Valid Inequalities for Optimal Transmission Switching

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of ISyE Georgia Tech 19th Combinatorial Optimization Workshop Aussois, France,

1 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

Dispatch Focus today is on a simple problem of meeting demand for power at

2 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

3 Power Background Notation Power Grid Networks Look Weird Many Authors (UW-Madison & Ga. Tech) IP for Transmission Switching Aussois 3 / 28

4 Power Background Notation It s Just a Network Power Network: (N, A) with... G N: generation nodes D N: demand nodes Load forecasts (MW) b i for i D Generation cost ($/MW) c i and capability p i (MW) for i G Peak load rating (MW) u ij for (i, j) A 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

5 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

6 Power Background Economic Dispatch Linear Program for (DC) Economic Dispatch s.t. j:(i,j) E min c i p i x,p,θ i G x ij j:(j,i) E x ji = u ij x ij u ij p i p i p i p i d i i G i D 0 i N \ G \ D (i, j) E i G x ij = α ij (θ i θ j ) x ij R p i R + θ i R (i, j) E i G i N (i, j) E 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

7 Power Background Tradeoffs MCNF++ This economic dispatch problem is just a min cost network flow problem with some additional potential constraints The potential drop (θ A θ D ) must be the same aloing the paths: A B D and A C D B A D C 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)

8 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% % % % 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,

9 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

10 Transmission Switching Formulations Switching Off Lines Regular Flow Constraints x ij = α ij (θ i θ j ) U ij x ij U ij (i, j) E (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

11 Transmission Switching Formulations MILP Formulation s.t. j:(i,j) E min x,p,θ,z x ij c i p i i G j:(j,i) E x ij = p i d i U ij z ij x ij U ij z ij α ij (θ i θ j ) x ij + M(1 z ij ) 0 α ij (θ i θ j ) x ij M(1 z ij ) 0 L i θ i L i p i p i p i z ij {0, 1} i G i D 0 i N \ G \ D (i, j) E (i, j) E (i, j) E i N i G (i, j) E Many Authors (UW-Madison & Ga. Tech) IP for Transmission Switching Aussois 11 / 28

12 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

13 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

14 Shagadelic Cycle Inequalities Intuition Key (Simple) Insight?! A Assume (WLOG) that α ij = 1 We can just set x ij = α ij x ij and scale u ij by α ij Then we have... B C x AB = θ A θ B 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

15 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)}: { C = (x, θ, z) R 2n {0, 1} n u ij x ij u ij (i, j) 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

$Shagadelic Cycle Inequalities Theorems Now We Do Math { C = (x, θ, z) R 2n {0, 1} n u ij x ij u ij (i, j) C } z ij (θ i θ j ) = x ij (i, j) C Theorem For S C such that u(s) > u(c\s), the$

16 Shagadelic Cycle Inequalities Theorems Now We Do Math { C = (x, θ, z) R 2n {0, 1} n u ij x ij u ij (i, j) C } z ij (θ i θ j ) = x ij (i, j) C Theorem For S C such that u(s) > u(c\s), the shagadeliccycle inequalities (sci) x(s) + a C β S az a b S (1) x(s) + a C β S az a b S (2) 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

17 Shagadelic Cycle Inequalities Theorems Shagadelic-Cycle Inequalities, Example x 1 2z 1 x 2 4z 2 x 3 3z 3x1 + x2 + z1 z2 + 3z3 6 S = {1, 2} x 1 + x 3 z 1 + z 2 2z 3 2 S = {1, 3} x 2 + x 3 + 5z 1 + z 2 + 2z 3 10 S = {2, 3} x 1 + x 2 + x 3 + 7z 1 + 5z 2 + 6z 3 18 S = {1, 2, 3} Logic Enforced For S = {1, 2}, if z 1 = z 2 = 1, then { 6 z3 = 0 x 1 + x 2 3 z 3 = 1 For S = {1, 3}, if z 1 = z 3 = 1, then { 5 z2 = 0 x 1 + x 3 4 z 2 = 1 Many Authors (UW-Madison & Ga. Tech) IP for Transmission Switching Aussois 17 / 28

$Theorem If S C, and u(c \ S) < u(s), then the shagadelic-cycle inequalities (sci) are facet-defining for cl conv(c).$

18 Shagadelic Cycle Inequalities Theorems Proofs! Yeah, Baby! Theorem If S C, and u(c \ S) < u(s), then the shagadelic-cycle inequalities (sci) are facet-defining for cl conv(c). Thus, all 2 n inequalities are necessary in the description of cl conv(c) Many Authors (UW-Madison & Ga. Tech) IP for Transmission Switching Aussois 18 / 28

$(x, θ, z) R 3n u ij z ij x ij u ij z ij (i, j) C z ij 1 (i, j) C x(s) + a C β S az a b S S C : u(s) > u(c \ S) x(s) + } β S az a b S S C : u(s) > u(c \ S) a C Many Authors (UW-Madison & Ga.$

19 Shagadelic Cycle Inequalities Theorems Even More Proofs Along with some other trivial inequalities, the shagadelic cycle inequalities are sufficient to describe the convex hull of C cl conv(c) = { (x, θ, z) R 3n u ij z ij x ij u ij z ij (i, j) C z ij 1 (i, j) C x(s) + a C β S az a b S S C : u(s) > u(c \ S) x(s) + } β S az a b S S C : u(s) > u(c \ S) a C Many Authors (UW-Madison & Ga. Tech) IP for Transmission Switching Aussois 19 / 28

$Given solution ^x R n +, ^z [0, 1] n, the separation problem for (sci) is where max C A:C is a cycle max {^x(s) + (β S ) ^z b S }, S C:2u(S) u(c) β S a = u(s \ a) u(c \ S) b S = (n 1)(2u(S) u(c)) a C$

20 Separation Heuristic Can We Use the sci? Given solution ^x R n +, ^z [0, 1] n, the separation problem for (sci) is where max C A:C is a cycle max {^x(s) + (β S ) ^z b S }, S C:2u(S) u(c) β S a = u(s \ a) u(c \ S) b S = (n 1)(2u(S) u(c)) a C Theorem The separation problem for (sci) is NP-Hard Theorem Something that seems like it must be true, but I can t prove it. Many Authors (UW-Madison & Ga. Tech) IP for Transmission Switching Aussois 20 / 28

21 Separation Heuristic Simple Observations Observation: If a C ^za C 1, then (^x, ^z) cannot be violated by any (sci) This suggests a two-phase separation heuristic. Separation Heuristic 1 Find a necessary cycle C such that a C ^za > C 1 2 Find S C in the given cycle Do (1) by (truncated) enumeration Given C, algebra shows that (2) is a knapsack problem: K C = 1 a C (1 ^za) 0 ^v a = ^x a + u a^z a 2u a( e C\a (1 ^ze)) ν = max ^v ay a u ay a 1 2 u(c) y {0,1} n a C a C If ν u(c)k C > 0, then (sci) is violated by (^x, ^z) Many Authors (UW-Madison & Ga. Tech) IP for Transmission Switching Aussois 21 / 28

$.. Not really, it is just that this specific knapsack problem is easy Take the items: S C = {a C ^x a u a^z a + 2u ak C > 0} Many Authors (UW-Madison & Ga.$

22 Separation Heuristic P=NP I Can Solve the Knapsack Problem in Polynomial Time! Since I have proved that P = NP, the Clay Mathematics Institute should pay me... Not really, it is just that this specific knapsack problem is easy Take the items: S C = {a C ^x a u a^z a + 2u ak C > 0} Many Authors (UW-Madison & Ga. Tech) IP for Transmission Switching Aussois 22 / 28

23 Computational Results Transmission Switching Standard IEEE Benchmark Instances Optimal Switching can make some difference in generation cost Generation Cost Instance No Switching With Switching case3les case6ww case case case ieee case case case case118b case case Many Authors (UW-Madison & Ga. Tech) IP for Transmission Switching Aussois 23 / 28

24 Computational Results Transmission Switching But These Are (Now) Too Easy no cuts with cuts instance time nodes time nodes case3les case6ww case case case ieee case case case case118b case case Solving the MIP model using CPLEX v12.5 Case118 was the instance that Ferris et al. report not being able to solve with CPLEX (version 9) Many Authors (UW-Madison & Ga. Tech) IP for Transmission Switching Aussois 24 / 28

25 Computational Results Transmission Switching Creating More Instances Modify the 118B instance by modifying the demands randomly. Create 15 new instances Comparing on Many 118B Instances Avg. Time Avg. Nodes No Cuts With Cuts Cuts show some promise We continue to work on pure transmission switching on larger instances (> 2000) nodes. These problems are still way too hard for CPLEX with and without cuts There are many alternative optimal solutions to the linear programming relaxation Which one(s)? should we cut off? Many Authors (UW-Madison & Ga. Tech) IP for Transmission Switching Aussois 25 / 28

26 The End Accomplishments Prove that transmission switching problem is NP-Complete Understand a cycle relaxation derived from the structure of the problem Give a complete description of the convex hull of the set with 2 n inequalities Also have an (integer) extended formulation in dimension 6n + 1 Even with initial implementation, we can significantly improve default CPLEX behavior Still To Do Working on effective mechanisms for using these inequalities for larger instances A special challenge for (pure) transmission switching is the extreme dual generacy of LP solutions so engineering effective cutting plane mechanisms is important Up Next Study more complicated structures besides cycles Try to include demands at nodes, for Flow-(sci) Extend to potential preserved, but nonlinear relationship between potential and flow Gas and Water Network design Many Authors (UW-Madison & Ga. Tech) IP for Transmission Switching Aussois 26 / 28

27 The End The Real Conclusion Many Authors (UW-Madison & Ga. Tech) IP for Transmission Switching Aussois 27 / 28

28 The End Thank you! For Giving Us a Great Foundation Many Authors (UW-Madison & Ga. Tech) IP for Transmission Switching Aussois 28 / 28

Valid Inequalities for Potential-Constrained Network Design

Valid Inequalities for Potential-Constrained Network Design Hyemin Jeon Jeff Linderoth Jim Luedtke Dept. of ISyE UW-Madison jrluedt1@wisc.edu MINLP 2014, CMU, Pittsburgh, PA June 3, 2014 Jeon, Linderoth,