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

22

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

33

Problem 1Formulation and Solution of a Discretely-Constrained MPEC

as a MIP

44

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).

55

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).

66

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

77

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

88

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

99

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)

1010

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

1111

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

1212

Electricity Market Model III: MILP I Logic of the various binary-related constraints

1313

Electricity Market Model III: MILP I Logic of the various binary-related constraints

1414

Electricity Market Model III: MILP I Logic of the various binary-related constraints

1515

Electricity Market Model IV: MILP II

Replacing ISO KKT conditions by disjunctive constraints yields a mixed integer linear problem (MILP)

1616

Fifteen-Node Network: Structure

1717

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

1818

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)

1919

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

2020

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)

2121

Problem/Approach 2Benders Method for DC-

MPECs

2222

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

2323

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

2424

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

2525

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)

2626

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

2727

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

2828

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

2929

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

3030

Example 1

c vector =0

Numerical Results

3131

Example 2

Numerical Results

3232

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

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

3434

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

3535

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

3636

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

3737

Possible Numerical Complications

3838

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

3939

Numerical Results-Power Models (Many Other Random Problems

Also Tested)

4040

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