenneth Arrow: ”The attainment of equilibrium requires a disequilibriumprocess.”
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, ... ]
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.
We will use the terms dynamics and learning dynamics in games interchangeably.
Dynamic Modeling Dynamical Market
Learning Dynamics in Games
Kenneth Arrow: ”The attainment of equilibrium requires a disequilibriumprocess.”
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, ... ]
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.
We will use the terms dynamics and learning dynamics in games interchangeably.
Dynamic Modeling Dynamical Market
Learning Dynamics in Games
Kenneth Arrow: ”The attainment of equilibrium requires a disequilibriumprocess.”
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, ... ]
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.
We will use the terms dynamics and learning dynamics in games interchangeably.
Dynamic Modeling Dynamical Market
Learning Dynamics in Games
Kenneth Arrow: ”The attainment of equilibrium requires a disequilibriumprocess.”
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, ... ]
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.
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
Dynamic Modeling State Based Games
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.
Dynamic Modeling State Based Games
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∑
j=1
µj(Rjihi (x)− cj)
Dynamic Modeling State Based Games
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∑
j=1
µj(Rjihi (x)− cj)
Dynamic Modeling State Based Games
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)
]+y
=
h(x , y) if y > 0,
max(0, h(x , y)) if y = 0.
Dynamic Modeling State Based Games
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)
]+y
=
h(x , y) if y > 0,
max(0, h(x , y)) if y = 0.
Dynamic Modeling State Based Games
Dynamic Model of Wholesale Market
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
Dynamic Modeling State Based Games
Dynamic Model of Wholesale Market
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
Dynamic Modeling State Based Games
Dynamic Model of Wholesale Market
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
Dynamic Modeling State Based Games
Dynamic Model of Wholesale Market
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
Dynamic Modeling State Based Games
Dynamic Model of Wholesale Market
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
Dynamic Modeling State Based Games
Dynamic Model of Wholesale Market
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
Dynamic Modeling State Based Games
Dynamic Model of Wholesale Market
τGi˙PGi = ρn(i) − cGiPGi − bGi
τDj˙PDj = cDjPDj + bDj − ρn(j)
τδn δn = −∑m∈Ωn
Bnm [ρn − ρm + γnm − γmn]
τρn ρn = −∑i∈θn
PGi +∑j∈ϑn
PDj +∑m∈Ωn
Bnm [δn − δm]
τnmγnm = [Bnm [δn − δm]− Pmaxnm ]+γnm
(1)
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
Dynamic Modeling State Based Games
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 State Based Games
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
0 0
] [x1(t)x2(t)
]+
[b
f2(x1, x2)
](2)
where
x1(t) =[PG PD δ ρ
]T(Ng+Nd +2N−1)×1
x2(t) =[γ1 . . . γNt
]TNt×1
A1 =
−τ−1
g cg 0 0 τ−1g AT
g
0 τ−1d
cd 0 −τ−1d
ATd
0 0 0 −τ−1δ
ATr BlineA
−τ−1ρ Ag τ−1
ρ Ad τ−1ρ ATBlineAr 0
A2 =[
0 0 −BTlineArτ
−1δ
0]T
b =[bTg τ
−1g bTd τ
−1d
0]T
f2(x1, x2) =[τ−1γ [BlineArRx1 − Pmax ]+x2
]
Dynamic Modeling Market Model: Stability Analsys
Market Model: Stability Analysis
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 Market Model: Stability Analsys
Market Model: Stability Analysis
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
Market Model: Stability Analysis
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
2
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.
Dynamic Modeling Asymptotic Stability
Market Model: Stability Analysis
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
2
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.
Dynamic Modeling Asymptotic Stability
Market Model: Stability Analysis
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
2
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.
Dynamic Modeling Asymptotic Stability
Market Model: Stability Analysis
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
2
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.
Simulation Results
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