Discretely-Constrained MPECs for Electricity Markets

1. Steven A. Gabriel1,2, FlorianLeuthold3 1 Dept. of Civil & Env. Engineering, Co-Director, Engineering and Public Policy Program, University of Maryland, USA 2German Institute for Economic Research (DIW), Berlin Germany 3Technische Universität Dresden, Dresden, Germany/Austrian Power Grid Discretely-Constrained MPECs for Electricity Markets Instituto de Investígación Tecnológica (IIT)Universidad Pontificia ComillasMadrid, Spain3 December 2010

2. 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

3. Problem 1Formulation and Solution of a Discretely-Constrained MPEC as a MIP

4. 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).

5. Electricity Market Modeling Approaches • 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).

6. 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 General Problem Formulation-DC MPEC

7. Disjunctive Constraints Lower-level problem as mixed complementarity problem relating to a market equilibrium Replacing perpendicular condition by disjunctive constraints Lower-level problem as Mixed Integer ProblemK is a constantr is a vector of binary variables

8. 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

9. 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)

10. Electricity Market Model II: Overall MPEC Profit maximization Leader’s generation cap (“x-only constraints”) ISO KKTs including fringe firm j • Problem: Objective bilinear (price*quantity)  Non-convex mixed integer problem

11. Electricity Market Model III: MILP I • Linearization of the objective function, bilinear term replaced by an approximation, discrete generation choices Relating price and associated binary variable Discrete generation levels for leader Binary variable logic • Parameterizing the output decisions of the strategic player

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

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

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

15. Electricity Market Model IV: MILP II Replacing ISO KKT conditions by disjunctive constraints yields a mixed integer linear problem (MILP)

16. Fifteen-Node Network: Structure

17. 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

18. 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)

19. Fifteen-Node Network: Results III • Problem size increases dramatically for strategic behavior runs • The size depends on the number of discrete production choice possibilities • The computation times is long but varies depending on the possible discrete choices

20. 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)

21. Problem/Approach 2Benders Method for DC-MPECs

22. 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

23. 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 • The discrete (often binary) upper-level variables makes this a hard problem in addition to the MPEC computational difficulties

24. 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

25. Theoretical Results • For clarity, assume the lower-level problem is an LP (x is constant) • 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

26. Theoretical Results For clarity, assume the upper-level problem is a DC-MPEC with binary variables Now add conditions that describe S(x)

27. Theoretical Results Now add conditions that describe S(x) But problem in 2 is equivalent to the following Upper-level problem has only the “complicating” variables x

28. Theoretical Results But problem in 2 is equivalent to the following Definition of αcan be transformed as follows

29. Theoretical Results Key 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

30. Numerical Results • Example 1 c vector =0

31. Numerical Results • Example 2

32. Numerical Results • 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

33. From x=-1, 2 Benders cuts From x=+1, 2 Benders cuts Numerical Results • Example 1 (Cont.)

34. Numerical Results • Example 1 (Cont.) • From x=+5, 3 Benders cuts

35. Numerical Results • Example 1 (Cont.) • Sort out N=7 Benders cuts in the increasing order of xj. • Compute intersection point Ik of two neighboring tangential lines.

36. Numerical Results • 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, -10x +1 and +1x +10, respectively or put into one large master problem with “if-then” logic

37. Possible Numerical Complications

38. Numerical ResultsPower Market Equilibrium EACH PRODUCER Maximizes profit subject to operational constraints Mixed integer linear program EACH CONSUMER Maximizes utility subject to minimum demand requirements Linear program Market equilibrium INDEPENDENT SYSTEM OPERATOR Maximizes social welfare subject to power balance Linear program

39. Numerical Results-Power Models (Many Other Random Problems Also Tested)

40. 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