1. Exact Convex Relaxation for Optimal Power Flow in Distribution Networks
Lingwen Gan1, Na Li1, Ufuk Topcu2, Steven Low1
1California Institute of Technology
2University of Pennsylvania

2. Optimal power flow in distribution 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

3. 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. A
Def: If every solution lies in A, then call the relaxation exact.

4. Outline Formulate OPF. Convex relaxation is in general not exact.
Propose a modified OPF.
Modified OPF has an exact convex relaxation.

5. An OPF example Volt/VAR control Potential controllable elements
Inverters of PV panels
Controllable loads (shunt capacitors, EVs)

6. Mathematical formulation
For simplicity, let’s look at one-line networks.
Bus 0
Bus 1
Bus n

7. Mathematical formulation
Bus 0
Bus 1
Bus n
f is strictly increasing in power loss
Nonconvex

8. Convex relaxation
OPF
SOCP A
Q: does every solution to SOCP lie in A?

9. Is SOCP exact? In general, no. 1 unit power generation,
p injected into the grid,
1-p gets curtailed.

10. What do we do when non-exact?
Modify OPF in order to obtain an exact convex relaxation. A’ A B’
We want
the grey area to be small;
the relaxation to be exact after modification.

11. The modification
OPF
OPF-m
What is vilin(p,q)?

12. What is vilin(p,q)? A’ An affine function of p and q.
is a linear constraint on p and q.
An upper bound on vi.
Smaller feasible set than OPF.

13. Grey area is empirically small and “bad”
is a good approximation of v
Grey area is small. A’ w
Grey area is “bad”.

14. Convex relaxation OPF-m SOCP-m A’
Q: Does every solution to SOCP-m lie in A’?

15. Exactness of SOCP-m Thm: If condition (*) holds, then
the SOCP-m relaxation is exact;
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.

16. Condition (*)
Depend only on parameters , not solutions of OPF-m or SOCP-m.
Impose small distributed generation.

17. How to check (*)?
much stricter than (*)!

18. (*) holds with significant margin
IEEE network: no distributed generation.
Worst case: maximizes
No load, all capacitors are switched on.

19. (*) holds with significant margin
SCE 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.

20. Thank you! 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!