discretely-constrained mpecs for electricity markets steven a. gabriel 1,2, florian leuthold 3 1...
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Discretely-Constrained MPECs for Electricity Markets
Steven A. Gabriel1,2, Florian Leuthold 3
1 1 Dept. of Civil & Env. Engineering, Co-Director, Engineering and Public Policy Program, Dept. of Civil & Env. Engineering, Co-Director, Engineering and Public Policy Program, University of Maryland, USAUniversity of Maryland, USA
22German Institute for Economic Research (DIW), Berlin GermanyGerman Institute for Economic Research (DIW), Berlin Germany33Technische UniversitTechnische Universität Dresden, Dresden, Germany/Austrian Power Gridät Dresden, Dresden, Germany/Austrian Power Grid
Instituto de Investígación Tecnológica (IIT)
Universidad Pontificia ComillasMadrid, Spain
3 December 2010
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Outline of Talk Overview and Motivation for Problem 1- MIP
Mathematical Formulation
Numerical Results
Problem/Approach 2- Benders Method for DC-MPEC
Conclusions
Reference for Problem 1:S.A. Gabriel, F.U. Leuthold , 2010. "Solving Discretely-Constrained MPEC Problems with Applications in Electric Power Markets," Energy Economics, 32, 3-14.
Reference for Problem/Approach 2:S.A. Gabriel, Y. Shim, A.J. Conejo, S. de la Torre, R. García-Bertrand. 2009. "A Benders Decomposition Method for Discretely-Constrained Mathematical Programs with Equilibrium Constraints with Applications in Energy,“ Journal of the Operational Research Society 61, 1404-1419
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Problem 1Formulation and Solution of a Discretely-Constrained MPEC
as a MIP
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Motivation: Market Structures in Europe
France: EDF has a market share of 80% Germany: EON+RWE 55% market
share; +Vattenfall+EnBW 85% market share
Liberalization of vertical integrated companies proceeds sluggish
Former integrated companies have information advantages in terms of geographical specifics and network knowledge
This gives rise potentially to one (or more) dominant players in the market, rest can be considered as “competitive fringe”
Need for modeling that takes this structure into account Source: EDF (2008), EON (2008), Google
Maps (2008), RWE (2008).
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Electricity Market Modeling Approaches
optimization models equilibrium models simulation models (e.g. agent based)
single firm profit
maximization
welfare maximization/
perfect competition
prices exogenous/
perfect competition
prices endogenous/
imperfect competition
imperfect competition
Bertrand
Cournot
Stackelberg
Conjectural Variations
SFE
Collusion
Simulation models do not follow a single mathematical formulation For the rest: The type of competition mostly defines the resulting model
– Perfect vs. imperfect competition Optimization vs. equilibrium models
– One stage vs. two/three stages approach Combining this with further characteristics of electricity markets can make models basically impossible to
solve
– Discrete variables (e.g., investments, start-up, unit commitment)
– Stochastic modeling (e.g., stochastic demand, stochastic wind generation)
Current focus of research: Solving discretely-constrained equilibrium models
Source: Day et al. (2002), Görner et al. (2008), Kahn (1998), Smeers (1997), Ventosa et al. (2005).
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General Problem Formulation-DC MPEC
x: dominant firm upper- level planning variables (e.g., generation), some may be discrete/some continuous
y: lower-level market/ISO variables, all continuous(e.g., market prices, phase angles)
quadratic objective function (e.g., min costs-revenue), willinvolve product of price and generation, bilinear (non-convex) term
Joint x-y constraints
x-only constraints
y-only constraints, includes lower-level problem solution set S(x) as a function of x
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Disjunctive Constraints
1,0r
Lower-level problem as mixed complementarity problem relating to a market equilibrium
Lower-level problem as Mixed Integer ProblemK is a constantr is a vector of binary variables
Replacing perpendicular condition by disjunctive constraints
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Electricity Market Model I: Fundamental Idea
Assumption: Stackelberg competition
– Leader makes output decision
– Follower decides taking the leaders decision as given
Leader: Strategic production company
– Maximizes individual profit under maximum generation constraints and non-negative production (upper-level problem)
– Takes into account followers’ decisions (lower-level problem)
Follower: ISO
– Maximizes social welfare
– Decides over the output decision of the competitive fringe
– Takes into account technical constraints such as maximum fringe generation, line flow, and energy balance constraints
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Electricity Market Model II: ISO Problem
Welfare maximization
Energy balance
Line flow cap
Generation cap
Voltage angle 0 for slack
Non-negative demand
Non-negative production
KKT conditions for the lower-level problem are necessary and sufficient, they are S(x)
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Electricity Market Model II: Overall MPEC
Problem: Objective bilinear (price*quantity) Non-convex mixed integer problem
Profit maximization
Leader’s generation cap (“x-only constraints”)
ISO KKTs including fringe firm j
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Electricity Market Model III: MILP I Linearization of the objective function, bilinear term replaced by an approximation, discrete generation choices
Parameterizing the output decisions of the strategic player
Discrete generation levels for leader
Binary variable logic
Relating price and associated binary variable
trueconditionsboth for when iablebinary var
selected is gfor when iablebinary var
0for when iablebinary var
,
insu,,
n
insu
insu
n
q
q
q
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Electricity Market Model III: MILP I Logic of the various binary-related constraints
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Electricity Market Model III: MILP I Logic of the various binary-related constraints
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Electricity Market Model III: MILP I Logic of the various binary-related constraints
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Electricity Market Model IV: MILP II
Replacing ISO KKT conditions by disjunctive constraints yields a mixed integer linear problem (MILP)
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Fifteen-Node Network: Structure
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Fifteen-Node Network: Results IGeneration (MWh)
We compare perfect competition (comp) to an imperfect competition (strat) run
It can be shown that under strategic behavior, the player produces in total less than in the competitive run
Why? Next slide
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Fifteen-Node Network: Results II
Because the player can influence the prices at nodes where it is profitable for him, in order to maximize individual profits
Also, a player can use network constraints in order to game (price differences)
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The computation times is long but varies depending on the possible discrete choices
Fifteen-Node Network: Results III
Problem size increases dramatically for strategic behavior runs
The size depends on the number of discrete production choice possibilities
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Future Work for Problem 1 Speeding up the solution of the DC-MPEC expressed as a mixed-integer program
– When RWE was the leader, solution time was 4 minutes– When EDF was the leader, solution time was 5 hours! (presumably due to the
fact that EDF had too many choices for how to generate power)– Need to add cuts to reduced search procedure time
Consideration of when the lower-level problem can also have integer variables– For example, ISO or competitive fringe go/no decisions to make– May use a variant of Benders decomposition to solve this (Gabriel et al., 2007)
Consideration of “n-1” problem for network resilience
Additional discrete variables– Investment decisions– Unit commitment decisions
Gauss-Seidel/SOR approach for solving related EPECs (top-level is an equilibrium problem)
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Problem/Approach 2Benders Method for DC-
MPECs
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Overview and Motivation Many problems in infrastructure planning involve
– Some central authority (e.g., ISO) making planning decisions– Users of the infrastructure then reacting to these decisions
This can be construed an instance of a Stackelberg game with the central authority as the leader and the users as the followers, i.e., an MPEC
min ( , )
. .
,
where
, are upper, lower-level variables, resp.
feasible region for upper-level problem
solution set of lower-level problem
f x y
s t
x y
y S x
x y
S x
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Overview and Motivation In this research, we focus on certain class of MPECs in which
– The central authority makes decisions on discrete (and possibly continuous) variables
– The users are modeled by optimization or complementarity problems
Feasible Region
= , | , ,
where
: ,
n
n m p p
x y x Z g x y a
g R R a R
The discrete (often binary) upper-level variables makes this a
hard problem in addition to the MPEC computational difficulties
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Overview and Motivation Electric Power Example
– ISO determines, via maximize welfare rules, which generators run or don’t run (binary planning variables)– Maximum profit rules involve the product of locational marginal prices and generation variables both being
lower-level variables y but depending on the upper-level planning variables x Telecommunications Planning Example
– Wireless Free Space Optical (FSO) ring topology which must be reconfigured in real-time due to changing atmospheric and other conditions
– Telecommunications planning involves which nodes and links are selected for on the fly configuration (and possibly link capacities), discrete upper level variables
– User load on a given network, lower-level variables
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Theoretical Results For clarity, assume the lower-level problem is an LP
LCP( ):
0 0
0 0
T
x
y e M z
z My Nx k
min
. .
0
Te y
s t
My k Nx
y
Note that lower-level problem is a function of upper-level planning variable vector x
Could start with a convex QP or LCP and still have a lower-level problem that is an LCP so this form is somewhat general
(x is constant)
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Theoretical Results
min
. .
0,1
T T
n
c x d y
s t
x
Ax By a
y S x
1. For clarity, assume the upper-level problem is a DC-MPEC with binary variables
2. Now add conditions that describe S(x)
min
. .
0,1
0 0
0 0
T T
n
T
c x d y
s t
x
Ax By a
y e M z
z My Nx k
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Theoretical Results
min
. .
0,1
0 0
0 0
T T
n
T
c x d y
s t
x
Ax By a
y e M z
z My Nx k
2. Now add conditions that describe S(x)3. But problem in 2 is equivalent to the
following
,
min
. .
0,1
where x min |
0 0
0 0
T
n
Ty z
T
c x x
s t
x
d y
Ax By a
y e M z
z My Nx k
Upper-level problem has only the“complicating” variables x
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Theoretical Results
,
min
. .
0,1
where x min |
0 0
0 0
T
n
Ty z
T
c x x
s t
x
d y
Ax By a
y e M z
z My Nx k
3. But problem in 2 is equivalent to the following
4. Definition of α can be transformed as follows
, , , x min |
0 1
0
0 1
0
0,1 , 0,1
T
y z b b
T
m k
d y
Ax By a
y C b
e M z Cb
z C b
My Nx k Cb
b b
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Theoretical ResultsKey Results It can be shown that α is a piecewise-linear (not necessarily
convex) function of the upper-level planning variables x Incorporating this result in 3 and 4 means that the DC-MPEC can
be solved by solving a sequence of mixed-integer linear programs with approximations for α (solved one problem with lower-level binary variables)
The results can at times be sensitive to choice of the constant “C” in the lower-level problem, need care in choosing this value
If α where a piecewise linear and convex function of x, could just use Benders method
So our approach is to use a variant of Benders within each subdomain of x that relates to a convex piece of α
Tricky part is to determine the domain decomposition for x relating to convex pieces of α, will use sampling of points
Need to be careful since subgradient information on α may be bad approximation
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Example 1
c vector =0
Numerical Results
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Example 2
Numerical Results
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Example 1 (Cont.)
– Step 1: Initial sampling points, x={-5,-1,+1,+5}
– Step 2: Generate/Collect all Benders cuts generated from each sampling point.
• From x=-5, 3 Benders cuts
x(0)= -5,alpha(-5)= 5,slope= -1
alpha >= 2.5alpha >= 5-(x+5)
x(2)= 1,alpha(1)= 2.5,slope= 0
alpha >= 2+0.33(x-10)
x=1
x=10
x(1)= 10,alpha(10)= 2,slope= 0.3333
Numerical Results
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3333
From x=-1, 2 Benders cuts
From x=+1, 2 Benders cuts
Example 1 (Cont.)
"enumeration code"x(0)= -1,alpha(-1)= 2.5,slope= 0
"enumeration code"x(1)= -10,alpha(-10)= 10,slope= -1
alpha >= 2.5
alpha >= 10-(x+10)
x=-10
x(0)= 1,alpha(1)= 2.5,slope= -2.3333
alpha >= 2.5-2.3333(x-1)
alpha >= 2+0.3333(x-10)
x=-10
x(1)= 10,alpha(10)= 2,slope= 0.3333
Numerical Results
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From x=+5, 3 Benders cuts
Example 1 (Cont.)
alpha >= 0.33+0.33(x-5)
x(1)= -10,alpha(-10)= 10,slope= -1
x(0)= 5,alpha(5)= 0.3333,slope= 0.3333
alpha >= 2.5
alpha >= 10-(x+10)
x=-10x=+1
x(2)= 1,alpha(1)= 2.5,slope= 0
Numerical Results
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Example 1 (Cont.)
– Sort out N=7 Benders cuts in the increasing order of xj.
– Compute intersection point Ik of two neighboring tangential lines.
Numerical Results
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Example 1 (Cont.)
– Observe slope change at each intersection point.• At the intersection point x=+1, the slope was changed from 0 to -2.333, which
implies non-convexity between the left side and the right side of x=+1.
– Solve the upper and lower level problem for each subdomain, -10x +1 and +1x +10, respectively or put into one large master problem with “if-then” logic
Numerical Results
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Possible Numerical Complications
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EACH PRODUCER
Maximizes profit subject to operational constraints
Mixed integer linear program
EACH CONSUMER
Maximizes utility subject tominimum demand requirements
Linear program
INDEPENDENT SYSTEM OPERATOR
Maximizes social welfare subject to power balance
Linear program
Market equilibrium
Numerical ResultsPower Market Equilibrium
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Numerical Results-Power Models (Many Other Random Problems
Also Tested)
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Future Work
Test Benders variant on a variety of planning problems– LP subproblem or– LCP subproblem
Extend results to include – Nonlinear subproblem objectives– Nonlinear linking constraints g(x,y)– Nonlinear upper-level problem objective– Try more problems with lower-level integer variables