# On possible sums from multiset of powers of $d$

Yizhou Guo guoyiz<EMAIL_ADDRESS>Saint-Petersburg State
University, Department of Mathematics and Computer Science

###### Abstract

Let $A$ be a finite multiset of powers of a positive integer $d>1$. We
describe the structure of the set $\mathrm{span}(A)$ of all sums of
submultisets of $A$, and in particular give a criterion of
$\mathrm{span}(A)=\mathrm{span}(B)$ for two multisets $A,B$.

##  1 Introduction

Let $A$ be a finite multiset wherein every element of $A$ is a positive
integer. Denote by $\mathrm{sum}(A)$ the sum of all elements of $A$
(respecting multiplicity), and by $\mathrm{span(A)}$ the set (not a multiset)
of all sums $\mathrm{sum}(C)$, where $C$ is a submultiset of $A$.

This setting describes a finite collection of tokens of positive integer
costs, and $\mathrm{span(A)}$ means the set of values which may be obtained
using this collection of tokens.

In the general case describing the structure of $\mathrm{span(A)}$ seems
difficult. Herefater we fix an integer $d>1$ and consider the case when each
element of $A$ is in the form $d^{k}$, for some integer $k\geqslant 0$.

A more convenient notation would be $A:=(a_{0},a_{1},\ldots,a_{k})$, where
$a_{i}\in\mathbb{Z}_{\geq 0}$ and $a_{k}\neq 0$, where $a_{i}$ is the
multiplicity of $d^{i}$ in $A$. We will from here on call this a
$d$-collection and let $\mathcal{A}_{d}$ denote the family of $d$-collections.
Let $\mathrm{comb}(A)$ denote
$\\{(c_{0},c_{1},\ldots,c_{k})\in\mathbb{Z}^{k+1}:0\leq c_{i}\leq a_{i}\\}$.

Then
$\mathrm{span(A)}=\\{\sum_{i=0}^{k}c_{i}d^{i}:(c_{0},c_{1},\ldots,c_{k})\in\mathrm{comb}(A)\\}$.

Here, we will describe and prove the correctness of an algorithm to decide for
arbitrary $A,B\in\mathcal{A}_{d}$ whether or not
$\mathrm{span(A)}=\mathrm{span(B)}$, as well as algorithm to decide for
arbitrary $n\in\mathbb{Z}_{\geq 0}$ if $n\in\mathrm{span}(A)$.

An elementary exchange of $(a_{0},a_{1},\ldots,a_{k})$ picks some $i$ such
that $a_{i}\geq d$, sets $a_{i}\leftarrow a_{i}-d$ and $a_{i+1}\leftarrow
a_{i+1}+1$. In other words, $d$ tokens of cost $d^{i}$ are replaced with one
token of cost $d^{i+1}$. We let $e_{i}$ denote the elementary exchange wherein
the $i$-th place is exchanged.

For
$A=(a_{0},a_{1},\ldots,a_{k}),B=(b_{0},b_{1},\ldots,b_{k})\in\mathcal{A}_{d}$,
we say that $A\preceq B$ if and only if $a_{i}\leq b_{i}$ for all $i$.

######  Proposition 1.1.

If $A\preceq B$, then $\mathrm{span}(A)\subseteq\mathrm{span}(B)$.

###### Proof.

That $A\preceq B$ implies $\mathrm{comb}(A)\subseteq\mathrm{comb}(B)$, from
which the result follows. ∎

######  Proposition 1.2.

$\mathrm{span}(e_{i}(A))\subseteq\mathrm{span}(A)$

###### Proof.

Instead of using the new token of cost $d^{i+1}$ you could use $d$ tokens of
cost $d^{i}$ which it was changed for. ∎

##  2 Condition for span invariance with respect to elementary exchanges

######  Proposition 2.1.

For $A=(a_{0},a_{1},\ldots,a_{k})$, if $a_{i}>2(d-1)$, then
$\mathrm{span}(A)=\mathrm{span}(e_{i}(A))$.

###### Proof.

It suffices to check that $\mathrm{span}(A)\subset\mathrm{span}(e_{i}(A))$,
since the opposite inclusion is proved in Proposition 1.2. Consider an
arbitrary submultiset $B\subset A$ and prove that
$\mathrm{sum}(B)\in\mathrm{span}(e_{i}(A))$. If $B$ contains at most $d-1$
tokens $d^{i}$, it is also a submultiset of $e_{i}(A)$, since $e_{i}(A)$
contains more than $2(d-1)-d=d-2$ tokens $d^{i}$. If $B$ contains at least $d$
tokens $d^{i}$, replace $d$ such tokens by a token $d^{i+1}$ and get a
submultiset $e_{i}(B)\subset e_{i}(A)$ with
$\mathrm{sum}(e_{i}(B))=\mathrm{sum}(B)$. Thus in both cases
$\sum(B)\in\mathrm{span}(e_{i}(A))$, as needed. ∎

From now on, we will call an elementary exchange at an $i$ where the
hypothesis of the above proposition is satisfied to be a proper elementary
exchange.

Algorithm 1 Proper elementary exchange sequence

procedure RunProperElementaryExchanges($A$)

for all $i,A[i]=a_{i}\cdot d^{i},0\leq a_{i}<d$

for all $i,A[i]\leq 2(d-1)$

$i\leftarrow 0$

while exists untraversed non-zero element of $A$ do

while $A[i]>2(d-1)$ do

$A[i]\leftarrow A[i]-d$

$A[i+1]\leftarrow A[i+1]+1$

$i\leftarrow i+1$ return A

######  Corollary 2.1.

Let $B$ be the $d$-collection returned by running Algorithm 1 on arbitrary
$d$-collection $A$. Then, $\mathrm{span}(A)=\mathrm{span}(B)$.

###### Proof.

Let $A,A_{1},A_{2},\ldots,A_{m},B$ be the sequence of $d$-collections obtained
from running the algorithm on $A$. This then follows from Proposition 2.1
applied to adjacent elements of this sequence along with transitivity of set
equality. ∎

The converse of Proposition 2.1 does not hold. A simple counterexample would
be for $d=2$, $A=(3,2,0)$ and an elementary exchange that results in
$A^{\prime}=(3,0,1)$ in which case, $2(d-1)=2\geq d-1$, but all $7$ values can
be assumed by both $A$ and $A^{\prime}$.

We say that $d$-collection $A$ is normal if and only if $a_{i}\leq 2(d-1)$ for
all $i$ and that $d$-collection $A$ is $j$-normal if and only if $d-1\leq
a_{i}\leq 2(d-1)$ for all $i<j$ and $a_{j}<d-1$. We also define the
normalization and $j$-normalization of a $d$-collection $A$ in the same manner
per Algorithm 1 (with the latter running a modified version that does not
exchange at any place $\geq j$), which we denote with $\mathrm{norm}(A)$ and
$\mathrm{norm}_{j}(A)$ respectively. $[\mathrm{norm}(A)]_{i}$ will denote the
value at its $i$th place (such subscripting can be applied to any expression
evaluating to a $d$-collection type).

######  Proposition 2.2.

If $\mathrm{norm}(A)=\mathrm{norm}(B)$, then
$\mathrm{span}(A)=\mathrm{span}(B)$.

###### Proof.

Follows from Proposition 2.1 along with transitivity of set equivalence. ∎

######  Lemma 2.1.

If a $d$-collection $A=(a_{0},a_{1},\ldots,a_{k})$ is such that $a_{i}\geq
d-1$ for all $i<k$, then $\mathrm{span}(A)=\\{0,1,\ldots,\mathrm{sum}(A)\\}$.

###### Proof.

We prove this by induction by $\mathrm{sum}(A)$. The base case follows
trivially. The inductive hypothesis applied to $A\setminus\\{d^{k}\\}$ implies
that all elements up to $M:=\mathrm{sum}(A)-d^{k}$ belong to
$\mathrm{span}(A)$. Thus so are all elements from $\mathrm{sum}(A)-M$ to
$\mathrm{sum}(A)$, since the set $\mathrm{span}(A)$ enjoys a symmetry
$x\to\mathrm{sum}(A)-x$. It remains to note that $\mathrm{sum}(A)\geq
d^{k}+(d-1)(1+d+\ldots+d^{k-1})=2d^{k}-1$, thus
$\mathrm{sum}(A)-M=d^{k}\leq\mathrm{sum}(A)-d^{k}+1=M+1$. ∎

######  Proposition 2.3.

If a $d$-collection $A=(a_{0},a_{1},\ldots,a_{k})$ is $j$-normal, then the
smallest value not in $\mathrm{span}(A)$ is $X:=1+\sum_{i=0}^{j}a_{i}d^{i}$,
which is necessarily $<d^{j+1}$.

###### Proof.

That $A$ is $j$-normal implies that

$\displaystyle X-1=a_{j}d^{j}+\sum_{i=0}^{j-1}a_{i}d^{i}$ $\displaystyle\leq$
$\displaystyle 2(d^{j}-1)+a_{j}d^{j}$ $\displaystyle\leq$ $\displaystyle
2(d^{j}-1)+(d-2)d^{j}=d^{j+1}-2.$

By Lemma 2.1, any $n\leq\sum_{i=0}^{j}a_{i}d^{i}=X-1$ is in
$\mathrm{span}(A)$. It remains to prove that $X\notin\mathrm{span}(A)$. Since
$X<d^{j+1}$, we can use only tokens not exceeding $d^{j}$ for collecting $X$,
but the sum of such tokens in $A$ equals $X-1$, thus indeed
$X\notin\mathrm{span}(A)$. ∎

##  3 Decomposition of normal $d$-collections

We now describe an algorithm for decomposing an arbitrary normal
$d$-collection $A$ to submultisets which partition $A$. Call an index $j\leq
k$ critical if either $j=k$ or $a_{j}<d-1$. Let $j_{1}<j_{2}<\ldots<j_{s}=k$
be all critical indices. Define the $d$-collections $A_{1},\ldots,A_{s}$ as
follows: $A_{i}$ consists of those tokens $d^{t}$ from $A$, for which
$j_{i-1}<t\leq j_{i}$ (for $i=1$, we put $j_{0}=-1$ by agreement). If $s=1$,
we say that $A$ is an _irreducible_ $d$-collection.

######  Theorem 3.1.

In above notations,

$\mathrm{span}(A)=\oplus_{i=1}^{s}d^{j_{i-1}+1}\cdot\\{0,1,\ldots,d^{-1-j_{i-1}}\mathrm{sum}(A_{j})\\},$

where the $\oplus$ sign means the direct sum: any element in
$\mathrm{span}(A)$ is uniquely represented as $x_{1}+\ldots+x_{s}$ for
$x_{i}\in
d^{j_{i-1}+1}\cdot\\{0,1,\ldots,d^{-1-j_{i-1}}\mathrm{sum}(A_{j})\\}$.

###### Proof.

It follows from Lemma 2.1 and scaling that

$\mathrm{span}(A_{i})=d^{j_{i-1}+1}\cdot\\{0,1,\ldots,d^{-1-j_{i-1}}\mathrm{sum}(A_{j})\\}.$

Also, by 2.3 we get $\mathrm{sum}(A_{i})\leq d^{j_{i}+1}-d^{j_{i-1}+1}$ for
all $i<s$. Since any element $x\in\mathrm{span}(A)$ may be represented as
$x=\sum x_{i}$ for $x_{i}\in\mathrm{span}(A_{i})$, it remains to prove the
direct sum condition. Assume that on the contrary that $x=\sum x_{i}=\sum
y_{i}$ for $x_{i},y_{i}\in\mathrm{span}(A_{i})$ but there exists $i$ for which
$x_{i}\neq y_{i}$. Choose minimal such $i$, obviously we have $i<s$. Then
$x_{i}-y_{i}=(y_{i+1}+\ldots+y_{s})-(x_{i+1}+\ldots+x_{s})$ is divisible by
$d^{j_{i}+1}$, but both $x_{i}$, $y_{i}$ belong to
$\mathrm{sum}(A_{i})\subset\\{0,1,\ldots,d^{j_{i}+1}-1\\}$, a contradiction. ∎

######  Proposition 3.1.

If two irreducible $d$-collections $A,B$ are distinct, then
$\mathrm{sum}(A)\neq\mathrm{sum}(B)$.

###### Proof.

Let $A=(a_{0},\ldots,a_{k})$, $B=(b_{0},\ldots,b_{l})$. Assume that $k\neq l$,
say, $l>k$. Then
$\mathrm{sum}(B)\geqslant(d-1)(1+d+\ldots+d^{l-1})+d^{l}=2d^{l}-1$ while
$\mathrm{sum}(A)\leq 2(d-1)(1+d+\ldots+d^{k})=2(d^{k+1}-1)\leq
2d^{l}-2<\mathrm{sum}(B)$. If $l=k$, then choose the minimal index $i$ for
which $a_{i}\neq b_{i}$. If $i=k$, then $\mathrm{sum}(A)\neq\mathrm{sum}(B)$
is clear. If $i<k$, then both $a_{i},b_{i}$ belong to $\\{d-1,\ldots,2d-2\\}$,
so $a_{i}$ and $b_{i}$ differ modulo $d$, thus $\mathrm{sum}(A)$ and
$\mathrm{sum}(B)$ differ modulo $d^{i+1}$. ∎

######  Proposition 3.2.

If $A$ and $B$ are two normal $d$-collections, and
$\mathrm{span}(A)=\mathrm{span}(B)$, then $A=B$.

###### Proof.

Let $A=A_{1}\sqcup A_{2}\sqcup\ldots\sqcup A_{s}$ be a decomposition of $A$,
$B=B_{1}\sqcup B_{2}\ldots\sqcup B_{t}$ that of $B$. The minimal non-element
of $\mathrm{span}(A)$ equals $\sum(A_{1})+1$ by Proposition 2.3 (if $s=1$,
apply Lemma 2.1 instead). Analogously for $B$, so we get
$\mathrm{sum}(A_{1})=\mathrm{sum}(B_{1})$ and thus $A_{1}=B_{1}$ by
Proposition 3.1. Now if $t$ or $s$ equals 1 we get from
$\mathrm{sum}(A)=\mathrm{sum}(B)$ that $s=t=1$ and $A=B=A_{1}$. If both $t,s$
are greater than 1, denote by $d^{j}$ the maximal element of $A_{1}$, note
that $\mathrm{sum}(A_{1})<d^{j+1}$ again by Proposition 2.3. Therefore, if we
consider the map $\psi:x\to\lfloor x/d^{j+1}\rfloor$ on the set
$\mathrm{span}(A)=\mathrm{span}(A_{1})+\mathrm{span}(A_{2})+\ldots+\mathrm{span}(A_{s})$,
its value does not depend on the component in $\mathrm{span}(A_{1})$, and we
get
$\psi(\mathrm{span}(A))=d^{-j-1}(\mathrm{span}(A_{2})+\ldots+\mathrm{span}(A_{s}))$.
Use the same for $B$ to get
$\mathrm{span}(A_{2})+\ldots+\mathrm{span}(A_{s})=\mathrm{span}(B_{2})+\ldots+\mathrm{span}(B_{t})$
and proceed by induction to conclude that $A_{2}=B_{2}$ and so on. ∎

######  Theorem 3.2.

$\mathrm{span}(A)=\mathrm{span}(B)$ if and only if
$\mathrm{norm}(A)=\mathrm{norm}(B)$

###### Proof.

Proposition 2.2 and Proposition 3.2 are the two directions. ∎

######  Corollary 3.1.

Start with arbitrary $d$-collection $A$ and apply elementary exchanges $e_{i}$
if $a_{i}\geqslant 2d-1$, in arbitrary order. Then this process terminates
after finitely many steps, and the resulted normalized $d$-collection does not
depend on the order.

###### Proof.

Note that we may get only finitely many different $d$-collections, since the
sum does not change. And there is a semiinvariant, for example, the sum of
squares of elements, which increases after every operation. Thus the process
terminates. By Proposition 2.1, these operations do not change the span. Since
normalized $d$-collection is uniquely determined by its span, the independence
of the order follows. ∎

We conclude with the criterion used to determine if an elementary exchange
breaks the desired span invariant.

######  Proposition 3.3.

$\mathrm{span}(e_{i}(A))=\mathrm{span}(A)$ if and only if
$[\mathrm{norm}_{i}(A)]_{i}>2(d-1)$

###### Proof.

$\Leftarrow$: Suppose $[\mathrm{norm}_{i}(A)]_{i}\leq 2(d-1)$, in which case a
run of Algorithm 1 would not perform any exchanges at $i$, which means that
$[\mathrm{norm}(A)]_{i}=A_{i}=[\mathrm{norm}_{i}(A)]_{i}>[\mathrm{norm}(e_{i}(A))]_{i}$,
from which follows by one direction of Proposition 3.2 that the spans differ.

$\Rightarrow$: If $[\mathrm{norm}_{i}(A)]_{i}>2(d-1)$, then Algorithm 1 which
returns $\mathrm{norm}(A)$ necessarily performs at least one elementary
exchange at $i$. Upon $e_{i}(A)$, we’ve performed the first elementary
exchange there doing so on the places less significant than $i$. Thus, once
that algorithm run on $\mathrm{e_{i}(A)}$ reaches $i$, the state of the
$d$-collection will be same as the state of $d$-collection right after the
first elementary exchange on $i$ when the algorithm is run on $\mathrm{A}$.
Thus, $\mathrm{norm}(e_{i}(A))=\mathrm{norm}(A)$ in this case, from which
follows by the other direction of Proposition 3.2 that the spans in this case
are the same. ∎

##  4 Acknowledgements

I would like to thank Professor Fedor Petrov of Saint-Petersburg State
University in the Department of Mathematics and Computer Science for
previewing this paper and further verifying the correctness of its proofs.
Moreover, it occurred to me to research this problem after I read parts of
[AE04] on the closely related topic of integer partitions.

## References

  * [AE04] George E. Andrews and Kimmo Eriksson.  Integer Partitions.  Cambridge University Press, 2004.


EEA Conference & Exhibition 2021,
30 June - 1 July, Wellington, New Zealand

Monitoring and Detection of Low-current High-Impedance Faults in Distribution
Networks

Anwarul Islam Sifat1, Fiona J. Stevens McFadden1, Ramesh Rayudu2, Joseph
Bailey1

1 Robinson Research Institute, Victoria University of Wellington

2 School of Engineering and Computer Science, Victoria University of
Wellington

Presenter:

Anwarul Islam Sifat

Fiona J. Stevens McFadden

Robinson Research Institute, Victoria University of Wellington

## Abstract

Faults in electricity distribution networks have the potential to ignite
fires, cause electrocution, and/or damage the system itself. High-current Low-
Impedance Faults (LIF) are typically detected and mitigated via over-current,
distance, directional relays, fuses, etc. In contrast, while High Impedance
Faults (HIF) are equally hazardous, they are much more challenging to detect
due to the fault current being much lower than load currents and their time-
varying and nonlinear behaviour. A solution for HIF detection will require,
firstly, the continuous observation of system current waveforms and, secondly,
the development of sophisticated pattern recognition algorithms to apply to
these measurements. However, New Zealand distribution networks are extensive
and largely unmonitored beyond the substation, and suitable HIF detection
schemes are still an ongoing research challenge.

Robinson Research Institute has been developing a solution to enable
widespread monitoring of overhead lines in distribution networks and detecting
faults. The sensing system we have developed and are evaluating utilizes
magnetic field sensors (specifically the Giant Magneto Resistive type), pole-
mounted at a distance from the overhead lines for ease of installation. These
sensors are an attractive option compared to traditional current transformers
(CTs) due to their non-contact sensing ability, wide frequency bandwidth, low
cost, miniature size, and ease of integration with required digital
componentry. Coupled with the algorithms we have developed, these sensors
allow observation of the individual overhead line currents and detect both Low
Impedance and High Impedance faults.

To date, we have built a physical test facility for power system fault
analysis and the development and evaluation of our sensing and fault detection
system. We have simulated LIF and HIF with different fault surface materials
and load-switching events. From the data collected, we have characterized the
unique fault behaviour for both LIF and HIF in 400V networks and trained a
Deep Learning classifier to recognize the type of fault present from its
unique signature. We have developed an outdoor pole-mountable sensing system
and have installed this in Wellington Electricity’s network for ongoing data
collection and evaluation.

This paper will describe the test facility and our experience developing and
implementing the sensing system. The widest range of HIF phenomena observed
was in the fault experiments involving the tree branch. For brevity,
therefore, this paper reports on the results of just these tree-branch
experiments. HIF faults on other surface materials will be reported elsewhere.
Finally, we will detail the pole-mountable sensing system installed in
Wellington Electricity’s network and the outcomes thus far.

##  I Introduction

Detection of power system faults is an ongoing challenge in the power
distribution industry. Faults overstress the existing infrastructure and push
operation close to its stability limit. In addition, faults can endanger human
life [1]. Broadly, faults can be classified as low (LIF) or high impedance
faults (HIF). Due to the resulting high current, LIFs are easily detectable by
conventional protection schemes, e.g. circuit breakers (CB) and protective
relays [2]. HIFs are when the impedance of the fault path is high and
stochastic in nature. These generate a much lower fault current that cannot be
easily recognized or reliably detected by conventional relays. A reliable and
generalized protection scheme is required that recognizes the patterns of all
types of fault.

The New Zealand distribution network is extensive, with about 98,549 km of
overhead lines, and are sparsely monitored despite them being an economically
significant part of the power networks [3]. Of these, 24,321 km are low
voltage (<1kV) distribution networks (LVDN) that have no monitoring equipment
at all. The networks are susceptible to unplanned outages. An average of 130
minutes of unplanned interruptions per customer was recorded during 2008-2020
period across New Zealand’s distribution networks, which cost distribution
companies around NZD $552,000/min [4]. In addition, these unplanned outages
incur a less easy to measure but significant cost to customers (the cost of
lost load to customers has been found to be twice that of a planned
interruption).

CTs have been the prevalent device in distribution systems for monitoring.
However, a CT has innate nonlinear magnetic core characteristics, high core
losses, narrow bandwidth, and is incapable of DC measurements. All generations
of CTs use a metal core, which makes them heavy and places an additional
burden on the overhead lines’ mechanical load. Also, the core saturates under
high-frequency (HF) transients [5], causing clipped-out instantaneous current
measurement. For these reasons plus their cost, the widespread use of CTs in
distribution networks as a solution for the lack of monitoring is not
attractive.

As an alternative, magnetic sensors, particularly Giant Magneto-Resistive
(GMR) sensors [6], are an attractive option due to their non-contact sensing
ability, low cost, miniature size, and wide frequency bandwidth [7]. A GMR
sensor implemented in the vicinity of overhead power lines can capture the
alternating magnetic field generated by the flow of current.

In this paper, we describe a purpose-built physical simulation test facility
that we have used to test the dynamic response of GMR sensors under fault
conditions and normal system events. The characteristics of tree branch (TB)
HIF from both current and magnetic measurements, are summarized and unique
features are observed. In addition to our focus on fault detection we have
investigated the use of GMR sensor measurements for overhead lines current
sensing and this evaluation is also presented. Finally, we detail a protype
pole mounted sensing system that we have built and installed in Wellington
Electricity’s network.

##  II Description of Robinson Research Test Facility

We have designed and implemented an indoor 400 V network test facility at
Robinson Research Institute. Its main intended use was to generate and
characterize the typical power system faults using current and magnetic field
measurements. The behaviour of a fault depends on the power system
configuration, its voltage-frequency level, and types of connected loads, and
therefore can only be fully realized through practical experiments. A
controllable and safe way to do such practical experiments is in such a
purpose-built power system test facility.

In addition to the use of this test facility for fault simulation studies, it
also has the capability to be used to assess other electrical and electronic
equipment. For example, the modular design of the test facility could allow a
user to characterize miniature circuit breakers (MCCB) under fault conditions,
evaluate conventional and new current sensors or perform micro-grid
simulations.

A schematic of the test facility is shown in Fig. 1, and key components are as
follows.

Figure 1: Test facility schematic diagram. The blue line indicates the fault
current return path (FCRP) to the neutral point of the transformer.

###  II-A Isolated Power Source

To protect the source supply from unwanted tripping during fault experiments
we use an isolation transformer. The transformer has no neutral connection in
its primary side and we did not connect the secondary side neutral to the
ground. All components of the facility are connected to the transformer
secondary winding neutral including the fault current return path (Fig. 1),
meaning the secondary side is a closed loop.

###  II-B Overhead Lines Representation

Our facility models the typical circuit length of up to 600m for a low voltage
(400 V) distribution network. The total circuit length is the combination of:

  * •

An actual spatial representation of overhead lines. Having a spatially correct
overhead line section was important to allow collection of data for the
development of a contact-less magnetic sensor unit for monitoring distribution
network overhead lines and detecting faults. In contrast to the typical span
length of 30m we use two 3m spans, as 80% of the total magnetic field is
generated within the first 3m of the line [8].

  * •

The remainder of the circuit length was physically implemented using a pi
model concept, which uses lumped resistance, inductance, and capacitance (Fig.
2 (b)). The modified Carson’s equation [9] was used to estimate the overhead
distribution line’s parameters (i.e., series impedance and mutual
capacitance). There are six pi model modules, each 100m to allow circuit
lengths of 100 to 600m of lines. A user can manually switch into any circuit
length using external switches.

Figure 2: Physical representation of overhead lines. (a) Overhead Lines
spatial arrangement. (b) circuit length simulator; a physical model of 600m
with a step size of 100m.

###  II-C Fault and Load Simulator:

The fault and load section includes components to simulate low and high
impedance single line to ground (SLG) faults under unbalanced load conditions.
Both broken and unbroken conductor faults can be simulated.

The fault apparatus is shown in Fig. 3. The apparatus is a plastic box with a
metal plate in the base that is connected to the transformer neutral via an
arrangement of current limiting resistors (Fig. 3 (c)). A bare conductor is
connected in parallel with the phase conductor to be faulted, with a relay
switch connecting these.

The load section presently includes two types of commonly used household
loads: heaters (radiant and fan) and an outdoor light unit (Fig. 4 (d)).
Therefore, the loads have a mix of resistive and inductive elements. Other
loads can readily be incorporated and we are currently expanding the range of
loads we can use.

Fault initiation and load switching are controlled by a computer-based (CPU)
graphic user interface (GUI) via a national instrument (NI) multiple
input/output (I/O) module that drives high-voltage mechanical relays through
solid state relays (SSR) (Fig. 4 (d)).

Figure 3: Fault staging apparatus (fault box) showing setup of (a) LIF; (b)
HIF with a tree branch used as a high impedance object (bottom figure). (c)
Current limiting resistor network connected in fault path connection. The
network resistance can be tuned to generate a fault current of 26 to 46 A
(RMS) using double pole switches. Figure 4: Sensor details and control and
data acquisition system architecture. (a) The three-axis GMR (AAH002 [10])
sensor, where the sensitive axes of each sensor were directed towards each of
the x, y, and z directions. (b) Voltage measurement using a step down
transformer incorporated with a voltage divider circuit to measure the
voltage. (c) Phase and neutral current measurements with four hall effect
current sensors (LEM HO 25-NP) [11] mounted on a PCB circuit board trace.(d)
Phase B load has two subsets, B1 and B2. The SLG fault is initiated by
switching F1, and the broken conductor to the ground fault is initiated by
switching F2. Both switches are connected to the fault box.

###  II-D Sensors and Data Acquisition

The test facility incorporates voltage, current, and magnetic sensors to allow
the power system faults and load events to be fully analyzed (Fig. 4). The
analog signals from these sensors were acquired synchronously using multiple
data acquisition units.

##  III Results: Characteristics of 400V Faults

In the test facility we have conducted experiments of low and high impedance
faults involving tree branch and the typical 400V system event of load
switching. HIFs were tested on different surface materials: sea sand, clay
soil and a tree branch. In total, 270 experiments have been conducted with
three different GMR sensor head positions and six circuit lengths. For
brevity, representative data from the tree branch fault experiments (TB-HIF)
is presented here for a GMR sensor head 0.6m below the lines and a 600-m
circuit length.

Figure 5: Snapshots on tree branch HIF stages (1st row) and measured
experimental data. The stages are profiled in the fault path current, which
excludes load current (2nd row), phase currents (3rd row), and magnetic flux
density measurements (5th row). The phases of non-linearity between voltage-
current are illustrated via v-i curves (4th row).

Four unique stages were observed in the TB-HIF experimental data (Fig. 5):

  * •

Low current initial stage, where the fault current is a saw-tooth wave-shape
(Fig. 5 (g)) and much lower than the load current (Fig. 5 (l)).

  * •

Sizzling stage, which has a persistent audible humming noise with visible
sparks. The fault current signal looks like a full-wave rectified wave-shape
(Fig. 5 (h)).

  * •

Negative half cycle arcing, where the arcs occur intermittently and only on
the negative half cycle. The arcing produces a crackling sound and burns the
tree branch. A significant distortion in both phase current and magnetic field
waveform is observed (Fig. 5 (n,x)).

  * •

Stable arcing, where distortion of both positive and negative half cycle
occurs. The v-i plot has an ’s’ shape typical of the non-linear characteristic
of HIF (Fig. 5 (t)).

For the low fault-current stages (initial stage and sizzling stage) the load
current dominates, and the fault current characteristics cannot be readily
observed in either the current or magnetic field measurements. A more detailed
analysis of this data is given in [12], where we have applied discrete Fourier
Transform and Hilbert Transform algorithms to characterize the frequency and
transeient characteristic of the TB-HIF stages.

##  IV Evaluation of GMR Sensors for Detection of HIFs

In summary, as shown in the example experimental results presented in section
III:

  * •

There is no one set of HIF features. Different unique stages can occur in HIFs
and their characteristics depend on the nature of the surface type which can
also change as the fault progresses. These unique stages exhibit different
fault current waveform shapes along with different visual and audible
phenomena.

  * •

Arcing does not always occur in HIFs; low current non-arcing stages also
occur.

  * •

Load current dominates over the low current non-arcing HIF fault behaviour.
These characteristics mean that HIF detection using simple algorithms is
likely to be problematic and sophisticated pattern recognition algorithms will
need to be applied to measurements in order to detect some HIFs.

We have developed a classification algorithm, based on Deep Learning, to
classify signals from the GMR sensors into 3 fault classes and one class of
’normal/no-fault’ (Fig. 6). When tested on the experimental data from the
fault test facility:

Figure 6: Fault pattern recognition framework.

  * •

100% of fault instances were classified as a fault of some type although 1.5%
of Arcing HIF were miss-classified as non-arcing HIF.

  * •

99.98% of normal system states were correctly classified as having no fault
present.

The next step is to collect and test this pattern recognition system on data
from real networks. Progress toward this is discussed in section VI.

##  V Evaluation of GMR Sensor for Non-contact Current Sensing

Figure 7: Comparison of load current measurement using two vertically placed
sensors at 0.42m and 1m from the overhead lines (b) Instantaneous difference
between actual and calculated current (c) Vertical combination of sensor heads
placed 0.58 m apart.

As shown in the comparison of the GMR versus Hall Effect current measurements
in section III, the phase current behaviour is strongly reflected in
measurements from a GMR sensor on the pole. As part of our project we have
therefore also evaluated the use of GMR sensors as non-contact current
sensors, in addition to their use for fault detection.

For non-contact current sensing we have evaluated the use of two 2D GMR sensor
heads (x- and z-axes) vertically located on the pole. We mathematically
decouple the overhead line current from the GMR sensors measurements. The
comparison between the current measured with the Hall Effect sensors versus
those calculated from the GMR sensor measurements shows a good match (Fig. 7
(a)). The instantaneous current error was less than 1 A (Fig. 7 (b)) and the
root mean square (RMS) error between the actual and calculated currents ranged
from 1.3-5.3%.

##  VI Pole Mounted Sensing System Prototype

Figure 8: Outdoor GMR sensor based current sensing unit.(a) Schematic of
electrical power pole and location of sensor head and control box. (b)
Connections between solar power system for data acquisition and sensor head
units. (c) Outdoor monitoring system installed in the Wellington Electricity
overhead line networks.

In parallel to our lab test facility experimental work, we have developed a
prototype outdoor sensing system and have installed it in the Wellington
Electricity (WE) network, near Gracefield, Lower Hutt substation (Fig. 8 (c)).
The outdoor prototype includes GMR sensor heads, and a data acquisition and
storage system. The system was designed as a standalone device, thus, it is
powered using off the shelf battery pack, solar panel, and charge controller
unit with maximum power point tracker (MPPT) capability (Fig. 8 (b)). With
this we are collecting data from two different locations on the pole to
observe the spatially cross-coupled fields from the four overhead lines (Fig.
8 (a)).

The data acquisition module was developed in-house using a embedded device,
Raspberry Pi integrated with an analog to digital converter (ADC). Data is
sampled at 27.7 kHz and transferred to the raspberry pi for storage via serial
peripheral interface (SPI) protocol (Fig. 8 (b)). The raspberry pi typically
runs on Debian operating system, which provides a graphical user interface
(GUI) to easily monitor and program the module on the fly. The module can also
be accessed over the wireless fidelity (WIFI) protocol.

We are currently continuously monitoring and collecting data from this outdoor
unit. We will use this data to confirm and refine our pattern recognition
algorithms for fault detection and to refine our hardware selections.

##  VII Conclusion

In electricity distribution networks, fault detection is an ongoing challenge,
particularly of High Impedance faults. GMR sensors are an attractive
proposition for more widespread monitoring and as a basis for fault detection
because of their broad bandwidth, low power consumption, miniature size,
ability to digitally interface with intelligent systems and low cost. Our
experiments in our purpose-built lab test facility have confirmed the dynamic
response of these sensors during fault events and load changes.

We have characterized the dynamic behaviour of fault currents and magnetic
fields. A number of new and unique stages of HIF faults have been identified,
dependent on surface material type and duration of the fault. FOr example as
shown here, a number of new and unique stages of TB-HIF faults have been
identified. Arcing does not always occur. It would be desirable to detect HIF
faults in real networks even during the non-arcing stages, but they cannot be
readily observed in raw current or magnetic field measurements.

In our work we have found that suitable signal processing techniques can
reveal the characteristic features of these non-arcing states and that Deep
learning algorithms can automatically detect and classify fault states versus
normal system events. Our current focus is to confirm and refine our
algorithms using data collected in real networks, starting initially with our
prototype outdoor sensing system installed in Wellington Electricity’s
network.

## Acknowledgment

This work was financially supported in part by the New Zealand Science for
Technological Innovation National Science Challenge (contract RTVU1702);
Ministry of Business, Innovation and Employment New Zealand (contract
RTVU1811) and Victoria University of Wellington Research Trust.

We would also like to acknowledge the support of Wellington Electricity, who
have installed a prototype version of our hardware in their network for
ongoing research data collection.

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