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!