exact convex relaxation for optimal power flow in distribution networks lingwen gan 1, na li 1, ufuk...
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Exact Convex Relaxation for Optimal Power Flow in Distribution Networks
Lingwen Gan1, Na Li1, Ufuk Topcu2, Steven Low1
1California Institute of Technology2University of Pennsylvania
Optimal power flow indistribution networks
• Optimal power flow (OPF) has been studied in transmission networks for over 50 years– DC (linear) approximation• Voltage close to nominal value• small power loss• small voltage angle
• Volt/VAR control, demand response problems motivate OPF in distribution (tree) networks– Have to solve nonlinear, nonconvex power flow
No longer true in distribution networks
A
How to solve nonconvex power flow?
• (Heuristic) nonconvex programming.– Hard to guarantee optimality.
• Convexify the problem.– Relax the feasible set A to its convex hull.• If solution is in A, then done.
– In general, don’t know when relaxation is exact.
Def: If every solution lies in A, then call the relaxation exact.
Outline
• Formulate OPF.• Convex relaxation is in general not exact.• Propose a modified OPF.• Modified OPF has an exact convex relaxation.
An OPF example
• Volt/VAR control• Potential controllable elements– Inverters of PV panels– Controllable loads (shunt capacitors, EVs)
Mathematical formulation
Bus 0 Bus 1 Bus n
For simplicity, let’s look at one-line networks.
Mathematical formulationBus 0 Bus 1 Bus n
f is strictly increasing in power loss
Nonconvex
A
Convex relaxationOPF SOCP
Q: does every solution to SOCP lie in A?
Is SOCP exact?In general, no.
1 unit power generation,p injected into the grid,1-p gets curtailed.
AA’
What do we do when non-exact?Modify OPF in order to obtain an exact convex relaxation.
We want• the grey area to be small;• the relaxation to be exact after modification.
B’
The modification
OPF OPF-m
What is vilin(p,q)?
A’
What is vilin(p,q)?
1. An affine function of p and q.• is a linear constraint on p and q.
2. An upper bound on vi.• Smaller feasible set than OPF.
A’
Grey area is empirically small and “bad”
w
is a good approximation of v
Grey area is small.
Grey area is “bad”.
Convex relaxationOPF-m SOCP-m
Q: Does every solution to SOCP-m lie in A’?A’
Exactness of SOCP-m
Thm: If condition (*) holds, then1. the SOCP-m relaxation is exact;2. the SOCP-m has a unique solution.
Condition (*)• can be checked prior to solving SOCP-m;• holds for all test networks (see later);• imposes “small” distributed generation.
Condition (*)
• Depend only on parameters , not solutions of OPF-m or SOCP-m.
• Impose small distributed generation.
How to check (*)?
much stricter than (*)!
(*) holds with significant margin
Worst case: maximizes .No load, all capacitors are switched on.
IEEE network: no distributed generation.
(*) holds with significant marginSCE network:5 PVs with 6.4MW nameplate generation capacity (11.3MW peak load).
Worst case: maximizes .No load, all PVs are generating at full capacity,all capacitors are switched on.
Summary
• The SOCP relaxation for OPF is in general not exact
• Propose OPF-m• The convex relaxation SOCP-m is exact if (*)
holds• (*) widely holds in test networks
Thank you!
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