Type Hierarchy

The original type hierarchy of PowerModels is used.

For details on GenericPowerModel, see PM.jl [documentation](https://lanl-ansi.github.io/PM.jl/stable/).

Formulations overview

Extending PowerModels, formulations for balanced OPF in DC grids have been implemented and mapped to the following AC grid formulations:

• ACPPowerModel
• ACRPowerModel
• DCPPowerModel
• LPACPowerModel
• SOCWRPowerModel
• SDPWRMPowerModel
• QCWRPowerModel
• QCWRTriPowerModel

Note that from the perspective of OPF convex relaxation for DC grids, applying the same assumptions as the AC equivalent, the same formulation (and variable space) is obtained for - SOCWRPowerModel, SDPWRMPowerModel, QCWRPowerModel and QCWRTriPowerModel. These are referred to as formulations in the AC WR(M) variable space.

Formulation details

The formulations are categorized as Bus Injection Model (BIM) or Branch Flow Model (BFM).

• Applied to DC grids, the BIM uses series conductance notation, and adds separate equations for the to and from line flow.
• Conversely, BFM uses series resistance parameters, and adds only a single equation per line, representing $P_{lij} + P_{lji} = P_{l}^{loss}$.

Note that in a DC grid, under the static power flow assumption, power is purely active, impedance reduces to resistance, and voltages and currents are defined by magnitude and direction.

Parameters used:

• \[g^{series}=\frac{1}{r^{series}}\]

  , dc line series impedance

• \[p \in \{1,2\}\]

  for single ($1$) or bipole ($2$) DC lines

• \[U_i^{max}\]

  maximum AC node voltage

• \[a\]

  constant power converter loss

• \[b\]

  converter loss proportional to current magnitude

• \[c\]

  converter loss proportional to square of current magnitude

Note that generally, $a \geq 0, b \geq 0, c \geq 0$ as physical losses are positive.

ACPPowerModel (BIM)

DC lines

• Active power flow from side: $P^{dc}_{ij}$ = $p \cdot g^{series}_{ij} \cdot U^{dc}_i \cdot (U^{dc}_i - U^{dc}_j)$.
• Active power flow to side: $P^{dc}_{ji}$ = $p \cdot g^{series}_{ij} \cdot U^{dc}_j \cdot (U^{dc}_j - U^{dc}_i)$.

ACDC converters

• Power balance: $P^{conv, ac}_{ij} + P^{conv, dc}_{ji}$ = $a + b \cdot I^{conv, ac} + c \cdot (I^{conv, ac})^2$.
• Converter current variable model: $(P^{conv,ac}_{ij})^2$ + $(Q^{conv,ac}_{ij})^2$ = $U_i^2 \cdot (I^{conv, ac})^2$.
• LCC converters, active /reactive power:

\[P^{conv, ac} = \cos\varphi_{c} \cdot S^{conv,ac,rated}\]

\[Q^{conv, ac} = \sin\varphi_{c} \cdot S^{conv,ac,rated}\]

ACRPowerModel (BIM)

DC lines

• Same model as ACP formulation

ACDC converters

Two separate current variables, $I^{conv, ac}$ and $i^{conv, ac, sq}$ are defined, the nonconvex relation $i^{conv, ac, sq} = (I^{conv, ac})^2$ is convexified.

• Linking both current variables: $(I^{conv, ac})^2$ $=$ $i^{conv, ac, sq}$
• Power balance: $P^{conv, ac}_{ij} + P^{conv, dc}_{ji}$ = $a + b\cdot I^{conv, ac} + c\cdot i^{conv, ac, sq}$.
• Converter current variable model: $(P^{conv,ac}_{ij})^2$ + $(Q^{conv,ac}_{ij})^2$ = $(U_{ri}^2+U_{ii}^2) \cdot i^{conv, ac, sq}$.
• LCC converters, active /reactive power: Same model as ACP formulation

DCPPowerModel (NF)

Due to the absence of voltage angles in DC grids, the DC power flow model reduces to network flow (NF) under the 'DC' assumptions

DC lines

• Network flow model: $P^{dc}_{ij}$ + $P^{dc}_{ji}$ = $0$

ACDC converters

Under the same assumptions as MATPOWER ($U_i \approx 1$), $P^{conv, ac}_{ij} \approx I^{conv, ac}$ allowing the converter model to be formulated as:

• Network flow model: $P^{conv, ac}_{ij}$ + $P^{conv, dc}_{ji}$ = $a + b P^{conv, ac}_{ij}$
• LCC converters, n.a.

AC WR(M) variable space. (BFM)

For the SDP formulation, the norm syntax is used to represent the SOC expressions below.

DC lines

The variable $u^{dc}_{ii}$ represents $(U^{dc}_{i})^2$ and $i^{dc}_{ij}$ represents $(I^{dc}_{ij})^2$.

• Active power flow from side: $P^{dc}_{ij} + P^{dc}_{ji}$ = $p \cdot r^{series} \cdot i^{dc}_{ij}$.
• Convex relaxation of power definition: $(P^{dc}_{ij})^2 \leq u^{dc}_{ii} \cdot i^{dc}_{ij}$.
• Lifted KVL: $u^{dc}_{jj} = u^{dc}_{ii} -2 p \cdot r^{series} P^{dc}_{ij} + (r^{series})^2 i^{dc}_{ij}$

ACDC converters

Two separate current variables, $I^{conv, ac}$ and $i^{conv, ac, sq}$ are defined, the nonconvex relation $i^{conv, ac, sq} = (I^{conv, ac})^2$ is convexified, using $U_i \leq U_i^{max}$:

• Power balance: $P^{conv, ac}_{ij} + P^{conv, dc}_{ji}$ = $a + b\cdot I^{conv, ac} + c\cdot i^{conv, ac, sq}$.
• Squared current: $(P^{conv, ac}_{ij})^2 + (Q^{conv, ac}_{ij})^2 \leq u_{ii} \cdot i^{conv, ac, sq}$
• Linear current: $(P^{conv, ac}_{ij})^2 + (Q^{conv, ac}_{ij})^2 \leq (U_i^{max})^2 \cdot (I^{conv, ac})^2$
• Linking both current variables: $(I^{conv, ac})^2$ $\leq$ $i^{conv, ac, sq}$
• LCC converters:

\[Q^{conv,ac} \geq Q^{1}_{c} + (P^{conv,ac} - P^{1}_{c})\frac{(Q^{2}_{c} - Q^{1}_{c})}{(P^{2}_{c} - P^{1}_{c})}\]

\[P^{1}_{c} = \cos \varphi_{c}^{\text{min}} \cdot S^{conv,ac,rated}\]

\[P^{2}_{c} = \cos \varphi_{c}^{\text{max}} \cdot S^{conv,ac,rated}\]

\[Q^{1}_{c} = \sin \varphi_{c}^{\text{min}} \cdot S^{conv,ac,rated}\]

\[Q^{2}_{c} = \sin \varphi_{c}^{\text{max}} \cdot S^{conv,ac,rated}\]

AC WR(M) variable space. (BIM)

For the SDP formulation, the norm syntax is used to represent the SOCs.

DC lines

The variable $u^{dc}_{ii}$ represents $(U^{dc}_{i})^2$ and $u^{dc}_{ij}$ represents $U^{dc}_{i}\cdot U^{dc}_{j}$.

• Active power flow from side: $P^{dc}_{ij}$ = $p \cdot g^{series} \cdot (u^{dc}_{ii} - u^{dc}_{ij})$.
• Active power flow to side: $P^{dc}_{ji}$ = $p \cdot g^{series} \cdot (u^{dc}_{jj} - u^{dc}_{ij})$.
• Convex relaxation of voltage products: $(u^{dc}_{ij})^2 \leq u^{dc}_{ii} \cdot u^{dc}_{jj}$.

ACDC converters

An ACDC converter model in BIM is not derived.