Sparse Polynomial Optimization for Urban Distribution Networks

Transcription

1 Sparse Polynomial Optimization for Urban Distribution Networks Martin Mevissen IBM Research - Ireland MINO/ COST Spring School on Optimization, Tilburg, March 11, 2015

2 The Team Brad Eck Bissan Ghaddar Chungmok Lee Jakub Marecek Martin Mevissen Nicole Taheri Susara van den Heever Rudi Verago

3 Motivation In many real-world decision problems, combined challenge of Nonconvex models and dynamics, e.g. energy conservation laws (friction induced headloss, AC power flow). Nonconvex objective functions, e.g. energy functionals, risk-averse optimization. Uncertainty in problem parameters, e.g. demands, prices, supply. However, we do have Samples of realizations for uncertain system parameters; e.g. collected iteratively by sensors and meters. Constraints and decision variables highly structured, e.g. sparsity of traffic, energy or water networks. Optimal decision for these hard, nonconvex real-world problems in high demand!

4 Motivation In many real-world decision problems, combined challenge of Nonconvex models and dynamics, e.g. energy conservation laws (friction induced headloss, AC power flow). Nonconvex objective functions, e.g. energy functionals, risk-averse optimization. Uncertainty in problem parameters, e.g. demands, prices, supply. However, we do have Samples of realizations for uncertain system parameters; e.g. collected iteratively by sensors and meters. Constraints and decision variables highly structured, e.g. sparsity of traffic, energy or water networks. Optimal decision for these hard, nonconvex real-world problems in high demand!

5 Motivation - AC Optimal Power Flow Economic dispatch of power generation is a critical problem for utility companies, { Production Cost [O Neill et al 2012]: Goal: Determine the optimal operating point of an electric power generation system. 519$bn Worldwide 112$bn USA Challenges: non-convex due to the non-linear power flow equations lack of global solver for generic power systems Approach: SDP provide strong bounds for ACOPF (Lavaei & Low 2010) Research on polynomial optimisation approach. Benefits: Even 1% improvement in dispatch would result in 1-5$bn savings for US (4-20$bn worldwide) [O Neill et al. 2012]

6 A FERC Study: Where do Leading Solvers Fail? Termination Conopt lpopt Knitro Minos Snopt Exceeded time limit with an infeasible solution 7.9% 22.2% 38.4% 0.0% 23.0% Exceeded time limit with a feasible solution 6.5% 0.9% 6.7% 0.0% 14.4% Early termination with an infeasible solution 0.0% 0.4% 0.6% 11.4% 0.0% Early termination with a feasible solution 31.2% 0.6% 1.8% 2.3% 0.2% Normal termination with an infeasible solution 10.5% 3.4% 0.0% 55.9% 1.6% Percentage of feasible 81.6% 74.1% 61.0% 32.7% 75.4% Percentage of best seen 43.8% 72.5% 52.5% 30.3% 60.8% Source: Anya Castillo and Richard O Neill, Staff Technical Conference, FERC, Washington DC, June 24 26, bus instances, 100 runs per instance and solver, 20 min runs, Xeon E GHz, 64 GB RAM

7 IBM Research - Ireland Motivation - Challenges in power distribution systems Optimization over power systems: Large-scale of power transmission and distribution networks. AC power flow. Integration of distributed, uncertain renewable supply.

8 Pressure management in water distribution networks Challenge: Leakage major problem in water distribution networks. Variables: Pressure p i at node i, and flow q i,j from node i to j. Parameters: Elevation e i and uncertain demand d i at node i. Decisions: Setting p j at pressure reducing valves.

9 Pressure management in water distribution networks Combination of challenges: Large-scale of water distribution networks Nonconvex friction-induced headloss Uncertain nodal demands.

10 Polynomial Optimization Polynomial Optimization Problem (POP) min p(x) s.t. g 1 (x) 0,..., g m (x) 0, x = (x 1,..., x n ) Hierarchy of SDP Relaxations for POP [Lasserre 2001] d-sdp ω min α p αy α s.t. M ω (y) 0, M ω ωj (g j y) 0, j = 1,..., m If K compact and archimedean, min(d-sdp ω ) min(pop) for ω. Challenge: Size of the matrix inequalities ( n + ω n ) grows rapidly!

11 Polynomial Optimization Polynomial Optimization Problem (POP) min p(x) s.t. g 1 (x) 0,..., g m (x) 0, x = (x 1,..., x n ) Hierarchy of SDP Relaxations for POP [Lasserre 2001] d-sdp ω min α p αy α s.t. M ω (y) 0, M ω ωj (g j y) 0, j = 1,..., m If K compact and archimedean, min(d-sdp ω ) min(pop) for ω. Challenge: Size of the matrix inequalities ( n + ω n ) grows rapidly!

12 Polynomial Optimization Polynomial Optimization Problem (POP) min p(x) s.t. g 1 (x) 0,..., g m (x) 0, x = (x 1,..., x n ) Hierarchy of SDP Relaxations for POP [Lasserre 2001] d-sdp ω min α p αy α s.t. M ω (y) 0, M ω ωj (g j y) 0, j = 1,..., m If K compact and archimedean, min(d-sdp ω ) min(pop) for ω. Challenge: Size of the matrix inequalities ( n + ω n ) grows rapidly!

13 Exploiting structure for efficient solution methods Decision optimization problem Mathematical optimisation model Structured Polynomial Optimization Problem Add valid inequalities to strengthen convexification. Exploit correlative sparsity of the POP. SDP relaxation for POP Exploit d- and r-space sparsity in the SDP relaxation. Efficient algorithms for solving SDPs.

14 Exploiting structure for efficient solution methods Decision optimization problem Mathematical optimisation model Structured Polynomial Optimization Problem Add valid inequalities to strengthen convexification. Exploit correlative sparsity of the POP. SDP relaxation for POP Exploit d- and r-space sparsity in the SDP relaxation. Efficient algorithms for solving SDPs.

15 Exploiting structure for efficient solution methods Decision optimization problem Mathematical optimisation model Structured Polynomial Optimization Problem Add valid inequalities to strengthen convexification. Exploit correlative sparsity of the POP. SDP relaxation for POP Exploit d- and r-space sparsity in the SDP relaxation. Efficient algorithms for solving SDPs.

16 Exploiting structure for efficient solution methods Decision optimization problem Mathematical optimisation model Structured Polynomial Optimization Problem Add valid inequalities to strengthen convexification. Exploit correlative sparsity of the POP. SDP relaxation for POP Exploit d- and r-space sparsity in the SDP relaxation. Efficient algorithms for solving SDPs.

17 Hydraulic model for WDN Service requirements p max p i p min i N, q max,i,j q i,j 0 (i, j) E. Conservation of mass q k,i q i,l = d i k l i N. Conservation of energy If q i,j > 0 : p i + e i p j e j = h l i,j (q i,j ), If q i,j = 0 : p i + e i p j e j 0, where h l i,j (q i,j ) the friction-induced head loss in pipe (i, j) in case of flow from i to j. Modeled as q i,j (p i + e i p j e j h l i,j (q i,j )) 0, p i + e i p j e j h l i,j (q i,j ) 0.

18 Hydraulic model for WDN Service requirements p max p i p min i N, q max,i,j q i,j 0 (i, j) E. Conservation of mass q k,i q i,l = d i k l i N. Conservation of energy If q i,j > 0 : p i + e i p j e j = h l i,j (q i,j ), If q i,j = 0 : p i + e i p j e j 0, where h l i,j (q i,j ) the friction-induced head loss in pipe (i, j) in case of flow from i to j. Modeled as q i,j (p i + e i p j e j h l i,j (q i,j )) 0, p i + e i p j e j h l i,j (q i,j ) 0.

19 Modeling the friction loss h l i,j (q i,j ) depends on roughness, diameter and length of (i, j). Explicit formulation (Hazen-Williams): h l i,j (q i,j ) = c i,j q i,j Implicit formulation (Darcy-Weisbach): { hl i,j (q i,j ) = c l i,j q i,j for laminar (i.e. small) flow regime, h l i,j (q i,j ) c (2) i,j q 2 i,j + c(1) i,j q i,j for turbulent (i.e. large) flow regime Piece-wise linear approximation of HW results in MILP. Global, quadratic lower and upper bounds of h l i,j (HW). [Sherali & Smith 1997]. Signpower for DW [Gleixner et al. 2011]. Smooth approximation of Signpower HW [Bragalli et al. 2011]. Our approach: Quadratic approximation of h l i,j according to either HW or DW!

20 Modeling the friction loss h l i,j (q i,j ) depends on roughness, diameter and length of (i, j). Explicit formulation (Hazen-Williams): h l i,j (q i,j ) = c i,j q i,j Implicit formulation (Darcy-Weisbach): { hl i,j (q i,j ) = c l i,j q i,j for laminar (i.e. small) flow regime, h l i,j (q i,j ) c (2) i,j q 2 i,j + c(1) i,j q i,j for turbulent (i.e. large) flow regime Piece-wise linear approximation of HW results in MILP. Global, quadratic lower and upper bounds of h l i,j (HW). [Sherali & Smith 1997]. Signpower for DW [Gleixner et al. 2011]. Smooth approximation of Signpower HW [Bragalli et al. 2011]. Our approach: Quadratic approximation of h l i,j according to either HW or DW!

21 Quadratic fit of HW or turbulent flow DW model Headloss (m) Hazen Williams Quadratic Fit 10% Error Relative Error Relative Error Head loss (m) Reynold's Number Darcy Weisbach Quadratic Fit 10% Error Relative Error Relative Error Flow (cmd) Flow (cms) HW: Reasonable approx over range of flows. DW: If friction factor constant, headloss would be quadratic.

22 How is the approximation at low flows? Reynold's Number Head loss (m) Darcy Weisbach Quadratic Fit 10% Error Flow (cms) Relative error is high, but absolute error is low Low flows were not problem in our tests [B. Eck, M. Mevissen, Quadratic Approximations for Pipe Friction, J. Hydroinformatics, 2014]

23 Validation of fit in hydraulic model Solution Using Quadratic Approx. Pressure (m) (a) 1200 Flow (L/s) Pressure (m) Flow (L/s) Epanet Solution using DarcyWeisbach 0 (b) Figure: Comparison of pressures and flows for the Exnet system as calculated by using quadradic headloss and EPANET 2.0

24 Cubic POP formulation for optimal valve setting Derive the POP (VS-PP3) min i N p i s.t. p max p i p min i N, q max,i,j q i,j 0 (i, j) E, k q k,i l q i,l = d i i N, q i,j (p i + e i p j e j h l i,j (q i,j )) 0 (i, j) E, p i + e i p j e j h l i,j (q i,j ) M v i,j 0 (i, j) E, (1 v j,i ) q max,i,j q i,j 0 (i, j) E, v i,j {0, 1} where h l i,j (q i,j ) = a i,j qi,j 2 + b i,j q i,j and valve placement indicator v i,j {0, 1} given for all (i, j) E. [B. Eck, M. Mevissen, Fast non-linear optimization for design problems on water networks. World Environmental and Water Resources Congress 2013]

25 Quadratic POP formulation for optimal valve setting Derive the POP (VS-PP2) min i N p i s.t. p max p i p min i N, q max,i,j q i,j q max,i,j (i, j) E, k q k,i l q i,l = d i i N, p j + e j p i e i + h l i,j (q i,j )) = 0 (i, j) E, if v i,j = 0, v i,j (p j + e j p i e i ) + h l i,j (q i,j )) 0 (i, j) E, if v i,j 0, v i,j q i,j 0 (i, j) E, v i,j { 1, 0, 1} where h l i,j (q i,j ) = a i,j q i,j q i,j +b i,j q i,j, valve placement indicator v i,j {0, 1} given for all (i, j) E and q i,j R. New quadratic model (with Mathieu Claeys, Bissan Ghaddar).

26 Exploit correlative sparsity [Waki et al. 2006] min 6 i=1 x i s.t. x 2 i 2 i x 1 + x 6 1, x 2 x 6 1, x 3 + x 4 1, x 3 x 6 1, x 4 + x 5 1, x 5 x (a) (b) (a) C 1 = {3, 4, 6}, C 2 = {4, 5, 6}, C 3 = {1, 6}, C 4 = {2, 6}

27 Sparse hierarchy Hierarchy of sparse SDP Relaxations for POP [Waki et al. 2006] s-sdp ω min α p αy α s.t. M ω (y, C k ) 0, k = 1,..., P M ω ωj (g j y, C k ) 0, j J k, k = 1,..., P min(s-sdp ω ) min(d-sdp ω ) If K compact, archimedean, and (C k ) P k=1 satisfy the running-intersection property, min(s-sdp ω ) min(pop) for ω. [Lasserre 2006] Reduction in size: Size of k matrix inequalities ( Ck +ω C k ) vastly smaller if C k << n.

28 Sparse hierarchy Hierarchy of sparse SDP Relaxations for POP [Waki et al. 2006] s-sdp ω min α p αy α s.t. M ω (y, C k ) 0, k = 1,..., P M ω ωj (g j y, C k ) 0, j J k, k = 1,..., P min(s-sdp ω ) min(d-sdp ω ) If K compact, archimedean, and (C k ) P k=1 satisfy the running-intersection property, min(s-sdp ω ) min(pop) for ω. [Lasserre 2006] Reduction in size: Size of k matrix inequalities ( Ck +ω C k ) vastly smaller if C k << n.

29 IBM Research - Ireland Valve setting POP satisfies structured sparsity: nz = Pescara: n = 270, P = 210, 3 Ck nz =

30 Sparse SDP relaxations for (VS-PP3) - Pescara nz = [Bragalli et al. 2011] PRV ssdp 2 SDPA IPOPT IPOPT to set Lbd Time Obj Gap Time , % , % , % 3

31 Sparse SDP relaxations for (VS-PP2) - Pescara [Bragalli et al. 2011] nz = SeDuMi SDPA ssdp 1 ssdp 2 ssdp 1 ssdp 2 PRV Obj Time FE Obj Time FE Obj Time FE Obj Time FE

32 Optimal Power Flow Problem Rectangular Power-Voltage ACOPF min k g ( c 2 k (P g k )2 + c 1 k Pg k + c0 k ) P k + j Q k = V k (Y k V k ) P min k P g k Pmax k, Qk min Q g k Qmax k (Vk min ) 2 RV 2 k + IV 2 k (Vk max ) 2 k N Plm 2 + Q2 lm S l,m max (l, m) L

33 Optimal Power Flow Problem POP formulation for ACOPF (OPF-PP4) min k G P min k Q min k ( ) ck 2 (Pd k + tr(y kxx T )) 2 + ck 1 (Pd k + tr(y kxx T )) + ck 0 P d k tr(y kxx T ) P max k P d k i N Q d k tr(ȳ k xx T ) Q max k Q d k i N (Vk min ) 2 tr(m k xx T ) (Vk max ) 2 i N (tr(y lm xx T )) 2 + (tr(ȳ lm xx T )) 2 (Slm max (l, m) L dsdp 2 : [Josz et al. 2013], [Molzahn et al. 2014]

34 Optimal Power Flow Problem POP formulation for ACOPF (OPF-PP2) min k G ( ) ck 2 (Pg k )2 + ck 1 (Pd k + tr(y kxx T )) + ck 0 Pk min Qk min tr(y k xx T ) + P d k Pmax k tr(ȳ k xx T ) + Q d k Qmax k (Vk min ) 2 tr(m k xx T ) (Vk max ) 2 Plm 2 + Q2 max lm (Slm )2 P g k = tr(y kxx T ) + P d k P lm = tr(y lm xx T ) Q lm = tr(ȳ lm xx T ) ssdp 1 and dsdp 1 equivalent to [Lavaei, Low 2010]

35 Moment approaches for (OPF-PP4) Limitations of Lavei & Low : [Lesieutre et al. 2011]; [Bukhsh et al. 2012] Dense Lasserre hierarchy for (OPF-PP4) [Josz et al. (RTE France) 2013], [Molzahn et al 2013] Solving feeders up to 10 buses to global optimality. Prominent instances: WB2, LMBM3, WB5,..

36 Exploit sparsity in matrix inequalities [Kim et al. 2011] min A 0 X s.t. M(X ) 0, X 0, where A 0 S n tridiagonal, X S n, and 1 X X X X 23. M(X ) = X X n 1,n 1 X n 1,n X 21 X 32 X X n,n 1 1 X nn Only X i,j with i j 1 are relevant for evaluating objective or matrix inequality constraint. The matrix inequality satisfies some sparsity pattern as well.

37 Sparsity pattern for the SDP example Domain-space sparsity: Range-space sparsity:

38 Positive semidefinite matrix completion Theorem [Grone 1984] Let C k (k = 1, 2,..., p) be the maximal cliques of a chordal graph G(N, E), N {1,..., n}. Suppose that X S n (E,?). Then X S n +(E,?) X (C k ) S C k + (k = 1,..., p) (a) (b) (a) C 1 = {3, 4, 6}, C 2 = {4, 5, 6}, C 3 = {1, 6}, C 4 = {2, 6}

39 Applying d-space and r-space conversion yields: Dense SDP min A 0 X s.t. M(X ) 0, X 0 Equivalent reduced SDP using PSD Completion [Grone 1984] min s.t. n 1 ( A 0 ii X ii + 2A 0 ) i,i+1x i,i+1 + A 0 nn X nn ( i=1 ) ( ) 1 0 X11 X 12 0, ( 0 0 ) ( X 21 z 1 ) 1 0 X ii X i,i+1 0 (i = 2, 3,..., n 2), ( 0 0 ) ( X i+1,i z i 1 z i ) 1 0 Xn 1,n 1 X n 1,n 0, ( 0 1 ) ( X n,n 1 X n,n + z n 2 ) 0 0 Xii X i,i+1 0 (i = 1, 2,..., n 1). 0 0 X i+1,i X i+1,i+1

40 Sparsity of the rank relaxation due to Lavaei-Low Sparsity of admittance matrix can be exploited. [Stott 1974]. 0 Exploit sparsity in SDP relaxation for OPF [Molzahn et al. 2013]. 50 Implementation of 0 [Kim 100 et 200 al ] : SparseCoLO [Fujisawa et al. nz = ]; SDP solver SeDuMi nz = nz = nz = 1597 n Obj Dual of dsdp 1 (OPF-PP2) Dim Time(Original) Time(SparseCoLO) x x x x sp 1.307x x57408 * nz = 1677

41 Dense Vs. sparse hierarchy - LMBM3 Table: Second order relaxation for LMBM3, where first order not exact. dsdp 2 (OPF-PP4) ssdp 2 (OPF-PP4) S23 max Bound Time Bound Time But: For > 10 buses, dense hierarchy intractable.

42 Dense Vs. sparse hierarchy - LMBM3 Table: Second order relaxation for LMBM3, where first order not exact. dsdp 2 (OPF-PP4) ssdp 2 (OPF-PP4) S23 max Bound Time Bound Time But: For > 10 buses, dense hierarchy intractable.

43 Exploiting sparsity in ACOPF and its SDP relaxation dual of dsdp 1 (OPF-PP2) primal of ssdp 2 (OPF-PP4) N Bound Time Bound Time 9mod mod mod * * * * * * 2736sp 1.307x * * Exploiting correlative sparsity of (OPF-PP4) allows to find global optima for instances with up to 40 buses. Ghaddar, Marecek, Mevissen, Optimal Power Flow as a Polynomial Optimization Problem, 2014, IEEE Trans. on Power Systems, To Appear

44 Conclusion and open questions Exploiting problem structure (sparsity, symmetry, decomposition...) in industrial decision optimisation problems crucial. Techniques available to scale-up powerful polynomial optimisation approaches. Combine advances in real algebra, polynomial optimisation and algorithms for semidefinite programs. Thanks for your attention!

45 Conclusion and open questions Exploiting problem structure (sparsity, symmetry, decomposition...) in industrial decision optimisation problems crucial. Techniques available to scale-up powerful polynomial optimisation approaches. Combine advances in real algebra, polynomial optimisation and algorithms for semidefinite programs. Thanks for your attention!

46 References B. Eck, M. Mevissen, Quadratic Approximations for Pipe Friction, J. Hydroinformatics, 2014, To Appear B. Eck, M. Mevissen, Fast non-linear optimization for design problems on water networks. World Environmental and Water Resources Congress 2013 Ghaddar, Marecek, Mevissen, Optimal Power Flow as a Polynomial Optimization Problem, 2014, IEEE Trans. on Power Systems, To Appear