Applications

GLOBAL OPTIMALITY FOR SINGLE-PHASE NETWORKS

2.6 Applications

by definition must be a global optimum as well. The argument on local optimality follows directly from the proof of Theorem 1.

The power flow equations are:

𝑣_𝑗 =𝑣_𝑘+2R𝑒(𝑧_{𝑗 𝑘}𝑆^H

𝑗 𝑘) − |𝑧_{𝑗 𝑘}|²ℓ_{𝑗 𝑘}, ∀(𝑗 , 𝑘) ∈ E (2.8a) 𝑣_𝑗 = |𝑆_{𝑗 𝑘}|²

ℓ_{𝑗 𝑘}

, ∀(𝑗 , 𝑘) ∈ E (2.8b)

𝑠_𝑗 = ∑︁

𝑘:𝑗→𝑘

𝑆_{𝑗 𝑘} − ∑︁

𝑖:𝑖→𝑗

(𝑆_{𝑖 𝑗} −𝑧_{𝑖 𝑗}ℓ_{𝑖 𝑗}), ∀𝑗 ∈ V. (2.8c)

Given a cost function 𝑓(𝑠) : C^𝑁 → R, we are interested in the following OPF problem:

minimize

𝑥=(𝑠,𝑣 ,ℓ,𝑆)

𝑓(𝑠) (2.9a)

subject to (2.8) (2.9b)

𝑣

𝑗 ≤ 𝑣_𝑗 ≤ 𝑣_𝑗 (2.9c)

𝑠

𝑗 ≤ 𝑠_𝑗 ≤ 𝑠_𝑗 (2.9d)

ℓ_{𝑗 𝑘} ≤ ℓ_{𝑗 𝑘}. (2.9e)

All the inequalities for complex numbers in this section are enforced for both the real and imaginary parts.

Definition 13. A function𝑔 :R→Risstrongly increasingif there exists real𝑐 > 0 such that for any𝑎 > 𝑏, we have

𝑔(𝑎) −𝑔(𝑏) ≥𝑐(𝑎−𝑏). We now make the following assumptions on OPF:

(i) The underlying graph Gis a tree.

(ii) The cost function 𝑓 is convex, and is strongly increasing in R𝑒(𝑠_𝑗)(or I𝑚(𝑠_𝑗)) for each 𝑗 ∈ V and non-decreasing in I𝑚(𝑠_𝑗)(or R𝑒(𝑠_𝑗)).

(iii) The problem (2.9) is feasible.

(iv) The line current limit satisfiesℓ_{𝑗 𝑘} ≤ 𝑣

𝑗|𝑦_{𝑗 𝑘}|².

Assumption (i) is generally true for distribution networks and assumption (iii) is typically mild. As for (ii), 𝑓 is commonly assumed to be convex and increasing in R𝑒(𝑠_𝑗) and I𝑚(𝑠_𝑗) in the literature (e.g., [31, 74]). Assumption (ii) is only

slightly stronger since one could always perturb any increasing function by an arbitrarily small linear term to achieve strong monotonicity. Assumption (iv) is not common in the literature but is also mild because of the following reason.

Typically𝑉_𝑗 = (1+𝜖_𝑗)𝑒^𝒊𝜃𝑗 in per unit where𝜖 ∈ [−0.1,0.1]and the angle difference 𝜃_{𝑗 𝑘} :=𝜃_𝑗−𝜃_𝑘 between two neighboring buses 𝑗 , 𝑘typically has a small magnitude.

Thus the maximum value of|𝑉_𝑗−𝑉_𝑘|²=| (1+𝜖_𝑗)𝑒^{𝒊𝜃𝑗 𝑘}− (1+𝜖_𝑘) |², which is equivalent toℓ_{𝑗 𝑘}/|𝑦_{𝑗 𝑘}|², should be much smaller than𝑣

𝑗 which is≈1 per unit.

Problem (2.9) is non-convex, as constraint (2.8b) is not convex. Denote byX the set of (𝑠, 𝑣 , ℓ, 𝑆) that satisfy (2.9b)-(2.9e), so (2.9) is in the form of (2.1). We can relax (2.9) by convexifying (2.8b) into a second-order cone [27]:

minimize

𝑥=(𝑠,𝑣 ,ℓ,𝑆)

𝑓(𝑠) (2.10a)

subject to (2.8a),(2.8c),(2.9c)−(2.9e) (2.10b)

|𝑆_{𝑗 𝑘}|² ≤ 𝑣_𝑗ℓ_{𝑗 𝑘}. (2.10c)

One can similarly regard ˆX as the set of (𝑠, 𝑣 , ℓ, 𝑆) that satisfy (2.10b), (2.10c). It is proved in [27] that if𝑠

𝑗 =−∞ −𝒊∞for all 𝑗 ∈ V, then (2.10) is exact, meaning any optimal solution of (2.10) is also feasible and hence globally optimal for (2.9).

Now we show that the same condition also guarantees that any local optimum of (2.9) is also globally optimal. This implies that a local search algorithm such as the primal-dual interior point method can produce a global optimum as long as it converges.

Theorem 5. If𝑠

𝑗 = −∞ −𝒊∞for all 𝑗 ∈ V, then any local optimum of (2.9) is a global optimum.

Proof. Our strategy is to construct appropriate 𝑉 and {ℎ_𝑥} and then prove such construction satsify both Condition (C) and Condition (C’). Let

𝑉(𝑥) := ∑︁

(𝑗 , 𝑘)∈E

𝑣_𝑗ℓ_{𝑗 𝑘} − |𝑆_{𝑗 𝑘}|². (2.11) Clearly,𝑉 is a valid Lyapunov-like function satisfying Definition 10.

For each 𝑥 = (𝑠, 𝑣 , ℓ, 𝑆) ∈ X \ Xˆ , let M be the set of (𝑗 , 𝑘) ∈ E such that

|𝑆_{𝑗 𝑘}|²< 𝑣_𝑗ℓ_{𝑗 𝑘}. For(𝑗 , 𝑘) ∈ M, the quadratic function 𝜙_{𝑗 𝑘}(𝑎) := |𝑧_{𝑗 𝑘}|²

4 𝑎²+ 𝑣_𝑗 −R𝑒(𝑧_{𝑗 𝑘}𝑆^H

𝑗 𝑘)

𝑎+ |𝑆_{𝑗 𝑘}|²−𝑣_𝑗ℓ_{𝑗 𝑘}

must have a unique positive root as 𝜙_{𝑗 𝑘}(0) < 0. We defineΔ𝑗 𝑘 to be this positive root if (𝑗 , 𝑘) ∈ Mand 0 otherwise.

Assumption (iv) impliesℓ_{𝑗 𝑘} ≤ 𝑣_𝑗|𝑦_{𝑗 𝑘}|², and therefore 𝑣_𝑗−R𝑒(𝑧_{𝑗 𝑘}𝑆^H

𝑗 𝑘) ≥ 𝑣_𝑗 − |𝑧_{𝑗 𝑘}||𝑆_{𝑗 𝑘}|

≥𝑣_𝑗− |𝑧_{𝑗 𝑘}|√︁

𝑣_𝑗ℓ_{𝑗 𝑘} ≥ 𝑣_𝑗− |𝑧_{𝑗 𝑘}|√︃

𝑣²

𝑗|𝑦_{𝑗 𝑘}|² = 0. It further implies𝜙_{𝑗 𝑘}(𝑎) is strictly increasing for𝑎 ∈ [0,Δ𝑗 𝑘].

Now consider the pathℎ_𝑥(𝑡) :=(𝑠˜(𝑡),𝑣˜(𝑡),ℓ˜(𝑡),𝑆˜(𝑡))for𝑡 ∈ [0,1], where

˜

𝑠_𝑗(𝑡) =𝑠_𝑗 − 𝑡 2

∑︁

𝑖:𝑖→𝑗

𝑧_{𝑖 𝑗}Δ𝑖 𝑗 − 𝑡 2

∑︁

𝑘:𝑗→𝑘

𝑧_{𝑗 𝑘}Δ𝑗 𝑘, (2.12a)

˜

𝑣_𝑗(𝑡) =𝑣_𝑗, (2.12b)

˜

ℓ_{𝑗 𝑘}(𝑡) =ℓ_{𝑗 𝑘} −𝑡Δ𝑗 𝑘, (2.12c)

˜

𝑆_{𝑗 𝑘}(𝑡) =𝑆_{𝑗 𝑘} − 𝑡

2𝑧_{𝑗 𝑘}Δ𝑗 𝑘. (2.12d)

Clearly we have ℎ_𝑥(0) =𝑥. It can be easily checked that ℎ_𝑥(𝑡)is feasible for (2.10) for𝑡 ∈ [0,1]andℎ_𝑥(1)is feasible for (2.9). Therefore,ℎ_𝑥 is indeed[0,1] →Xˆ and ℎ_𝑥(1) ∈ X.

Since 𝑧_{𝑗 𝑘} > 0, both real and imaginary parts of ˜𝑠_𝑗(𝑡) are strictly decreasing for (𝑗 , 𝑘) ∈ M and stay unchanged otherwise. By assumption (ii), 𝑓(𝑠˜(𝑡)) is also strictly decreasing. To show𝑉(ℎ_𝑥(𝑡)) is also decreasing, we notice that𝑉(ℎ_𝑥(𝑡)) equals

∑︁

(𝑗 , 𝑘)∈E

˜

𝑣_𝑗(𝑡)ℓ˜_{𝑗 𝑘}(𝑡) − |𝑆˜_{𝑗 𝑘}(𝑡) |²

= ∑︁

(𝑗 , 𝑘)∈M^c

𝑣_𝑗ℓ_{𝑗 𝑘} − |𝑆_{𝑗 𝑘}|²+ ∑︁

(𝑗 , 𝑘)∈M

˜

𝑣_𝑗(𝑡)ℓ˜_{𝑗 𝑘}(𝑡) − |𝑆˜_{𝑗 𝑘}(𝑡) |²

= ∑︁

(𝑗 , 𝑘)∈M^c

𝑣_𝑗ℓ_{𝑗 𝑘} − |𝑆_{𝑗 𝑘}|²− ∑︁

(𝑗 , 𝑘)∈M

𝜙_{𝑗 𝑘}(𝑡Δ𝑗 𝑘).

As𝜙_{𝑗 𝑘}(𝑎)is strictly increasing for𝑎 ∈ [0,Δ𝑗 𝑘], we conclude that𝑉(ℎ_𝑥(𝑡))is strictly decreasing for𝑡 ∈ [0,1].

By Corollary 1, the set{ℎ_𝑥}_𝑥∈X\Xˆ is uniformly bounded and uniformly equicontin- uous as allℎ_𝑥(𝑡)are linear functions in𝑡. In summary, Condition (C) is satisfied.

Finally, we show Condition (C’) also holds. By assumption (ii), there exists some real𝑐 >0 independent of𝑥 such that for any 0≤ 𝑎 < 𝑏 ≤ 1,

𝑓(𝑠˜(𝑎)) − 𝑓(𝑠˜(𝑏))

≥ 𝑐

∑︁

𝑗∈V

R𝑒(𝑠˜_𝑗(𝑎) −𝑠˜_𝑗(𝑏)) +I𝑚(𝑠˜_𝑗(𝑎) −𝑠˜_𝑗(𝑏))

=𝑐∥𝑠˜(𝑎) −𝑠˜(𝑏) ∥_m. where ∥ · ∥_m is defined as ∥a∥_m :=Í

𝑖|R𝑒(𝑎_𝑖) | + |I𝑚(𝑎_𝑖) | over the complex vector space. It is easy to check∥ · ∥_mis a valid norm.

On the other hand, by (2.12) we have ∥𝑣˜(𝑎) −𝑣˜(𝑏) ∥_m ≡0 and

∥ℓ˜(𝑎) −ℓ˜(𝑏) ∥_m ≤ 1 min

(𝑗 , 𝑘)∈E

{∥𝑧_{𝑗 𝑘}∥_m}∥𝑠˜(𝑎) −𝑠˜(𝑏) ∥_m,

∥𝑆˜(𝑎) −𝑆˜(𝑏) ∥_m ≤ 1

2∥𝑠˜(𝑎) −𝑠˜(𝑏) ∥_m. Therefore,

∥ℎ_𝑥(𝑎) −ℎ_𝑥(𝑏) ∥_m ≤ 3

2+ 1

min

(𝑗 , 𝑘)∈E

{∥𝑧_{𝑗 𝑘}∥_m}

∥𝑠˜(𝑎) −𝑠˜(𝑏) ∥_m

and there exists ˆ𝑐 > 0 independent of𝑥 , 𝑎, 𝑏 such that

𝑓(𝑠˜(𝑎)) − 𝑓(𝑠˜(𝑏)) ≥𝑐ˆ∥ℎ_𝑥(𝑎) −ℎ_𝑥(𝑏) ∥_m.

Therefore Condition (C’) is also satisfied, and by Theorem 2, any local optimum of (2.9) is a global optimum.

The results in this subsection only apply to radial networks, which serve as the underlying network for balanced distribution power systems. For transmission systems and unbalanced distribution systems, networks are usually highly meshed.

It has been found that for most of meshed networks, both convex relaxation and local search algorithms can also yield the global optimum for most of testcases [34, 41].

Thus Theorem 3 suggests that there may also exist similar Lyapunov-like function and paths for meshed networks. Finding such Lyapunov-like function and paths would be an interesting future work to extend our results in this chapter.

Low Rank Semidefinite Program

This subsection proves a known result in [17] but using a different approach. Adopt- ing the same notations as [17], we have the following problem.

minimize

𝑋≥0 tr(𝐶 𝑋) (2.13a)

subject to tr(𝐴_𝑖𝑋) =𝑏_𝑖, 𝑖 =1,· · · , 𝑚 (2.13b)

rank(𝑋) ≤𝑟 . (2.13c)

Here, 𝐶, 𝐴_𝑖, 𝑋 are all 𝑛-by-𝑛 matrices. We assume the problem is feasible and {𝑋 ≥ 0|(2.13b)}is compact.

Theorem 6. If(𝑟+1) (𝑟+2)/2> 𝑚+1, then any local optimum of (2.13)is either a global optimum or a pseudo local optimum.

Before proving Theorem 6, we consider the convex relaxation of (2.13) as minimize

𝑋≥0 tr(𝐶 𝑋) (2.14a)

subject to tr(𝐴_𝑖𝑋) =𝑏_𝑖, 𝑖=1,· · · , 𝑚 . (2.14b) As a side note, the results in [5, 61] show that if(𝑟 +1) (𝑟 +2)/2> 𝑚, then (2.14) is weakly exact to (2.13). While our theorem is the same as in [17], some insights to find𝑉 and{ℎ_𝑋}are also from the structures first raised in [5, 61].

Proof. Clearly, (2.13) can be reformulated in the form of (2.1) by setting 𝑓(𝑋) = tr(𝐶 𝑋),X ={𝑋 ≥ 0|(2.13b),(2.13c)}and ˆX ={𝑋 ≥ 0|(2.13b)}. Define𝑉 as

𝑉(𝑋) :=

𝑛

∑︁

𝑖=𝑟+1

𝜆_𝑖(𝑋),

where𝜆_𝑖(𝑋)is the𝑖^theigenvalue of𝑋(in decreasing order). This function𝑉satisfies Definition 10 and is concave.

For fixed𝑋 ∈X \ Xˆ , we denote rank(𝑋)as𝑟₀ > 𝑟. We first construct𝑟₀−𝑟 paths labeled as ℎ₁, ℎ₂,· · · , ℎ_𝑟

0−𝑟. When we construct ℎ_𝑖, if𝑖 > 1 then we assume path ℎ_𝑖−1 has already been constructed and let 𝑋_𝑖−1 := ℎ_𝑖−1(1). We let 𝑋₀ = 𝑋. For 𝑖 ≥ 1, if rank(𝑋_𝑖−1) ≤ 𝑟₀−𝑖 then we let ℎ_𝑖(𝑡) ≡ 𝑋_𝑖−1 for 𝑡 ∈ [0,1]. Otherwise, we decompose 𝑋_𝑖−1as𝑈Σ𝑈^HwhereΣis a𝑘-by-𝑘positive definite diagonal matrix with𝑘 =rank(𝑋_𝑖−1) > 𝑟₀−𝑖. The linear system

tr(𝐶𝑈𝑌 𝑈^H) =0

tr(𝐴_𝑖𝑈𝑌 𝑈^H) =0, 𝑖=1,· · · , 𝑚 (2.15)

must have a non-zero solution for Hermitian matrix𝑌 ∈C^𝑘×𝑘. To see this, we have 𝑘 ≥𝑟₀−𝑖+1≥ 𝑟+1, and thus𝑘(𝑘+1)/2 ≥ (𝑟+1) (𝑟+2)/2> 𝑚+1. As a result, (2.15) has more unknown variables than equations. We simply denote this non-zero solution as𝑌 and for any𝛼∈R,𝛼𝑌 is also a solution to (2.15). The concavity of𝑉 also implies that𝑉(𝑈(Σ+𝛼𝑌)𝑈^H) is concave in𝛼when𝑈 andΣare fixed. Since Σ > 0, one of the following two scenarios must be true.

• ∃𝑎 < 0 such that𝑉(𝑈(Σ+𝛼𝑌)𝑈^H)is non-decreasing, rank(𝑈(Σ+𝛼𝑌)𝑈^H) ≤ 𝑘 for𝛼∈ [𝑎,0] and rank(𝑈(Σ+𝑎𝑌)𝑈^H) ≤ 𝑘−1.

• ∃𝑏 > 0 such that𝑉(𝑈(Σ+𝛼𝑌)𝑈^H)is non-increasing, rank(𝑈(Σ+𝛼𝑌)𝑈^H) ≤ 𝑘 for𝛼∈ [0, 𝑏] and rank(𝑈(Σ+𝑏𝑌)𝑈^H) ≤ 𝑘 −1.

Without loss of generality, we suppose𝑉(𝑈(Σ+𝛼𝑌)𝑈^H)is non-increasing for𝛼∈ [0, 𝑏](otherwise we take−𝑌instead). We then constructℎ_𝑖asℎ_𝑖(𝑡) =𝑈(Σ+𝑡 𝑏𝑌)𝑈^H for 𝑡 ∈ [0,1]. By construction, 𝑉(ℎ_𝑖(𝑡)) is non-increasing and 𝑓(ℎ_𝑖(𝑡)) stays a constant.

Finally, we construct ℎ_𝑋 as the concatenation of pathsℎ₁,· · · , ℎ_𝑟

0−𝑟. That is to say, ℎ_𝑋(𝑡) :=ℎ_𝑖( (𝑟₀−𝑟)𝑡−𝑖+1)for𝑡 ∈ h 𝑖−1

𝑟₀−𝑟 ,

𝑖 𝑟₀−𝑟

i .

It is easy to see ℎ_𝑋 is continuous and ℎ_𝑋(0) = ℎ₁(0) = 𝑋. To seeℎ_𝑋(1) ∈ X, we prove that rank(𝑋_𝑖) ≤𝑟₀−𝑖. We first have rank(𝑋₀) =rank(𝑋) =𝑟₀. For𝑖 ≥ 1, we have rank(𝑋_𝑖) =rank(𝑋_𝑖−1) if rank(𝑋_𝑖−1) ≤ 𝑟₀−𝑖 and rank(𝑋_𝑖) ≤ rank(𝑋_𝑖−1) −1 otherwise. By induction, we can prove rank(𝑋_𝑖) ≤𝑟₀−𝑖always holds. As a result, rank(ℎ_𝑋(1)) = rank(ℎ_𝑟

0−𝑟(1)) ≤ 𝑟 and thus ℎ_𝑋(1) ∈ X. By construction, ℎ_𝑖(𝑡) never violates (2.13b) and thus is in ˆX, so is ℎ_𝑋(𝑡) for all 𝑡. Functions 𝑉(ℎ_𝑖(𝑡)) and 𝑓(ℎ_𝑖(𝑡)) being non-increasing implies that 𝑉(ℎ_𝑋(𝑡)) and 𝑓(ℎ_𝑋(𝑡)) are also non-increasing. Therefore, (C3) is satisfied. By Corollary 1 (C2) also holds for {ℎ_𝑋}. It completes the proof (by Theorem 4).

Remark 4. In [17], Theorem 3.4 claims that any local optimum of (2.13) should also be globally optimal, unless it is harbored in some positive-dimensional face of SDP. The result in this chapter further asserts that if it is indeed harbored in such a face, then there must be some point on the edge of this face whose cost can be further reduced in its neighborhood (i.e., the local optimum is in the same situation as point𝑐rather than𝑑 as in Fig. 2.1).

Table 2.2: Sufficient and necessary conditions Condition Relaxation exactness Local optimality Sufficient conditions: ⇒

(C1), (C2) Strong exactness l.o. is p.l.o. or g.o.

(C3), (C2) Weak exactness l.o. is p.l.o. or g.o.

(C’) Strong exactness l.o. is g.o.

Necessary condition: ⇐

(C1), (C2) Strong exactness l.o. is g.o.