Problem Formulation

Consider a power system network with n nodes (buses). The admittance-to-ground at bus i, for 1 ≤ i ∈ _Z ≤ n, is yii and the admittance of the line between connected nodes i and

j (denoted by i ∼ j) is yij = gij −ibij. Typically, gij ≥ 0 and bij ≥ 0, i.e., the lines are

resistive and inductive. Define the corresponding n×n admittance matrixY as

Yij =                yii+ X j∼i yij, if i=j, −yij, if i6=j and i∼j, 0 otherwise.

Remark 5. Y is symmetric but not necessarily Hermitian.

The remaining circuit parameters and their relations are defined as follows.

• V and I are n-dimensional complex voltage and current injection vectors, whereVk, Ik

denote the nodal voltage and the injection current at bus 1≤k ≤n respectively. The voltage magnitude |Vk| is bounded as

0< W_k≤ |Vk|2 ≤Wk.

• Sk = Pk +iQk is the complex power injection at node 1 ≤ k ≤ n, where Pk and Qk,

respectively, denote the real and reactive power injections and

• PD

k and QDk are the real and reactive power demands at bus 1 ≤ k ≤ n, which are

assumed to be fixed and given.

• PG

k and QGk are the real and reactive power generation at bus 1 ≤ k ≤ n. They are

decision variables that satisfy the constraints PG_k ≤P_kG≤PG_k and QG

k ≤Q G k ≤Q

G k.

Power balance at each bus 1 ≤ k ≤ n requires P_kG = P_kD +Pk and QGk =Q D k +Qk, which leads us to define P_k := PG_k −P_kD, Pk := P G k −PkD Q k := Q G k −Q D k, Qk := Q G k −Q D k.

The power injections at each bus 1≤k ≤n are then bounded as P_k≤Pk≤Pk, Q_k ≤Qk ≤Qk.

The branch power flows and their limits are defined as follows.

• Sij = Pij +iQij is the sending-end complex power flow from node i to node j, where

Pij and Qij are the real and reactive power flows respectively. The real power flows

are constrained as |Pij| ≤ Fij where Fij is the line-flow limit between nodes i and j

and Fij =Fji.

• Lij =Pij+Pji is the power loss over the line between nodesiandj, satisfyingLij ≤Lij

where Lij is the thermal line limit and Lij =Lji. Also, observe that since Lij ≥0, we

have|Pij| ≤Fij,|Pji| ≤Fji if and only if Pij ≤Fij, Pji≤Fji.

For 1 ≤ k ≤ n, let Jk = ekeHk where ek is the k-th canonical basis vector in Cn. Define

Yk :=ekeHkY. Substituting these expressions into (3.7) yields

Sk = eHkV I H_e k = tr V VH(YHekeHk) =VHY_kHV = VH YH k +Yk 2 | {z } =:Φ V +i VH YH k −Yk 2i | {z } =:Ψ V,

where Φk and Ψk are Hermitian matrices. Thus, the two quantities VHΦkV and VHΨkV

are real numbers and

Pk =VHΦkV, Qk =VHΨkV.

The real power flow fromi to j can be expressed as a quadratic form as follows. Pij = Re {Vi(Vi−Vj)HyijH}=V

H_Mij_V,

where Mij is an n×n Hermitian matrix.

The thermal loss of the line connecting buses i and j is Lij =Lji =Pij +Pji =VHTijV

whereTij _=Tji _:=Mij_+Mji _{0. The entries of the matrices Ψ}

k, Φk, 1≤k ≤n,Mij,Tij,

i∼j are described in detail in the appendix.

We can now write the OPF problem. Given a Hermitian n×n matrix C0, we have

Optimal power flow problemOP F: minimize

V∈Cn

VHC0V

subject to: P_k ≤VHΦkV ≤Pk, 1≤k ≤n, (3.8a)

Q k ≤V H_Ψ kV ≤Qk, 1≤k ≤n, (3.8b) W_k≤VHJkV ≤Wk, 1≤k≤n, (3.8c) VHMijV ≤Fij, i∼j, (3.8d) VHTijV ≤Lij, i∼j, (3.8e)

where (3.8a)–(3.8e) are, respectively, constraints on the real and reactive powers, the voltage magnitudes, the line flows and thermal losses.

power qP2

ij +Q2ij on each branch i ∼ j because such constraints are not quadratic in the

voltages and hence beyond the scope of our model.

Remark 6 (Objective Functions). We can consider different optimality criteria by changing

C0 as follows:

(i) Voltages: C0 =In×n (identity matrix) where we aim to minimize kVk2 =P_k|Vk|2.

(ii) Power loss: C0 = (Y +YH)/2where we aim to minimize

P

igii|Vi|2+

P

i<jgij|Vi−Vj|2.

(iii) Production costs: C0 =P_kckΦk where we aim to minimize P_kckPkG, ck ≥0.