Valid Inequalities for Optimal Transmission Switching Hyemin Jeon Jeff Linderoth Jim Luedtke Dept. of ISyE UW-Madison Burak Kocuk Santanu Dey Andy Sun Dept. of ISyE Georgia Tech 19th Combinatorial Optimization Workshop Aussois, France, January, 2015 Many Authors (UW-Madison & Ga. Tech) IP for Transmission Switching Aussois 1 / 28
Power Background Notation This is a “Power Systems” Talk But I don’t know much about power systems Hopefully you will find the mathematics interesting Economic Dispatch Focus today is on a simple problem of meeting demand for power at minimum cost Many Authors (UW-Madison & Ga. Tech) IP for Transmission Switching Aussois 2 / 28
Power Background Notation Power Grid Networks Look Weird Many Authors (UW-Madison & Ga. Tech) IP for Transmission Switching Aussois 3 / 28
Power Background Notation It’s Just a Network 14 Power Network: ( N, A ) with... 12 11 G ⊂ N : generation nodes 13 10 D ⊂ N : demand nodes 8 6 7 9 Load forecasts (MW) b i for i ∈ D Generation cost ($/MW) c i and 5 4 capability p i (MW) for i ∈ G Peak load rating (MW) u ij for 3 ( i, j ) ∈ A 1 2 Economic Dispatch Problem Determine power generation levels for i ∈ G and power transmission levels for ( i, j ) ∈ A to meet demands b i , i ∈ D , at minimum cost Many Authors (UW-Madison & Ga. Tech) IP for Transmission Switching Aussois 4 / 28
Power Background Optimal Power Flow Power Flow Electric power grids follow the laws of physics, characterized by nonlinear, nonconvex equations Direct control is difficult—We cannot dictate how power will flow. Making many assumptions, we can model the real power flow using the Direct Current (DC) model. DC Power Flow Assumption The (real) power x ij transmit over line ( i, j ) ∈ A is proportional to angle differences ( θ i − θ j ) at the endpoint nodes i ∈ N and j ∈ N . x ij = α ij ( θ i − θ j ) Many Authors (UW-Madison & Ga. Tech) IP for Transmission Switching Aussois 5 / 28
Power Background Economic Dispatch Linear Program for (DC) Economic Dispatch � c i p i min x,p,θ i ∈ G p i ∀ i ∈ G � � s.t. x ij − x ji = d i ∀ i ∈ D 0 ∀ i ∈ N \ G \ D j :( i,j ) ∈ E j :( j,i ) ∈ E − u ij ≤ x ij ≤ u ij ∀ ( i, j ) ∈ E p i ≤ p i ≤ p i ∀ i ∈ G x ij = α ij ( θ i − θ j ) ∀ ( i, j ) ∈ E x ij ∈ R ∀ ( i, j ) ∈ E p i ∈ R + ∀ i ∈ G θ i ∈ R ∀ i ∈ N x , θ need not be ≥ 0 Bounds on x , but no a priori bounds on θ (Usually derived from bounds on x ) Many Authors (UW-Madison & Ga. Tech) IP for Transmission Switching Aussois 6 / 28
Power Background Tradeoffs MCNF++ This economic dispatch problem is just A a min cost network flow problem with some additional “potential” constraints B C The potential drop ( θ A − θ D ) must be D the same aloing the paths: A → B → D and A → C → D “Braess Paradox” If line ( C, D ) didn’t exist, I wouldn’t have to enforce this potential balance constraint. Thus, removing lines of the transmission network may actually increase the efficiency of delivery. Many Authors (UW-Madison & Ga. Tech) IP for Transmission Switching Aussois 7 / 28
Power Background Tradeoffs Transmission Switching Tradeoff Having Lines Allows You to Send Flow: − U ij ≤ x ij ≤ U ij ∀ ( i, j ) ∈ E Having Lines Induces Constraints in the Network: x ij = α ij ( θ i − θ j ) ∀ ( i, j ) ∈ E Fisher, O’Neill & Ferris (’08) show that efficiency improved by switching off transmission lines Lines Off % Improvement 1 6.3% 2 12.4% 3 19.9% 4 20.5% 24.9% ∞ Many Authors (UW-Madison & Ga. Tech) IP for Transmission Switching Aussois 8 / 28
Transmission Switching Complexity Problem Complexity DC Transmission Switching Given: A network G = ( N, A ) with arc capacities and susceptances ( u ij , α ij ) ∀ ( i, j ) ∈ A , generation levels p i ∀ i ∈ G , demand levels b i ∀ i ∈ D . Does there exist a subset of arcs S ⊆ A such that deactivating arcs in S leads to a feasible DC power flow? Theorem DC Transmission Switching is NP-Complete Reduction from subset-sum The problem remains hard... Even if there are a polynomial number of cycles in the network Even on a series-parallel graph with only one supply/demand pair Many Authors (UW-Madison & Ga. Tech) IP for Transmission Switching Aussois 9 / 28
Transmission Switching Formulations Switching Off Lines Regular Flow Constraints x ij = α ij ( θ i − θ j ) ∀ ( i, j ) ∈ E − U ij ≤ x ij ≤ U ij ∀ ( i, j ) ∈ E Let z ij ∈ { 0, 1 } ∀ ( i, j ) ∈ A , Switched Flow Constraints x ij = α ij z ij ( θ i − θ j ) ∀ ( i, j ) ∈ E If (and only if) there are finite bounds on flow, then one can write a MILP formulation (Fisher, O’Neil, and Ferris ’08). z ij = 1 ⇔ line ( i, j ) ∈ A is used Many Authors (UW-Madison & Ga. Tech) IP for Transmission Switching Aussois 10 / 28
Transmission Switching Formulations MILP Formulation � min c i p i x,p,θ,z i ∈ G p i ∀ i ∈ G � � s.t. x ij − x ij = d i ∀ i ∈ D 0 ∀ i ∈ N \ G \ D j :( i,j ) ∈ E j :( j,i ) ∈ E − U ij z ij ≤ x ij ≤ U ij z ij ∀ ( i, j ) ∈ E α ij ( θ i − θ j ) − x ij + M ( 1 − z ij ) ≥ 0 ∀ ( i, j ) ∈ E α ij ( θ i − θ j ) − x ij − M ( 1 − z ij ) ≤ 0 ∀ ( i, j ) ∈ E − L i ≤ θ i ≤ L i ∀ i ∈ N p i ≤ p i ≤ p i ∀ i ∈ G z ij ∈ { 0, 1 } ∀ ( i, j ) ∈ E Many Authors (UW-Madison & Ga. Tech) IP for Transmission Switching Aussois 11 / 28
Transmission Switching The Challenge Throwing Down the Gauntlet Hedman, Ferris, O’Neill, Fisher, Oren, (2010) state “When solving the transmission switching problem, ... the techniques for closing the optimality gap, specifically improving the lower bound, are largely ineffective.” So they resort to a variety of heuristic, ad-hoc techniques to get good solutions to the MILP they propose My good colleague and continuous optimizer Michael Ferris impugns the good name of integer programmers by stating that we are not smart enough to solve DC transmission switching You will later see that CPLEX v12 is already orders of magnitude better than CPLEX v9 on DC transmission switching instances But still it’s not good enough for large-scale networks... Thus we have... Many Authors (UW-Madison & Ga. Tech) IP for Transmission Switching Aussois 12 / 28
Transmission Switching The Challenge Ferris’s Challenge to Integer Programmers Solve realistically-sized DC transmission switching instances to provable optimality As integer programmers, we would like to rise to the challenge, and improve these “ineffective” lower bound techniques. The Padberg Way Let’s study the mathematical structure of the problem, create a useful relaxation of the problem, and improve our description of the relaxation through cutting planes (facets) Many Authors (UW-Madison & Ga. Tech) IP for Transmission Switching Aussois 13 / 28
Shagadelic Cycle Inequalities Intuition Key (Simple) Insight?! Assume (WLOG) that α ij = 1 We can just set x ij = α ij x ′ ij and scale u ij by α ij A Then we have... x AB = θ A − θ B B C x BC = θ B − θ C x CA = θ C − θ A x AB + x BC + x CA = 0 The potential constraints essentially (only) enforce that flow around a cycle is zero. If you didn’t forget everything from your introductory electrical engineering class (like I did), then you will recognize this as Kirchoff’s Voltage Law. Insight We should focus on what goes on around a cycle and try to model this in a better way Many Authors (UW-Madison & Ga. Tech) IP for Transmission Switching Aussois 14 / 28
Shagadelic Cycle Inequalities Intuition The Padberg Way Simple IP People (like me) Like Simple Sets Directed cycle G = ( V, C ) , with V = [ n ] , C = { ( i, i + 1 ) | ∀ i ∈ [ n − 1 ] } ∪ { ( n, 1 ) } : � ( x, θ, z ) ∈ R 2n × { 0, 1 } n | − u ij ≤ x ij ≤ u ij ∀ ( i, j ) ∈ C C = � z ij ( θ i − θ j ) = x ij ∀ ( i, j ) ∈ C The inequalities in this set model the potential drop across each arc in a cycle Even though C has the “nonlinear” equations z ij ( θ i − θ j ) = x ij , it is the union of 2 n polyhedra, so cl conv ( C ) is a polyhedron. The Padberg Way—Structure, Structure, Structure! Even though C is just a relaxation of the true problem, we hope that by generating valid inequalities for C , we can improve performance of IP approaches Many Authors (UW-Madison & Ga. Tech) IP for Transmission Switching Aussois 15 / 28
Shagadelic Cycle Inequalities Theorems Now We Do Math � ( x, θ, z ) ∈ R 2n × { 0, 1 } n | − u ij ≤ x ij ≤ u ij ∀ ( i, j ) ∈ C C = � z ij ( θ i − θ j ) = x ij ∀ ( i, j ) ∈ C Theorem For S ⊆ C such that u ( S ) > u ( C \ S ) , the shagadelic- cycle inequalities ( sci ) � β S a z a ≤ b S x ( S ) + (1) a ∈ C � β S a z a ≤ b S − x ( S ) + (2) a ∈ C are valid for C , where β S a = u ( S \ a ) − u ( C \ S ) ∀ a ∈ C b S = ( n − 1 )( 2u ( S ) − u ( C )) Many Authors (UW-Madison & Ga. Tech) IP for Transmission Switching Aussois 16 / 28
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