Power Flow Equations: Complexity, Approximations and Relaxations · 2015. 1. 14. · NICTA...
Transcript of Power Flow Equations: Complexity, Approximations and Relaxations · 2015. 1. 14. · NICTA...
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NICTA Copyright 2014 From imagination to impact
Power Flow Equations:
Complexity, Approximations and Relaxations
Pascal Van Hentenryck
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NICTA Copyright 2014 From imagination to impact
Credits‣ Power systems
– PSCC’11, PES’12 (2), PSCC’14 (5) – IJEPES (to appear)
‣ Operations research – IJOC 2014, Mathematical Programming 2015
‣ Submissions – complexity results (under submission)
‣ People – C. Coffrin, A. Grastien, H. Hijazi, K. Lehmann, T. Mak – R. Bent (LANL) – D. Hill and I. Hiskens
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Outline
‣ Motivation ‣ Power Flow Formulations ‣ Complexity ‣ The LPAC Approximation ‣ Case Studies for LPAC ‣ Convex Relaxations ‣ Case Studies for Convex Relaxations ‣ Beyond Steady States
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Power Systems‣ Biggest machine on earth
– 400 billions of electricity
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Why fix it?‣ New challenges
– challenging existing assumptions ‣ New applications
– requiring new technology ‣ New enabling technologies
– enabling new functionalities
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Motivation.
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‣ Category 3 – August 21-28, 2011
‣ Fatalities – 49 direct (+ 7)
‣ Damages – ~ $15 billions
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Hurricane Irene
‣ Strategic Stockpiling ‣ Repositioning ‣ Power Restoration
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San Diego Blackout ‣ Causes
– Tripping of a line between Arizona and California – Cascading effect (not supposed to happen)
‣ Effects – >4 millions people without power, Sept. 8-9, 2011
‣ Economic Losses over US$ 100 million – Opportunity losses: $70 million – Overtime workers: $20 million – Spoiled food: $18 million
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Joint Repair and Power Restoration
‣The challenge - Schedule a fleet of repair crews to repair the grid and
minimize the overall size of the blackout after a disaster ‣Two fundamental aspects
- Scheduling the repairs - Scheduling the power restoration - Both are challenging in their own right
‣Assumptions for Last-Mile Restoration - Steady state behavior of the power grid - Ability to dispatch load and generation continuously
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Joint Repair and Power Restoration
Component repair
Increase in served demand
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Joint Repair and Power Restoration
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• 2-Step Approach (PSCC’11)
• Restoration Ordering Problem (ROP)
• Pickup and Delivery Routing with Precedences • Randomized Adaptive Decomposition over LNS over CP
Joint Repair and Power Restoration
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Restoration Ordering Problem (ROP)
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Initial Steady State Steady State 1 Steady State 2 Final Steady
State N
Fix item x1 Fix item x2
…
Fix item x3, x4,…, xN
‣Find the best sequence of restoration [x1,x2,x3,…,xn] so that the size of the blackout is minimised
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The ROP Problem
‣ Extremely challenging computationally – generalizes transmission switching
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Computational Challenge
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presolved problem has 1960 variables and 2186 constraints 510 constraints of type <varbound> 1475 constraints of type <linear> 201 constraints of type <logicor> Presolving Time: 0.52 time | node | dualbound | primalbound | gap 1.0s | 1 | 1.602999e+02 |-0.000000e+00 | 100.00% 14.3s | 1 | 1.602999e+02 | 1.213604e+02 | 24.29% 779s | 12012 | 1.602999e+02 | 1.228179e+02 | 23.38% 812s | 12658 | 1.602999e+02 | 1.238971e+02 | 22.71% 827s | 12858 | 1.602999e+02 | 1.263328e+02 | 21.19% 1019s| 18895 | 1.602999e+02 | 1.324728e+02 | 17.36% 1147s| 24867 | 1.602999e+02 | 1.353018e+02 | 15.59% 1276s| 33091 | 1.602999e+02 | 1.362924e+02 | 14.98% 1294s| 34851 | 1.602999e+02 | 1.419186e+02 | 11.47% 1958s| 91300 | 1.602999e+02 | 1.447137e+02 | 9.72% 5882s|435468 | 1.602999e+02 | 1.460944e+02 | 8.86% 202m| 1038k | 1.602999e+02 | 1.464802e+02 | 8.62% 253m| 1344k | 1.602999e+02 | 1.464802e+02 | 8.62%
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The ROP Problem
‣ Extremely challenging computationally – generalizes transmission switching
‣ Braess paradox – restoring a line may decrease the network flow
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Braess Paradox
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Joint Repair and Power Restoration
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The ROP Problem
‣ Modeling the power system – Line capacities (thermal constraints) – Constraints on real and reactive power injections – Power Flow equations
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Power Flow Equations‣ Ohm’s law for each line (i,j)
‣ Kirchhoff’s current law for each bus i
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pij “ gijv2i ´ gijvivj cosp✓i ´ ✓jq ´ bijvivj sinp✓i ´ ✓jq (1)
qij “ ´bijv2i ` bijvivj cosp✓i ´ ✓jq ´ gijvivj sinp✓i ´ ✓jq (2)
pi “ÿ
pi,jqPEpij
qi “ÿ
pi,jqPEqij
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The Core ROP Subproblem‣ Push as much load as possible in the network
while satisfying the power flow and side constraints – nonlinear – nonconvex – discrete
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The Linear DC Model
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Ignore reactive power
conductance is much smaller than susceptance
sin(x) close to x when x is small
pij “ gijv2i ´ gijvivj cosp✓i ´ ✓jq ´ bijvivj sinp✓i ´ ✓jq (1)
qij “ ´bijv2i ` bijvivj cosp✓i ´ ✓jq ´ gijvivj sinp✓i ´ ✓jq (2)
Voltage magnitudes are close to 1.0
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pi “ÿ
pi,jqPEpij
The Linear DC Model‣ Ohm’s law for each line (i,j)
‣ Kirchhoff’s current law for each bus i
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pij “ ´bijp✓i ´ ✓jq
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pij § bijp✓i ´ ✓jq ` Mp1 ´ zijqpij • bijp✓i ´ ✓jq ´ Mp1 ´ zijq
‣ The power equation becomes
‣ It is nonlinear: can be linearized since zi is a 0/1 variable
In the ROP
pij “ ´bijzijp✓i ´ ✓jq
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Case Studies
‣ 15 large disaster scenarios – with damage size in 50-1000
‣ Generated using, – US Transmission and Transportation Infrastructure – State-of-the-art disaster simulation tools (NHC, FEMA)
‣ Study quality over 0.5 - 8 hours – Average of 10 runs
‣ Comparing with best practices in the field – utilisation heuristics + greedy routing
‣ Lower bound – assuming infinitely many crews
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Case Studies
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Restoration Timeline − BM2 S16
Time
Powe
r Flo
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292835.177
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Restoration Timeline − BM2 S16
Time
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532409.453
Case Studies
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0 500 1000 1500 2000
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Restoration Timeline − BM2 S14
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366219.884
Case Studies
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Case Studies
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Restoration Timeline − BM2 S9
Time
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567393.666
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Case Studies
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Restoration Timeline − BM2 S3
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981149.0342
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Case Studies
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Power Restoration
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q A fundamental open question ‣Is this “optimal” restoration plan “feasible” operationally?
Power Restoration
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q A fundamental open question ‣Is this “optimal” restoration plan “feasible” operationally?
‣ These are not “normal operating” conditions -“Maddeningly difficult” to find an AC solution in cold start contexts [Overbye et al, 2004]
‣ The ROP is stressing the network
Power Restoration
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AC Power/Load Flow ‣ Seed a power flow study
– with the optimization results ‣ Different information for different buses
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Power Restoration
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N-3 Contingencies (IEEE-30)
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IEEE PES’12
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N-10 Contingencies (IEEE-30)
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Expansion Planning
‣ Simplest problem formulation – starting point
‣ Network design problem – add lines to meet the increased load
‣ Under – voltage constraints – thermal limits – constraints on real and reactive power injection
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Expansion Planning
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New Challenges
‣ Power applications that – are mixed nonlinear optimisation problems – require accurate (coupled?) models of the power
system • congestion
‣ Observe that – the discrete nature precludes some technology
• integrating discrete optimization and homotopy methods is not easy
– need for relaxations! • performance guarantees
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Outline
‣ Motivation ‣ Power Flow Formulations ‣ Complexity ‣ The LPAC Approximation ‣ Case Studies for LPAC ‣ Convex Relaxations ‣ Case Studies for Convex Relaxations ‣ Beyond Steady States
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Power Flows‣ Complex Number Formulation
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Power Flows‣ Rectangular formulation
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Power Flows‣ Hybrid formulation
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Power Flows‣ Side constraints
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NICTA Copyright 2014 From imagination to impact
W-Formulation‣ Define
‣ Power Flow becomes
47
Wij = ViV⇤j
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NICTA Copyright 2014 From imagination to impact
W-Formulation
48
pij = gijwRii � gijwR
ij � bijwIij
qij = �bijwRii + bijwR
ij � gijwIij
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NICTA Copyright 2014 From imagination to impact
Dist-Flow Relaxation
‣ M. E. Baran and F. F. Wu, “Network reconfiguration in distribution systems for loss reduction and load balancing,” 1989.
49
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NICTA Copyright 2014 From imagination to impact
Outline
‣ Motivation ‣ Power Flow Formulations ‣ Complexity ‣ The LPAC Approximation ‣ Case Studies for LPAC ‣ Convex Relaxations ‣ Case Studies for Convex Relaxations ‣ Beyond Steady States
50
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NICTA Copyright 2014 From imagination to impact
Complexity of AC-Feasibility‣ AC-Feasibility is NP-Hard
– D. Bienstock and A. Verma (2006)
51
‣ What are bananas? – small circuits that create discontinuities
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NICTA Copyright 2014 From imagination to impact
Complexity of AC-Feasibility‣ AC-Feasibility is NP-Hard
– D. Bienstock and A. Verma (2006) – Introducing discontinuities
52
max
X
i
pdi
subject to
pgi � pdi =
X
(i,j)
pij
pij = �bijsin(✓i � ✓j)
|✓i � ✓j | ✓u
pgli pgi pgu
i
pdli pgi pdu
i
Only extension to LDC needed to be NP-
hard
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NICTA Copyright 2014 From imagination to impact
Complexity of AC-Feasibility‣ AC-Feasibility on Acyclic Networks is NP-Hard
– K. Lehmann, A. Grastien, and P. Van Hentenryck
53
solve
pgi � pd
i =
X
(i,j)
pij
qgi � qd
i =
X
(i,j)
qij
pij = gij(1� cos(✓i � ✓j))� bij sin(✓i � ✓j)
qij = �bij(1� cos(✓i � ✓j))� gij sin(✓i � ✓j)
|✓i � ✓j | ✓u
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NICTA Copyright 2014 From imagination to impact
AC-Feasibility on Acyclic Networks‣ Star Network
‣ Key ideas
54
L G
G
G
G
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NICTA Copyright 2014 From imagination to impact
Outline
‣ Motivation ‣ Power Flow Formulations ‣ Complexity ‣ The LPAC Approximation ‣ Case Studies for LPAC ‣ Convex Relaxations ‣ Case Studies for Convex Relaxations ‣ Beyond Steady States
55
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NICTA Copyright 2014 From imagination to impact 56
The AC/DC Conundrum
‣Find an approximation of AC power flows that - is more accurate than the LDC model - is useful outside normal operating conditions - reasons about voltage magnitudes and reactive power - can be embedded in discrete optimization solvers
• mixed integer programming solvers
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NICTA Copyright 2014 From imagination to impact
Trigonometric Functions
‣ Two approximations – sin(x) is replaced by x – cosine is replaced by its piecewise linear relaxation
57
pij = gijv2i � gijvivj ccos(✓i � ✓j))� bijvivj(✓i � ✓j)qij = �bijv2i + bijvivj ccos(✓i � ✓j))� gijvivj(✓i � ✓j)
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NICTA Copyright 2014 From imagination to impact
Trigonometric Functions
58
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NICTA Copyright 2014 From imagination to impact
Voltage Magnitudes
59
‣ Understanding power flows [Grainger, 94]
– Phase angle differences determine active power
– Voltage magnitude differences determine reactive power
‣ Experiments
– Per unit system
– Look at how the equations behave when
• g = 0.2 and b = 1.0 vi “ 1.0, vj P p0.8, 1.2q, p✓i ´ ✓jq P p´⇡{6,⇡{6q
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NICTA Copyright 2014 From imagination to impact
Voltage Magnitudes
60
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NICTA Copyright 2014 From imagination to impact
v “ v̂ ` �
Voltage Magnitudes
61
‣ Key ideas
- Substitute into the power flow equations - First-order Taylor expansion to remove quadratic terms
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NICTA Copyright 2014 From imagination to impact
Outline
‣ Motivation ‣ Power Flow Formulations ‣ Complexity ‣ The LPAC Approximation ‣ Case Studies for LPAC ‣ Convex Relaxations ‣ Case Studies for Convex Relaxations ‣ Beyond Steady States
62
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NICTA Copyright 2014 From imagination to impact
Experimental Settings‣ Bonmin (Bonami 2008)
– heuristics for solving MINLPs. – outer approximation method for convex MINLPs
‣ QP and SOCP – CPLEX 12.8 or Gurobi
‣ Ipopt (Waechter and Biegler 2006) – NLP
63
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NICTA Copyright 2014 From imagination to impact
Experimental Results
64
‣ Wide variety of IEEE and MATPOWER Benchmarks
- ieee14,mp24,ieee30,mp30,mp39,ieee57,ieee118,ieedd17,mp300
- Small benchmarks are easy in general - IEEE 118 is also easy
- All LPAC models solved almost instantly (LPs)
‣ Comparison with an AC Solver
- LDC and LPAC solutions versus an AC solution
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NICTA Copyright 2014 From imagination to impact
Line Active Power
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NICTA Copyright 2014 From imagination to impact
Line Active Power
DC Model LPAC Model
66
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NICTA Copyright 2014 From imagination to impact
Bus Angles
LPAC ModelDC Model
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NICTA Copyright 2012 From imagination to impact
Importance of cos: Reactive Power
Cold-Start LPAC Model (cos=1) Cold-Start LPAC Model
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NICTA Copyright 2014 From imagination to impact
Power Restoration
69
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NICTA Copyright 2014 From imagination to impact
Power Restoration
70
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NICTA Copyright 2014 From imagination to impact
Power Restoration
71
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NICTA Copyright 2014 From imagination to impact
DC versus LPAC in Restoration
72
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NICTA Copyright 2014 From imagination to impact
DC versus LPAC in Restoration
73
How good is the DC ordering?
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NICTA Copyright 2014 From imagination to impact
Expansion Planning
‣ Simplest problem formulation – starting point
‣ Network design problem – add lines to meet the increased load
‣ Under – voltage constraints – thermal limits – constraints on real and reactive power injection
74
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NICTA Copyright 2014 From imagination to impact
Expansion Planning
75
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NICTA Copyright 2014 From imagination to impact
Heuristic AC Method (HAC)
‣ Many proposals
‣ Destructive heuristics – start with all the possible lines (feasible solution) – consider each line l in turn for removal
• if feasible in AC model, remove the line l – order the line by increasing relative load
‣ Outperforms the state of the art – constructive methods
76
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NICTA Copyright 2014 From imagination to impact
Case Studies
‣ Traditional benchmarks
‣ New benchmarks – MathPower benchmarks – load and generation scaled by a factor of 3 – reaction injection is half of the real injection – cost is 1
77
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NICTA Copyright 2014 From imagination to impact
Measuring Accuracy
‣ Thermal limits
‣ Voltage magnitudes
78
max(
pp2nm + q2nmSunm
,
pp2mn + q2mn
Sumn
)
max(0,vun � vn, vn � vl
n)
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NICTA Copyright 2014 From imagination to impact
Core Results
79
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NICTA Copyright 2014 From imagination to impact
Constraint Tightening (10%)
80
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NICTA Copyright 2014 From imagination to impact
VAr Compensation
‣ VAr compensation is cheaper – can be used to meet voltage bounds
‣ Case study: Perfect Voltage Profile (PVP) – unlimited Var compensation at each bus – bus becomes synchronous condenser
• unlimited reactive power injection, voltage set-point at 1.0
– DC model • VAr compensation used in second step (cross-over)
– LPAC model • a single, integrated model
81
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NICTA Copyright 2014 From imagination to impact
VAr Compensation
82
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NICTA Copyright 2014 From imagination to impact
VAr Compensation
83
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NICTA Copyright 2014 From imagination to impact
VAr Compensation
84
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NICTA Copyright 2014 From imagination to impact
Outline
‣ Motivation ‣ Power Flow Formulations ‣ Complexity ‣ The LPAC Approximation ‣ Case Studies for LPAC ‣ Convex Relaxations ‣ Case Studies for Convex Relaxations ‣ Beyond Steady States
85
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NICTA Copyright 2014 From imagination to impact
SDP Relaxation‣ Lavaei, J., Low, S.: Zero duality gap in optimal power flow
problem. (2012)
86
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NICTA Copyright 2014 From imagination to impact
SOCP Relaxation‣ Jabr, R.: Radial distribution load flow using conic
programming. (2006) ‣ Sojoudi, S., Lavaei, J.: Network Topologies Guaranteeing
Zero Duality Gap for Optimal Power Flow Problem.(2012)
87
wii wij
wji wjj
�⌫ 0
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NICTA Copyright 2014 From imagination to impact
SOCP Relaxation‣ Jabr, R.: Radial distribution load flow using conic
programming. (2006) ‣ Sojoudi, S., Lavaei, J.: Network Topologies Guaranteeing
Zero Duality Gap for Optimal Power Flow Problem.(2012)
88
wii � 0wiiwjj � (wR
ij)2 + (wI
ij)2
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NICTA Copyright 2014 From imagination to impact
Dist-Flow Relaxation
‣ Define
89
only appear squared
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NICTA Copyright 2014 From imagination to impact
Dist-Flow Relaxation
‣ M. Farivar et al: Inverter var control for distribution systems with renewables, 2011
90
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NICTA Copyright 2014 From imagination to impact
Transcendental Relaxation‣ Key ideas
– compositional, structural relaxation – relax trigonometric functions – exploit the narrow bounds in power systems
– dual modelling ‣ Resulting optimization model
– quadratic and convex
91
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NICTA Copyright 2014 From imagination to impact
Power Flow Equations‣ Ohm’s law for each line (i,j)
‣ Kirchhoff’s current law for each bus i
92
pij “ gijv2i ´ gijvivj cosp✓i ´ ✓jq ´ bijvivj sinp✓i ´ ✓jq (1)
qij “ ´bijv2i ` bijvivj cosp✓i ´ ✓jq ´ gijvivj sinp✓i ´ ✓jq (2)
pi “ÿ
pi,jqPEpij
qi “ÿ
pi,jqPEqij
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NICTA Copyright 2014 From imagination to impact
Convex Quadratic Relaxation‣ Quadratic convex relaxation of
– cosine function – square function
‣ Polyhedral relaxation of – sine function
‣ MacCormick relaxation of – multi-linear terms
93
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NICTA Copyright 2014 From imagination to impact 94
QP Relaxation of Cosine
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NICTA Copyright 2014 From imagination to impact
‣Convex quadratic relaxation of cosine
‣Proposition:
|cosp✓q “ 1 ´˜1 ´ cosp✓uq
p✓uq2
¸✓2.
@✓ P r´✓u, ✓us : |cosp✓q • cosp✓q.
QP Relaxation of Cosine
95
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NICTA Copyright 2014 From imagination to impact
QP Relaxation of Square
96
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NICTA Copyright 2014 From imagination to impact
QP Relaxation of Square‣ Convex quadratic Relaxation
97
qv2 • v2
qv2 §`vu ` vl
˘v ´ vuvl
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NICTA Copyright 2014 From imagination to impact
QP Relaxation of Sine
98
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NICTA Copyright 2014 From imagination to impact
Polynomial Relaxation of Sine
‣ Polynomial relaxation
‣ Proposition:
99
|sinp✓q § cos
ˆ✓u
2
˙ ˆ✓ ´ ✓u
2
˙` sin
ˆ✓u
2
˙
|sinp✓q • cos
ˆ✓u
2
˙ ˆ✓ ` ✓u
2
˙´ sin
ˆ✓u
2
˙
@✓ P r´✓u, ✓us : cos
`✓u
2
˘ `✓ ´ ✓u
2
˘` sin
`✓u
2
˘• sinp✓q.
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NICTA Copyright 2014 From imagination to impact
MacCormick Relaxations
‣ Sequential Bilinear Relaxations
100
}vivj • vlivj ` vljvi ´ vlivlj
}vivj • vui vj ` vuj vi ´ vui vuj
}vivj § vlivj ` vuj vi ´ vlivuj
}vivj § vui vj ` vljvi ´ vui vlj
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NICTA Copyright 2014 From imagination to impact
The Initial QC Model
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NICTA Copyright 2014 From imagination to impact
Dual Modelling‣ Key idea
– Use several models of the optimisation problem – The solution set is the intersection
‣ Heavily used in optimisation for decades – constraint programming (early 1990s)
• improving propagation – mathematical programming
‣ In nonlinear optimisation – Formalized by [Liberty, 04] to strengthen the relaxation – Generalized reduction constraints [Ruiz & Grossmann, 11]
• intersecting several formulations based on physical interpretation of the problem
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NICTA Copyright 2014 From imagination to impact
Redundancy in MINLP
103
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NICTA Copyright 2014 From imagination to impact
Redundancy in MINLP
104
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NICTA Copyright 2014 From imagination to impact
Power Loss Formulation‣ Power loss on line (i,j)
105
Current magnitude squared
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NICTA Copyright 2014 From imagination to impact
The QC-NLP Model
106
‣ Power loss on line (i,j)
‣ Proposition:
How to prove this?
Use the perspective z f(x/z,y/z) where f(x,y) = x2 + y2
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NICTA Copyright 2014 From imagination to impact
The QC-SOCP Model
107
‣ Second-Order Cone Formulation – [Farivar, M., Clarke, C., Low, S., Chandy, K. 2011]
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One/Off Constraints
108
‣ Can we avoid big-M transformations?
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NICTA Copyright 2014 From imagination to impact
On/Off Constraints
109
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On/Off Constraints
110
‣ Can I represent the convex hull in the original space of variables?
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NICTA Copyright 2014 From imagination to impact
On/Off Constraints: Point Case‣ When is a point,
the convex hull becomes
111
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NICTA Copyright 2014 From imagination to impact
On/Off Constraints: quadratic‣ Consider the function
‣ The convex hull becomes
112
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NICTA Copyright 2014 From imagination to impact
On/Off Constraints‣ The idea can be generalised for intervals and
monotone functions: e.g. linear constraints
113
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NICTA Copyright 2014 From imagination to impact
On/Off Constraints‣ The idea can be generalised for intervals and
monotone functions: e.g. linear constraints
114
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NICTA Copyright 2014 From imagination to impact
On/Off Constraints in Power Systems‣ When a line is switched off, the phase angle
difference must increase. ‣ We can then apply the above results to each
constraint
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NICTA Copyright 2014 From imagination to impact
On/Off Constraints in Power Systems‣ Sine constraint
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NICTA Copyright 2014 From imagination to impact
On/Off Constraints in Power Systems
117
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NICTA Copyright 2014 From imagination to impact
Outline
‣ Motivation ‣ Power Flow Formulations ‣ Complexity ‣ The LPAC Approximation ‣ Case Studies for LPAC ‣ Convex Relaxations ‣ Case Studies for Convex Relaxations ‣ Beyond Steady States
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NICTA Copyright 2014 From imagination to impact
Experimental Settings‣ Instances: [Matpower 2011]
‣ Congested Instances
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NICTA Copyright 2014 From imagination to impact
Optimal Power Flows
‣ The problem – Minimize the cost of generation for a given load – Intensively studied – Purest
120
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NICTA Copyright 2014 From imagination to impact
Optimal Power Flows
121
quadratic production cost
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NICTA Copyright 2014 From imagination to impact
Optimal Power Flows
122
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NICTA Copyright 2014 From imagination to impact
Optimal Power Flows
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NICTA Copyright 2014 From imagination to impact
Optimal Power Flows
124
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NICTA Copyright 2014 From imagination to impact
Optimal Power Flows
125
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NICTA Copyright 2014 From imagination to impact
Optimal Power Flows: Summary‣ Dual modelling is critical ‣ QC-NLP dominates the SOCP relaxation
– especially on congested benchmarks ‣ QC-NLP is orders of magnitude faster than the
SDP relaxation with minimal loss in accuracy ‣ QC-NLP is orders of magnitude faster than QC-
SOCP
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NICTA Copyright 2014 From imagination to impact
Outline‣ Motivation ‣ Complexity ‣ The LPAC Approximation
– The model – Power restoration
‣ The QC Relaxation – The model – Optimal power flow – Optimal line-switching power flow – Capacitor placement
‣ Conclusion and future work
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NICTA Copyright 2014 From imagination to impact
Optimal Line-Switching Power Flows‣ The Problem
– same as OPF but allows to switch off lines – exploit Braess paradox in power systems – very hard computationally
‣ Prior work – [Fisher, O’Neil, Ferris, 07], [Bienstock, 12] – shows the cost benefits of switching lines off – based on the LDC model
‣ This work – First attempt at provable quality bounds
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NICTA Copyright 2014 From imagination to impact
Optimal Line-Switching Power Flows
129
‣ The AC Model
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NICTA Copyright 2014 From imagination to impact
Optimal Line-Switching Power Flows‣ AC-MINLP
– the AC version ‣ DC
– the original version: Fisher, O’Neill, Ferris, 2008 ‣ QC
– QC-NLP – QC-SOP
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NICTA Copyright 2014 From imagination to impact
Optimal Line-Switching Power Flows
131
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NICTA Copyright 2014 From imagination to impact
Optimal Line-Switching Power Flows
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NICTA Copyright 2014 From imagination to impact
Optimal Line-Switching Power Flows
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NICTA Copyright 2014 From imagination to impact
Optimal Line-Switching Power Flows
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NICTA Copyright 2014 From imagination to impact
Optimal Line-Switching Power Flows‣ The DC model is overly optimistic
– infeasible configurations – suboptimal solutions when crossed-over
‣ QC bounds shows the quality of heuristic B&B – small gaps: 0.13% and 1.95% on traditional
benchmarks at the root node – gaps can reached 40% on congested benchmarks at
the root node – gaps can be reduced to 4% by solving the mixed
integer version
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NICTA Copyright 2014 From imagination to impact
Outline‣ Motivation ‣ Complexity ‣ The LPAC Approximation
– The model – Power restoration
‣ The QC Relaxation – The model – Optimal power flow – Optimal line-switching power flow – Capacitor placement
‣ Conclusion and future work
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NICTA Copyright 2014 From imagination to impact
Capacitor Placement‣ Capacitors inject reactive power ‣ The problem is to place capacitors in a network
to improve the voltage profile – well-studied problems – almost always tackled by heuristics
• need to reason about reactive power and voltage magnitudes
‣ The goal is to minimize the number of capacitors
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NICTA Copyright 2014 From imagination to impact
Capacitor Placement
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NICTA Copyright 2014 From imagination to impact
Capacitor Placement‣ Key Messages
– MINLP heuristics give near optimal solutions – QC gives tight lower bounds
• justifying the first message – The QC model scales well to medium-sized instances
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NICTA Copyright 2014 From imagination to impact
Capacitor Placement
140
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NICTA Copyright 2014 From imagination to impact
Reconfiguration Problems‣ Find out the network topology (by opening and
closing switches) to – minimize losses – or maximize balances
‣ while satisfying operational constraints – thermal limits – phase angle and voltage magnitude constraints – …
‣ Relaxation – Dist-flow for radial networks, QC for meshed networks
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NICTA Copyright 2014 From imagination to impact
Loss Minimization
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NICTA Copyright 2014 From imagination to impact
Loss Minimization
143
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NICTA Copyright 2014 From imagination to impact
Reconfiguration Problems‣ Key messages
– Zero-gap relaxation on radial networks – Tight on meshed networks
• loses accuracy on the largest one – Scalability
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NICTA Copyright 2014 From imagination to impact
Outline
‣ Motivation ‣ Power Flow Formulations ‣ Complexity ‣ The LPAC Approximation ‣ Case Studies for LPAC ‣ Convex Relaxations ‣ Case Studies for Convex Relaxations ‣ Beyond Steady States
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NICTA Copyright 2014 From imagination to impact
Restoration Ordering Problem (ROP)
146
Initial Steady State Steady State 1 Steady State 2 Final Steady
State N
Fix item x1 Fix item x2
…
Fix item x3, x4,…, xN
‣Find the best sequence of restoration [x1,x2,x3,…,xn] so that the size of the blackout is minimised
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NICTA Copyright 2014 From imagination to impact
Key Question‣ Can I move from steady states to steady states?
‣ Capturing the dynamics
147
State VariablesDynamic
Variables
Network State
Dynamics
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NICTA Copyright 2014 From imagination to impact
Transient Analysis‣ Can you correct the steady states to ensure
transient stability? – rotor angle stability – Swing equation
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NICTA Copyright 2014 From imagination to impact
Transient Analysis‣ Can you correct the steady states to ensure
transient stability? – rotor angle stability – Swing equation
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NICTA Copyright 2014 From imagination to impact
Transient Stability‣ Angle for the center of inertia
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NICTA Copyright 2014 From imagination to impact
Transient Stability‣ A generator is modeled as an internal bus
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NICTA Copyright 2014 From imagination to impact
Network State
152
‣ Power flow equations before closing
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NICTA Copyright 2014 From imagination to impact
Network State
153
‣ Power flow equations after closing
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NICTA Copyright 2014 From imagination to impact
Network State
154
‣ Voltage Stability Constraints
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NICTA Copyright 2014 From imagination to impact
Minimal Transient-Stable Correction
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Overall Procedure
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NICTA Copyright 2014 From imagination to impact
Overall Procedure
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NICTA Copyright 2014 From imagination to impact
Overall Procedure
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NICTA Copyright 2014 From imagination to impact
Power Restoration with Transient Stability
‣ Minimize the distance to the steady-state dispatch while constraining the rotor angle
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NICTA Copyright 2014 From imagination to impact
Power Restoration with Transient Stability
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‣ Minimizing the rotor angle while maintaining a certain dispatching distance
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NICTA Copyright 2014 From imagination to impact
Beyond Steady States‣ Using the dynamics to correct the steady state
– hierarchical approach – feedback loop
‣ The power flow equations – need to be solved exactly – the relaxations are “cheating”
‣ Scalability is the next frontier – more realistic dynamic models – thermal limits – scaling performance
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NICTA Copyright 2014 From imagination to impact
Open Issues‣ Power Flows
– Can we get scalable approximations with performance guarantees on these problems?
– Can we get scalable relaxations? – Can we define what normal operating conditions
means? ‣ Dynamics
– Can we get scalable optimization approaches? ‣ Stochastic and robust optimisation
– Generalizing the deterministic case
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