Study of the Paper “Efficient AC Optimal Power Flow and Global

Optimizer Solutions” part2, The Angular Cut

Yuyang Chen

Presentation AgendaI. Background introduction , review of part 1’s

theory

II. Primary theoretical studies: how does the angular cut in D&C work, Detail explanations

III. Secondary theoretical studies: the effects of additional constraints.

IV. Conclusions

I. Background introduction , review of part 1’s theory• In Bai’s paper , SDP relaxation of OPF gives some promising results. But it

offers no help when the rank of SDP solution matrix is larger than 2, which is physically meaningless.

• In Mitsubishi's B&B method it attempts to address such problem by D&C algorithm type approach.

• This paper’s 1st contribution is the introduction of a Novel angular cut, which increase the effectiveness of D&C method.

• This paper’s 2nd contribution is to guarantee the infeasibility of original OPF if no solution is feasible from GO method.

• This paper ‘s method does not required CONOPF , a Non-Linear solver.

Note: for simplicity we will refer to the (SDP, D&C , angular cut )method used in this paper as “GO method”

I. Background introduction , review of part 1’s theory

• Previously , we gave a frame work of GO method : through SDP relaxation of OPF and Divide and Conquer method we can find the Global Optimizer.

• Let’s take a closer look at how a efficient branching from parent problem into child problems is achieve through a novel angular cut introduced in this paper

II. Primary theoretical studies:What is the effect of applying D&C , its visualization on OPF feasible region and SDP solution space.

𝜃𝑖∗ = tan−1 𝑦𝑖

∗

𝑥𝑖∗ = tan−1 𝑥𝑖

∗𝑦𝑖∗

𝑥𝑖∗2 = tan−1 𝑊𝑖,𝑖+𝑁

∗

𝑊𝑖,𝑖∗

but Also , 𝜃𝑖∗ = tan−1 𝑦𝑖

∗2

𝑥𝑖∗𝑦𝑖

∗ = tan−1 𝑊𝑖+𝑁 ,𝑖+𝑁∗

𝑊𝑖,𝑖+𝑁∗

if rank W >1 tan−1 𝑊𝑖+𝑁 ,𝑖+𝑁∗

𝑊𝑖,𝑖+𝑁∗ ≠ tan−1 𝑊𝑖,𝑖+𝑁

∗

𝑊𝑖,𝑖∗

• GO method can be more efficient because it exploit the SDP solution when dividing parent problem but B&B did not.

• In D&C type approach ( this include both B&B and Go method) ,If parent node solution overlap with child node solution then suchalgorithm is less efficient and can be improved:

• In B&B method parent node solution and child node solutions over lap. In GO method they do not.

• Go method do not use CONOPF , a non-linear solver.

• Adding more constrains to the same set of variables makes the SDP relaxation tighter and therefore easier to compute.

III . Secondary theoretical studies:Effects of adding additional constraints without adding additional variables

Conclusions & ending remarks