Measurement Conversion

Introduction

Any network formulation has a specific variable space, e.g., ACP includes vm, va, px and qx^[1]. w = vm^2 is the lifted voltage variable native to branch flow conic and linear forms. The conversions for the reduced formulations work identically as their non-reduced equivalent.

-	vm	va	cmx	cax	crx	cix	px	qx	vr	vi	w
ACP	N	N	SF	X	F	F	N	N	X	X	X
ACR	S	PP	SF	X	MF	MF	N	N	N	N	X
IVR	S	PP	S	PP	N	N	M	M	N	N	X
SDP	X	X	X	X	X	X	N*	N*	X	X	N
LD3F	S	X	SF	X	X	X	N	N	X	X	N

where:

F: conversion of type Fraction
M: conversion of type Multiplication
MF: conversion of type MultiplicationFraction
N: native to the network formulation
PP: conversion of type Tangent
S: conversion of type Square
SF: conversion of type SquareFraction
X: not provided

The N* in the SDP formulation indicates that those variable are only available for generators, loads and other devices/extensions, but not for measurements that refer to branch flows, yet.

Conversions

Certain measurement variables may not be natively supported in the formulation space. Consequently, it becomes necessary to convert them into that specific space. This is accomplished through the inclusion of an additional constraint(s). The different types of conversion constraints are enumerated in what follows.

Tangent

The conversion type Tangent allows to include va measurements in the ACR and IVR formulation, and cax measurements in the IVR formulation, respectively through:

\[\begin{eqnarray} \tan(\text{va}) &= \frac{\text{vi}}{\text{vr}} \\ \tan(\text{cax}) &= \frac{\text{cix}}{\text{crx}} \end{eqnarray}\]

These are non-linear equality constraints, modeled using @NLconstraint.

Fraction

The conversion type Fraction allows to include crx and cix measurements in the ACP formulation, respectively through:

\[\begin{eqnarray} \text{crx} &= \frac{\text{px}\cdot\cos(\text{va})+\text{qx}\cdot\sin(\text{va})}{\text{vm}} \\ \text{cix} &= \frac{\text{px}\cdot\sin(\text{va})-\text{qx}\cdot\cos(\text{va})}{\text{vm}} \end{eqnarray}\]

These are non-linear equality constraints, modeled using @NLconstraint.

Multiplication

The conversion type Multiplication allows to include px and qx measurements in the IVR formulation, respectively through:

\[\begin{eqnarray} \text{px} &= \text{vr}\cdot\text{crx} + \text{vi}\cdot\text{cix} \\ \text{qx} &= \text{vi}\cdot\text{crx} - \text{vr}\cdot\text{cix} \end{eqnarray}\]

These are quadratic equality constraints, modeled using @constraint.

MultiplicationFraction

The conversion type MultiplicationFraction allows to include crx and cix measurements in the ACR formulation, respectively through:

\[\begin{eqnarray} \text{crx} &= \frac{\text{px}\cdot\text{vr}+\text{qx}\cdot\text{vi}}{\text{vr}^{2}+\text{vi}^{2}} \\ \text{cix} &= \frac{\text{px}\cdot\text{vi}-\text{qx}\cdot\text{vr}}{\text{vr}^{2}+\text{vi}^{2}} \\ \end{eqnarray}\]

These are non-linear equality constraints, modeled using @NLconstraint.

SquareFraction

The conversion type SquareFraction allows to include cmx measurements in the ACP and ACR formulation, through:

\[\begin{equation} \text{cmx}^{2} = \frac{\text{px}^{2} + \text{qx}^{2}}{\text{vm}^{2}} \end{equation}\]

If the conversion is applied to the LinDist3Flow formulation, then vm^2 is replaced by w. These are non-linear equality constraints, modeled using @NLconstraint.

Square

The conversion type Square allows to include vm measurements in the ACR and IVR formulation, and cmx measurements in the IVR formulation, respectively through:

\[\begin{eqnarray} \text{vm}^{2} &= \text{vi}^{2} + \text{vr}^{2} \\ \text{cmx}^{2} &= \text{cix}^{2} + \text{crx}^{2} \end{eqnarray}\]

These are quadratic equality constraints, modeled using @constraint.

No conversion provided

As displayed in the Table, some conversions are not provided. This is because the measured quantities are either unlikely to take place in practice, e.g., w, or tend to appear in pairs, e.g., cmx and cax with PMUs. In the latter case, it is more efficient to transform cax and cmx into rectangular variables a priori and then use them, for instance, with IVR.

1The x in px, qx, cmx, cax, crx and cix indicates that these variables exists for branches (~), generators (g) and loads (-). In order to capture the variable for a specific element it should be rewritten, e.g., "px" respectively becomes "p", "pg" and "pd".