2.2 Bus injection model and conic relaxations
2.2.6 Proof of Theorem 1
We now prove that the feasible sets of OPF and P1,Pch,P2 are equivalent when restricted
to the underlying G-partial matrices. Similarly, the feasible sets of their relaxations are equivalent when G is a tree. When any of the relaxations are exact we can construct an n-dimensional complex voltage vector V ∈V that optimally solves OPF.
To define the set of G-partial matrices associated with P1,Pch,P2 suppose F is a graph
onn nodes such thatG is a subgraph of F, i.e., IG ⊆IF. An F-partial matrix WF is called
anF-completion of the G-partial matrixWG if
[WF]ij = [WG]ij for all (i, j)∈IG ⊆IF,
i.e., WF agrees with WG on the index set IG. If F is Cn, the complete graph on n nodes,
F-completion if in addition WF 0. WF is a rank-1 F-completion if rank WF = 1. It can
be checked that if WG 6 0 then WG does not have a psd F-completion. If rank WG 6= 1
then it does not have a rank-1 F-completion. Define
W1 :={WG |WG satisfies (2.7a)−(2.7b),
∃ psd rank-1Cn-completion of WG}.
Recall that for W, an n ×n matrix, W(G) is the G-partial matrix corresponding to the IG entries of W. Given an n×n psd rank-1 matrix W that is feasible for P1, W(G) is in
W1. Conversely given a WG ∈ W1, its psd rank-1 Cn-completion is a feasible solution for
P1. HenceW1 is the set of IG entries of all n×n matrices feasible forP1 and is nonconvex.
Define
W+1 :={WG | WG satisfies (2.7a)−(2.7b),
∃ psd Cn-completion ofWG}.
W+1 is the set of IG entries of all n×n matrices feasible for R1. It is convex and contains
W1.
Similarly define the corresponding sets for Pch and Rch:
Wch:={WG |WG satisfies (2.7a)−(2.7b),
∃ psd rank-1Gch-completion ofWG},
W+ch:={WG |WG satisfies (2.7a)−(2.7b),
∃ psd Gch-completion ofWG}.
Wch and W+ch are the sets ofIG entries of Gch-partial matrices feasible for problemsPch and
P2 and R2 define:
W2 :={WG | WG satisfies (2.7a)−(2.7b) and (2.8),
WG(e)0, rank WG(e) = 1 for all e∈E},
W+2 :={WG | WG satisfies (2.7a)−(2.7b),
WG(e)0 for alle ∈E}.
Informally the sets W1,W+1,Wch,W+ch,W2 and W+2 describe the feasible sets of the various
problems restricted to the IG entries of their respective partial matrix variables.
To relate the sets to the feasible set of OPF, consider the map f from Cn to the set of
G-partial matrices defined as:
f(V) :=WG where [WG]kk=|Vk|2, k∈N,and
[WG]jk =VjVkH, (j, k)∈E.
Also, let f(V) := {f(V) | V ∈V}.
The sketch of the proof is as follows. We prove Theorem 1(a) in Lemma 3 and then Theorem 1(b) in Lemma 4 below. Theorem 1(c) then follows from these two lemmas. Lemma 3. f(V) =W1 =Wch =W2.
Proof. First, we show that f(V) = W1. Consider V ∈ V. Then W = V VH is feasible
for P1 and hence the G-partial matrix W(G) is in W1. Thus, f(V) ⊆ W1. To prove
W1 ⊆f(V), consider the rank-1 psd Cn completion of a G-partial matrix in W1. Its unique
spectral decomposition yields a vector V that satisfies (2.3)–(2.4) and hence is inV. Hence, f(V) = W1.
Now, fix a chordal extension Gch of G. We now prove:
W1 ⊆ Wch ⊆ W2 ⊆ W1.
it is easy to check that W(Gch) is feasible for Pch and hence WG is in Wch as well.
To show Wch ⊆W2 consider aWG ∈Wch and its psd rank-1Gch-completionWch. Since
every edgee ofG is a 2-clique inGch,WG(e) = Wch(e) is psd rank-1 by the definition of psd
and rank-1 for Wch. We are thus left to show that WG satisfies the cycle condition (2.8).
Consider the following statement Tk for 3≤k≤n:
Sk: For all cycles (n1, n2, . . . , nk) of length k inGch we have:
∠[Wch]n1n2 +∠[Wch]n2n3 +. . .+∠[Wch]nkn1 = 0 mod 2π.
Fork = 3, a cycle (n1, n2, n3) defines a 3-clique inGch and thus Wch(n1, n2, n3) is psd rank-1
and Wch(n1, n2, n3) =uuH for some u:= (u1, u2, u3)∈C3. Then
∠[Wch]n1n2 +∠[Wch]n2n3 +∠[Wch]n3n1
=∠(u1uH2 )(u2uH3 )(u3uH1 )
= 0 mod 2π.
LetTr be true for all 3≤r≤k and consider a cycle (n1, n2, . . . , nk+1) of length k+ 1 inGch.
Since Gch is chordal, this cycle must have a chord, i.e., an edge between two nodes, say,n1
andnk0, that are not adjacent on the cycle. Then (n1, n2, . . . , nk0) and (n1, nk0, nk0+1, . . . , nk)
are two cycles in Gch. By hypothesis,Tk0 and Tk−k0+2 are true and hence ∠[Wch]n1n2 +∠[Wch]n2n3 +. . .+∠[Wch]nk0n1
=∠[Wch]n1nk0 +∠[Wch]nk0nk0+1+. . .+∠[Wch]nkn1
= 0 mod 2π.
We conclude that Tk+1 is true by adding the above equations and using ∠[Wch]n1nk0 = −∠[Wch]n
k0n1 mod 2π since Wch is Hermitian. By induction, Wch satisfies the cycle condi-
tion. Also, WG=Wch(G) satisfies the cycle condition and hence inW2. This completes the
proof of Wch⊆W2.
WG to show WG ∈W1. Define θ ∈ Cn as follows. Letθ1 := 0. For j ∈ N \ {1} let (1, n2),
(n2, n3),. . . ,(nk, j) be any path from node 1 to node j. Define
θj :=−(∠[WG]1n2 +∠[WG]n2n3 +. . .+∠[WG]nkj) mod 2π.
Note that the above definition is well-defined: if there is another sequence of edges from node 1 to node j, the above relation still defines θj uniquely because WG satisfies the cycle
condition. Let V := hp[WG]11eiθ1, · · · p [WG]nneiθn i .
Then it can be verified that W := V VH is a psd rank-1 Cn-completion of WG. Hence
WG∈W1. This completes the proof of the lemma.
Lemma 4. W+1 =W+ch⊆W+2. If G is acyclic, then W+1 =W+ch=W+2. Proof. It suffices to prove
W+ch ⊆ W + 1 ⊆ W + ch ⊆ W + 2. (2.10) To show W+ch ⊆ W+
1, suppose WG ∈ W+ch. Let Wch be a psd Gch-completion of WG
for a chordal extension Gch. Since any psd partial matrix on a chordal graph has a psd
Cn-completion [74, Theorem 7],Wchhas a psdCn-completion. Obviously, any psdCn-completion
ofWchis also a psdCn-completion ofWG, i.e., WG∈W+1. The relationW + 1 ⊆ W + ch ⊆ W + 2
follows a similar argument to the proof of Lemma 3.
If G is acyclic, then G is itself chordal and hence WG has a psd Cn-completion, i.e.,
W+2 ⊆W+1. This impliesW+1 =W+ch=W
+ 2.
To prove Theorem 1(c) note that parts (a) and (b) imply
Hence R1 is exact (p∗1 = r ∗ 1) iff Rch is exact (p∗ch = r ∗ ch). If R2 is exact, i.e., p∗2 = r ∗ 2, then
both inequalities above become equalities, proving Theorem 1(c). This completes the proof of Theorem 1.