This article has been accepted for publication in a future issue of this journal, but has not been

fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TPWRD.2020.3016717, IEEE
Transactions on Power Delivery
JOURNAL OF LATEX CLASS FILES, VOL. , NO. , MONTH YEAR 1

PMU-based Cable Temperature Monitoring and

Thermal Assessment for Dynamic Line Rating
Ravi Shankar Singh, Student Member, IEEE, Sjef Cobben and Vladimir Ćuk

Abstract—The aim of this paper is to present a novel phasor However, many cable sections such as cables connecting
measurement unit (PMU) data-based cable temperature moni- off-shore wind park to land substation or cables connecting so-
toring method with an intended application towards facilitating lar and wind parks to the grid transport intermittent renewable
dynamic line rating. First part of the paper presents the method
to estimate and track the temperature of a 3-phase cable segment. power. Many urban load centers have time-dependent peaks.
The benefit of this temperature monitoring method is that no In such cases, if steady-state current rating is applied then due
additional temperature measurement sensors are required to to the thermal inertia of the cable system, the cable may never
be placed along the cable. The method is based on a novel approach its thermal limits. This results in under-utilization of
algorithm which gives accurate resistance estimates for 3-phase the loading capacity of the cables. To utilize the cables more
cable segments even in the presence of random and bias errors
in the grid measurements. The performance of the method is optimally, dynamic loading models are required.
demonstrated by utilizing data from PMUs in a distribution Increasing the loadability of the cables to maximize the
grid. The results from the grid data show that the method is accommodation of the intermittent peaks of power flows would
capable of monitoring the cable temperature up to an accuracy help acquire more clean energy and deliver more power to
of ±5◦ C. The later part of the paper presents a system to utilize load centers using the existing cable infrastructure. Time based
the temperature estimates given by the monitoring method to
predict the dynamic thermal state of the cable for forecasted flexibility in loading limits of power lines, also known as
power-flow scenarios. This is demonstrated by using the available dynamic line rating (DLR) has been a topic of interest in
temperature estimates to initialize and solve the system of the recent past. Authors in [2] presented a case where DLR
equations given by the thermoelectric equivalent (TEE) model applied to a 132 kV overhead line section enabled connection
of the cable. of up to 50% extra wind power. Results from a large number
Index Terms—Cable Temperature Monitoring, Cable TEE of simulations presented in [3] investigating the application of
Model, Dynamic Line Rating, PMU Application. DLR on overhead conductors connecting wind farms showed
DLR to have a significant economic potential.
For underground power cables, two ways to decide the
I. I NTRODUCTION flexible loading limits are the cyclic and the emergency rating
OST-OPTIMIZED generation and distribution of elec- for cables which are presented in the standard IEC 60853:2
C trical energy from eco-friendly sources has become the
objective of modern power network operation. Distribution
[1]. Cyclic rating of the cables can be used when cables are
exposed to a daily cyclic pattern. However, no such pattern
networks are being reinforced with more decentralized and is required to calculate the emergency rating which gives the
renewable power generation sources. Electrical power demand amount of current a cable can carry for a specified time period
of urban areas is also increasing continuously. This increasing before the temperature limit is breached. The emergency
generation and demand of electrical energy puts growing stress rating is calculated by studying the dynamic thermal response
on the network assets including the distribution cables. One of the cable system in presence of a load step. The state
of the constraints for routing the extra power away from the variables of the TEE cable thermal model are the conductor,
source centers or towards the load centers is the capacity or screen, jacket and soil temperature. Initial temperature of these
loadability of the cables. The loadability is dependent on the state variables are necessary to calculate the dynamic thermal
thermal rating of the cables and the ambient conditions. For response and hence the emergency rating.
cables, the insulation especially is very sensitive to the tem- A method to estimate the time-dependent thermal state of
perature higher than the recommended maximum temperature. the power cables utilizing the TEE model was presented in
For example, thermal limit of XLPE cable is 90 ◦ C. [4]. A finite element model (FEM) based method was used to
IEC standard 60287 presents a method to calculate the compute dynamic rating in [5]. However both methods require
steady-state rating of a cable system [1]. Steady-state cable temperature measurements to initialize the TEE model. This
ratings are suitable for cables under high load factor. This requires one or more temperature sensing measurements in-
means that the ratio of daily average of hourly load to the stalled along the cable path. Information about the temperature
daily maximum load is close to unity. The standard uses the could be achieved using the distributed temperature sensing
thermoelectric equivalent (TEE) model for cables which has (DTS) equipment [6]. However, installation of sensors for DTS
lumped thermal resistance and capacitance parameters in a in existing cables requires retrofitting the cable system with
thermal ladder network. fiber-optic cables and could be a major challenge in multiple
ways. This paper presents a solution to this challenge using
the presented temperature estimation method.

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In absence of temperature measuring devices, data from Previous methods also did not provide any confidence interval
phasor measurement units (PMUs) at both ends of a cable (CI) for the calculated resistance and temperature parameters.
could be used to estimate the cable resistance and eventually CIs around the results would make the application more
the cable temperature in real-time. However, for medium and trustworthy when tracking the temperature of a critical cable-
short length cable sections, achieving temperature estimates section in real-time. To overcome these drawbacks, this paper
with the desired accuracy and precision could be challenging. presents a new method which is capable of giving accurate
It is also important to get accurate temperature estimates and reliable temperature estimates for 3-phase cable systems
continuously using the real-time data. In the past, work has in presence of bias and random errors.
been presented to showcase the feasibility of this idea albeit
mostly in a simulation environment and only for high-voltage
long-distance overhead lines. A. Paper Contribution
Authors in [7] proposed a method to estimate parameters
and temperature of overhead line conductors. However the This paper presents a solution to perform thermal assess-
algorithm is presented only for single phase line and no ment of cable sections to implement DLR without using
measurements errors were considered. A review of different specific temperature measurement infrastructure. The focus is
methods to estimate the parameters of a 3-phase overhead line on a relatively new domain of MV-distribution networks where
is presented in [8] and [9]. Single and double measurement the feeder lengths are relatively short.
methods and multiple measurement method using linear and At first, a new method to provide temperature estimates in
non-linear regression were compared. It was shown that the real time is presented. Unlike the other methods, this method
multiple measurement method using linear regression per- uses a 3-phase cable model to give accurate temperature
formed the best for short lines. However, quantitative effect estimates of all the conductors, even in the presence of random
of random errors and systematic bias errors present in the and systematic bias errors. This paper presents the modelling
measurement chain was not studied. The results from a field of a 3-core cable to make the impedance matrix and discusses
experiment using a robust estimator showed that further in- how an error in the model is a contributing factor to the
vestigation of the uncertainty caused by bias errors is required errors in resistance and temperature estimates. To facilitate the
[10]. According to the authors, the uncorrected systematic bias small thermal time-constant of the cables and small duration
present in the measurement chain could cause the algorithm power-flow transients a much shorter and continuously sliding
to give incorrect results. 1 hour data window was used. To complete the temperature
A calibration method to accurately estimate the line param- estimation process, CI around the estimates are computed. The
eters of a 1-phase line segment along with the bias errors was performance of the algorithm is demonstrated using two days
presented in [11]. This method uses simplification based an long field PMU data. This long period is useful in observing
assumptions that the phase errors in the current and voltage any trends in the change of cable temperature along with the
transformers (CTs and VTs) are smaller that 0.530◦ . This trends in the power flow.
however, could be untrue for real cases. An optimization- Subsequently, application of the proposed temperature mon-
based method was presented in [12] which estimated the bias itoring method in thermal assessment of a cable in presence
errors along with other unknown parameters of a 1-phase of load forecasts is presented. A flowchart describing the
line segment which would minimize the difference function whole process of temperature estimation and its utilization for
between the measured and estimated phasors at one end of advance dynamic thermal assessment of cables is presented in
the line. Fig. 1. The capability to track cable conductor temperature in
A review of methods to enable PMU based thermal moni- real-time also becomes an important tool for monitoring and
toring of overhead transmission lines is presented in [13]. Real a safe implementation of a DLR scheme.
PMU data from a 400 kV overhead line was utilized and the
results based on methods presented in [8], [11] and [12] are
compared. It was concluded that only the optimization based B. Paper Structure
method could give reasonably accurate results. However, the
data-window to calculate the parameters was 6 hours long, The remainder of the paper is arranged as follows: Sec-
while the duration of power-flow transients and the thermal tion II discusses the requirements in terms of accuracy of
time constant for the cables could be as low as 30 minutes. the resistance estimates which in turn help to estimate the
Such a long window may give smoother average results but temperature of the cable conductors within a desired range.
might miss the vital transients in the temperature. Section III presents the resistance estimation algorithm in
Both calibration and optimization methods presented in detail. The process of cable modelling to identify significant
[11] and [12] calculate the line parameters and bias error parameters and the process of uncertainty calculation is also
correction coefficients considering a 1-phase line model. The presented. Section IV presents the process of estimation of the
impedance model used did not include any mutual impedance temperature and associated uncertainty. Demonstration of the
parameters which might be present in a 3-phase line segment. method utilizing the field PMU data is presented in Section
Estimating the parameters for a 3-phase system could be more V. The intended application of thermal assessment using TEE
challenging if it includes additional mutual impedance and models of underground cable is discussed in Section VI. The
admittance parameters depending on the nature of the system. conclusions are drawn in Section VII.

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Start
TABLE II
ACCURACY R EQUIREMENTS OF R ESISTANCE E STIMATES .

Real-time
Temperature Uncertainty Resistance Uncertainty
Resistance
Cu Al

± 3 ◦C < 1.12 (%) < 1.21 (%)

Cable Conductor
Temperature ± 5 ◦C < 1.85 (%) < 2.01 (%)
± 10 ◦ C < 3.74 (%) < 4.03 (%)
Cable Thermal
Initialize Unknown
Model for
Variables
Transients

Temperature Monitoring
The change in the DC resistance of the copper coil in the
temperature range 10 - 50 ◦ C was used to calculate α using a
New Loading Cable Temperature
Scenario Response linear regression model. The hypothesis for the linear model
was found correct in the measured temperature range and the
value of α with uncertainty up to 3 standard deviations was
found to be 0.003742 ±2.7914 × 10−4 ◦ C−1 . The uncertainty
No Satisfies Thermal
Constraint? calculated for α was used as a contributing factor for uncer-
tainty in the final temperature estimates. The value of α for
Yes aluminum conductor was taken to be 0.00403 ◦ C −1 [1]. Table
II shows the accuracy requirement of resistance estimates
Dispatch corresponding to different range of accuracy of estimation of
the Cu and Al conductor temperature. It serves the purpose
Fig. 1. Flowchart showing the process of utilizing the resistance estimates to
of a reference maximum level of uncertainty budget we have
assess flexible loading limits. for the resistance estimates to achieve a certain desired range
of accuracy in the temperature estimates. It is presented as
maximum allowed uncertainty because there are several other
II. R EQUIRED ACCURACY OF R ESISTANCE E STIMATES sources of uncertainties as well. It is calculated using the
This section discusses about the required accuracy in resis- value of α in (1). So, for monitoring method to determine
tance estimates for the cable temperature estimation method. the temperature of a cable conductor made of aluminum with
The fundamental factor is the desired accuracy range of the an accuracy of ± 5 ◦ C, the errors in resistance estimates must
temperature of the cable being monitored. This temperature be less than 2.01% of the true resistance.
range could then be translated into the accuracy range for
resistance estimates. According to IEC-60287-1-1, the AC III. R ESISTANCE E STIMATION
resistance of a conductor at temperature Ti is given by [14]: This section presents the cable resistance parameter es-
Ri = R0 (1 + α(Ti − T0 ))(1 + ys + yp ) (1) timation which is the core of the temperature estimation
process. The resistance estimation process was divided into
where α is the temperature coefficient (◦ C −1 ) of the resistivity three parts. First a correct model of the cable system was
for a given material, R0 is the DC resistance of the conductor made to identify the other unknown parameters needed to
at temperature T0 , and ys and yp are the skin and proximity be estimated along with the resistance. After estimating the
coefficients. The above constants depend upon the particular parameters, uncertainty of the estimates are evaluated. The
conductor material which is typically copper or aluminum. three parts are presented in the following subsections.
A test in the laboratory was performed to investigate the
effect of heat on the resistance of a copper coil. The DC
A. Cable System Modelling
resistance was measured at temperature ranging 10-50 ◦ C. At
each temperature point, 200 readings were taken. The mean Accurate modelling of the cable system impedance and
and uncertainty up to three standard deviations of the measured admittance matrix is of prime importance as it facilitates the
values are presented in Table I. selection of significant parameters to estimate. Impedance and
admittance models of overhead line and a cable for 3-phase
parameter estimation is shown in [10] and [15] respectively.
TABLE I The cable section in the grid was a 3-core cable whose cores
R ESISTANCE MEASUREMENT AT VARIOUS TEMPERATURES are arranged in a trefoil arrangement. A cross-section with
representational construction details of the cable is presented
Temperature (◦ C) DC Resistance (Ω)
in Fig. 2. As the PMUs are measuring current and voltage at
10.425 ±0.1 2.3561 ±0.0420 the conductors, only core-core sub-matrices of the complete
19.972 ±0.1 2.4458 ±0.0432 cable impedance and admittance model are used to select the
29.753 ±0.1 2.5365 ±0.0417 significant parameters [16].
39.587 ±0.1 2.6288 ±0.0426
Unless a very low current, the percentage current unbalance
49.000 ±0.1 2.7245 ±0.0426
in the grid cable was found out to be between 1-2%. Thus

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where i is the effective electric permittivity insulation be-

Conductor
Core Insulation tween the core and jacket [16]. The conductivity of the insu-
Belt Insulation
Screen lation is very small and is considered zero [19]. Considering
Jacket
Dab = Dbc = Dac = Ds :
 
1 Ds r
Vab = qa ln + qb ln (8)
2πi r Ds
Similarly,
 
1 Ds r
Fig. 2. Representational construction details of the field cable. Vac = qa ln + qc ln (9)
2πi r Ds
Using the balanced power-flow condition, qa + qb + qc = 0,
for making the impedance and admittance matrix, a balanced  
current was considered. In 3-core cables, the return path 1 Ds r
Vab + Vac = 2qa ln − qa ln (10)
of the current is via the other phases. However, there is 2πi r Ds
no return current in a balanced system. Thus, the mutual
For a three phase network it can be shown that, Vab + Vac =
components of the resistance were neglected. It can also be
3Van , where Van is the voltage of phase a with respect to the
shown that for a trefoil core arrangement and balanced power-
neutral. Further simplification of (10) leads to the result for
flow conditions, the mutual couplings between the voltages
phase a:
and currents of the three phases through the reactance and 1 Ds
susceptance components are also zero. 3Van = 3qa ln (11)
2πi r
The flux linkage of phase a conductor with radius r is given
by the sum of internal and external flux [16]: and the charge and voltage relation using the capacitance
  matrix can be written as:
µ0 1 1 1
λa = Ia ln 0 + Ib ln + Ic ln (2)
    
qa Caa 0 0 Van
2π r Dab Dac  qb  =  0
0
Cbb 0   Vbn  (12)
where, Ia , Ib and Ic are the currents in each phase, r is the qc 0 0 Ccc Vcn
0 −1
self geometric mean radius (r = re 4 ), Dij is the distance
where,
between two conductors and the permeability of non-magnetic 2π
material is taken as µ0 which is the permeability of free space Caa = Cbb = Ccc =  i (13)
Ds
[17]. ln
µ0 r
Let 2π ln r10 is denoted as self inductance component (Ls )
µ0
and 2π ln D1s as mutual inductance component (Lm ). For a are the self capacitances of the conductors. This shows that
trefoil core arrangement the distances are: there is no off-diagonal element in the admittance matrix for
3-core trefoil cable with balanced power-flow.
Dab = Dbc = Dac = Ds . (3) Thus the parameters identified to be estimated for the 3-
For balanced power-flow conditions, the sum of currents is: phase cable system are the self resistance, reactance and
susceptance of each phase (raa , rbb , rcc , xaa , xbb , xcc , baa , bbb
Ia + Ib + Ic = 0. (4) and bcc ) where, x and b are the reactance and susceptance
1
Using 2, (3) and (4), it can be shown that: given by jωL and jωC and ω is the angular frequency.
     However, this model is valid only for balanced power-flow
λa Ls − Lm 0 0 Ia
in the cable. With the increase of unbalance, the significance
 λb  =  0 Ls − Lm 0   Ib  (5)
of off-diagonal impedance and admittance components also
λc 0 0 Ls − Lm Ic
increases. Identification of correct set of parameters to estimate
where, is important and its effect on the performance of the algorithm
µ0 Ds
Ls − Lm =
ln 0 . (6) is discussed in the next subsection.
2π r
Thus there is no off-diagonal element in the inductance matrix
for a trefoil cable in the condition of balanced power-flow. B. Estimation Algorithm
Thus the impedance matrix of the cable system does not have The resistance estimation algorithm is aimed to give accu-
any off-diagonal elements. rate and reliable estimates. The algorithm takes into account
Similarly it can be shown that the off-diagonal elements of the presence of bias errors in the measurement of current
the admittance matrix are also zero for a trefoil cable. For a and voltage phasors. The bias errors are caused due to error
system of three conductors carrying charge q coulombs/meter in or unavailability of correction coefficients for ratio and
each, the relationship between the charge and the voltages can phase errors in the CTs and VTs. Extra parameters in the
be written in form [18]: linear regression model were added which would model these
bias errors. These bias errors are assumed to be constant for
duration of one window length of sampled data. It can be
 
1 Dab r Dbc
Vab = qa ln + qb ln + qc ln (7) shown that for a sinusoidal signal of the form V = |M |ejθ ,
2πi r Dab Dab

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IL the range between 0 to 1 %. Similarly the phase angle errors

IS IR
of the CTs and VTs were varied between 0 to 1 degrees.
Z=r+jX
The coefficients K1 − K6 were calculated based on these
VS B/2 B/2 VR ratio and phase errors and are substituted in (17) and (18).
Deviations in estimates of the elements of B and Z matrices
were calculated. For the simulated system, it was observed in
Fig. 3. Nominal Pi model for a medium length medium voltage cable. [15] that the error in B is most sensitive to K1 . It was found
that ignoring coefficients K1 alone could lead to a maximum
error of 15% in B estimates. However, ignoring K2 and K3
the measured signal Vm with a magnitude error of γ% and only caused about 0.5% error each. Similar analysis showed
phase error of δθ can be written as: that the resistance R is sensitive to K4 resulting in maximum
Vm = V (1 + γ)ejδθ (14) error of more than 50% on ignoring it, while ignoring K5
and K6 resulted in maximum errors of around 3% and 0.25%
The correction coefficients for the bias errors can be rep- respectively. Hence only the adjusted correction coefficients
±jδθ
resented as e1+γ . In Cartesian coordinates, this could be K1 and K4 were included into the system of linear equations.
represented in the form a±jb. where a is the real part given by
cos(±jδθ)
and b being the imaginary part given by sin(±jδθ) . B
I S − K1 I R = VS +VR


1+γ 1+γ (19)
Using the correction coefficients at both ends, voltage and 2
current phasors at both ends of a 3-phase system represented 
B R

by a pi-model shown in Fig. 3 can be written in the following V S − K4 V R =Z V + IR (20)
2
form:
B To solve for the parameters of B and Z matrices, (19)
CiS I S − CiR I R = Cv S V S + Cv R V R


(15) and (20) were written as two separate equations for real
2
  and imaginary parts. This was done for all the three phases
B R R
Cv S V S − Cv R V R = Z Cv V + CiR I R (16) resulting in twelve equations. Now, the parameter estimation
2 process was divided in two parts. All the measured data was
where, CiS , CiR , Cv S , Cv R are the complex three phase arranged according to the six equations from (19) forming a
correction coefficients for the ratio and phase errors of CTs set of over-determined system of linear equations. The set of
and VTs at both ends of the line. Superscripts S and R dis- linear equations can be written as:
tinguish the sending and receiving ends of the cable. Voltage
Y = Aβ + E (21)
and current phasors are represented as complex numbers in
Cartesian coordinates. Diagonal matrices Z and B contain the where, Y is a vector made up of multiple measurements of
three phase impedance (r + jx) and shunt susceptance (jb) reIas , reIbs , reIcs , imIas , imIbs and imIcs , the parameter
elements of the cable. Using a new set of Adjusted Correction vector β is [reK1a reK1b reK1c imK1a imK1b imK1c
Coefficients, (15) and (16) can be rewritten as: Baa Bbb Bcc ]T . Square matrix A is the relationship matrix
made of real and imaginary components of measured I R , V S
B
I S − K1 I R = K2 V S + K3 V R and V R phasors while E is the error vector.


(17)
2 Parameters K1 , and B for all phases were estimated using
V S − K4 V R = Z K5 I R + K6 V R


(18) the analytical solution for solving the LS problem minβ ||Z −
Aβ||2 . The most optimal estimate (β̃) is given as:
where,
CiR Cv S Cv R β̃ = (AT A)−1 AT Y (22)
K1 = , K2 = , K 3 = ,
CiS CiS CiS Now, estimated susceptance parameters Bii were substi-
R R
Cv Ci B tuted in (20) where the new parameter vector β is
K4 = , K5 = , K6 = K4
Cv S Cv S 2 [reK4a reK4b reK4c imK4a imK4b imK4c raa rbb rcc xaa
are the adjusted complex coefficients. xbb xcc ]T . A new relationship matrix A and vector Y was
In this way, (17) and (18) represent (15) and (16) such that formed and the parameter vector β was estimated using (22).
it allows the treatment of the measurements at one end of If the cable system was modelled to include the off-diagonal
the cable (the sending end in this case) as error free. The components of the capacitance and inductance matrices then
measurements of the receiving end are corrected using the the resulting matrix A would be near rank-deficient and hence
adjusted correction coefficients. ill-conditioned. The near rank deficiency would be caused by
To reduce the number of unknowns, a sensitivity analysis the extra almost linearly dependent columns of voltage and
was performed to identify the most prominent correction current phasors used to find out the non-existent off-diagonal
coefficients. For this, an MV cable of 20 kV and length 10 components. The condition of the matrix A is quantified by the
km was simulated. A power flow profile recorded from the condition number. The higher the condition number the more
50 kV network was used. Now all the CT and VT ratio ill-conditioned the matrix is. Solutions given by (22) using ill-
errors were varied as per a random uniform distribution in conditioned A matrix would vary significantly even in case of

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small errors in the elements of A and hence would have high K5 and K6 . The total standard deviation in the resistance
variance. estimates is given by:
This is an important realization because all the variables q
used to form the LS problem shown in (21) are measurements u(Ri ) = u2rnd + u2b (25)
with some amount of error. Hence for a given power-flow The following section presents the process of calculating
condition, the accuracy of the resistance parameters estimate the temperature estimates and associated uncertainty of the
would depend on the validity of the cable model along with estimates.
the accuracy of the measurement devices.
After estimation of the cable parameters, uncertainty in the IV. T EMPERATURE E STIMATION
estimates was quantified in terms of CI associated with each
parameter. Since cable resistance has a direct relationship with The resistance estimates from the solution given by (22) are
the cable temperature, accuracy of the resistance estimates was used in (1) to achieve the temperature estimates. The uncer-
of prime concern. tainty in the temperature estimates comes from the individual
uncertainty associated with the resistance estimates (Ri ), the
measured DC resistance at 20 ◦ C (R0 ) and the used coefficient
C. Uncertainty in Estimates of resistivity (α). Treating these individual uncertainties as
The deviation in the resistance estimates was calculated independent from each other, the combined uncertainty in the
in two parts. One part of the deviation was due to random temperature estimates is then given by:
errors in PMU estimates and the other part of the deviations XN  ∂f 2
2
was caused by the bias errors in the CTs, VTs and the u (Ti ) = u2 (xi ) (26)
i=1 ∂xi
phasor estimates given by the PMUs. The uncertainty due
to random errors in PMU estimates were taken as per the where, u(Ti ) is the standard deviation of each temperature
specification provided by the manufacturer. The random errors estimate Ti , f is the function given in (1) and each u(xi ) is the
were the absolute maximum errors distributed uniformly with standard deviation associated with all the parameters xi . The
zero probability of errors outside the range. The standard skin and proximity effects however, were ignored in this paper
deviation caused by the random errors (urnd ) of the impedance because the values of harmonic currents are limited in the
estimates were derived based on the co-variance of solution analyzed system 50 kV network. The total harmonic distortion
of the LS problem. It can be shown that expected variance in (THD) of the current ranges between 5-10 % with about 95
the impedance estimates is given by: % of the contribution by the lower order fifth harmonic (250
Hz). This makes the impact of the skin effect very limited, if
0
 0 not negligible.
u2rnd := V ar[b|A] = (A A)−1 (23)
n−K The following section V demonstrates the results using
PMU data from the mentioned 50 kV ring network in the
where,  is the residual vector, n is length of the vector Y and
Netherlands.
K is the number of parameters.
The CT and VT correction coefficients are used by the
V. R ESULTS FROM FIELD DATA
PMUs while estimating the voltage and current phasors.
However, to cater any change in correction coefficients and This paper takes data from a 50 kV ring distribution network
minimize the effect of bias errors in the measurements, promi- in the Netherlands provided via the Dutch National Metrology
nent adjusted coefficients K1 and K4 were added. However, Institute (VSL) [21]. The ring network has five substations and
neglecting the coefficients of K2 , K3 , K5 (assumed 1) and six PMUs. One of the intended research goal for installing
assuming K6 as B2 causes deviation in the cable impedance PMUs in the network was application of PMU data to estimate
parameters from their true values. This deviation is quantified the cable impedance and explore possibilities of implementing
as a bias error (ub ). The deviation in the impedance parameters DLR. Hence the cable between substations Oosterland and
caused by the bias errors in measurements is estimated by Tholen has two PMUs (one at each end). The monitored cable
calculating the combined uncertainty calculation as specified between substations Oosterland and Tholen is 15.313 km long
in the Guide to the Expression of Uncertainty in Measurement and has an AC resistance of 1.98 Ω at 20 ◦ C [22]. The current
(GUM) [20]. In this work the magnitude and phase errors were rating of the cable per phase is 350 A.
assumed to vary normally with a standard deviation of ±10% Voltage and current phasors at both sides of the monitored
from their last calibrated values. The uncertainty (variance u2b ) cable were collected for 40 hours at a rate of 5 phasor estimates
in resistance estimate due to believed bias in the measurements
was quantified by:
TABLE III
 2 U NCERTAINTY SPECIFICATIONS OF USED PMU S
XN ∂f
u2b = u2 (xi ) (24)
i=1 ∂xi Entity Uncertainty

where, f is the analytical function to calculate the resistance voltage magnitude ±0.02%
current magnitude ±0.03%
and is given by the real part of (18). Each u(xi ) is the believed
voltage and current phase ±0.01◦
standard deviation in real and imaginary parts of coefficients

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per second. The random errors associated with PMUs are

mentioned in Table III. One end of the cable has VTs and
CTs of accuracy class 0.5 and 0.1 respectively while the other
end has both VTs and CTs of accuracy class 1. Before utilizing
the measurement data in the estimation algorithm, the data was
processed using the first (static) stage of the Kalman filter
presented in [23] to avoid any bad-data points. The model
was initiated by a measurement noise covariance given by the
Table III and an assumed stable process covariance. These
covariance matrices could be adapted dynamically depending
upon the measurements.
The resistance estimates along with reactance and suscep-
tance were calculated. An expanded CI with 3 times the
standard deviation (3σ = 99.7 % confidence) was calculated
using (25). The resistance estimates were translated in the
temperature estimates. The uncertainty in the temperature
estimates were calculated using (26). Maximum uncertainty
in the constants α and R0 was taken to be ± 0.1 % each. The
resistance and temperature estimates for three phases of the
monitored cable are presented in Fig. 4. The data window
was 1 hour long and was sliding every 5 minutes. Hence
a resistance and temperature estimate is achieved every 5
minutes using the past 1 hour of recorded data. The confidence Fig. 4. Estimated resistance and conductor temperature for the three phases
intervals for both the resistance values and the temperature are of a field cable for a time duration of 40 hours (Feb 1-3,2020). The believed
AC resistance for the cable at 20 ◦ C is 1.98 Ω.
shown as the shaded area around the mean expected values.
Most part of the uncertainty is contributed by the expected
bias errors in the measurements. parameters of thermal resistance and capacitance. The temper-
It can be observed that the total CI expands at certain time ature estimates from the monitoring method are proposed to
periods. Just like at other time periods, this expansion of CI is set the initial temperature of layers of the cable and the soil.
also caused mostly by the expansion of uncertainty caused by Fig. 5 presents a TEE model of a single core cable for
the bias errors (ub ). It was observed that the sensitivity of the of a long-duration transient where the soil layer has been
impedance estimation function (real part of (18)) increases to divided into three layers with different thermal resistance and
the ignored bias error coefficients when the current is low. The capacitance. Wc is the ohmic joule loss in the conductor
effect of random errors is also more observable at low current caused by current flowing and the real time resistance given
levels. A reason for this observation could be that in low by (1). Wd is the dielectric loss in the insulator which has
current conditions, assumptions like perfectly balanced power- been divided into two equal parts. Ws is the joule loss in the
flow and a negligible THD level of current might become less screen of the cables. Qc , Qi , Qscr , Qc and Qsi are the thermal
valid. In such conditions, these reasons would make the model capacitances of the conductor, insulator, screen, jacket and ith
of the cable used in this paper less accurate. The voltage of layers of the surrounding soil.
the system does not vary so much when compared to the To keep the temperature gradient within the layer small,
current. Thus this expansion is observed when the current components like insulation and surrounding soil must be
flowing through the cable is low. A delay is observed for subdivided into smaller layers [1]. The thermal capacitance
this uncertainty expansion in comparison to the current levels of the insulator has been divided into two parts using the van
because the data window holds one hour long data. It is also Wormer coefficient p. For a specific cable system in [4], the
worth noting that from the point of view of application in soil layer of the TEE model was sub-divided into 100 layers
DLR, the uncertainty in temperature estimates at very low to get accurate results comparable to FEM based method. In
current levels are less critical than uncertainty at higher current this paper also, the soil layer is divided into 100 equally
levels. It was observed that the uncertainty of the estimator is
less at higher current levels.
The following section discusses the process of utilizing real- T1 T3 Ts1 Ts2 Ts3

time temperature estimates of a cable to calculate it’s dynamic

thermal response to power-flow forecasts. Wc(t) Qc
Wd1
pQi
Wd2 Wscr
Qj Qs3
(1-p)Qi Qscr Qs1 Qs2

VI. T HERMAL A SSESSMENT USING TEE MODEL Cable Soil

A TEE model of the cable is created as per recommen- Fig. 5. Cable TEE model with lumped resistance and capacitances for long
dations in [1] and [4]. In the TEE model, various layers of duration transient. Cable and soil components are shown in two boxes. The
the cable and its surroundings are represented using lumped soil is divided into three layers for representational purpose.

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Transactions on Power Delivery
JOURNAL OF LATEX CLASS FILES, VOL. , NO. , MONTH YEAR 8

thick layers. Thermal resistance of the insulator, jacket and where parallel thermal capacitances are added together such
the surrounding soil is represented by T1 , T3 and Tsi . The that Q1 = Qc + pQi and Q2 = Qj + Qscr + (1 − p)Qi .
modelled cable in Fig. 5 has no armor hence the thermal However, this system of equation implies that the resistance
resistance of armor (T2 ) is ignored. of the cable remains constant. The heat generated by joule
heating is dependent on varying current values but a constant
A. Parameters of the TEE model resistance. This is contradictory to realistic case where the
resistance of the cable also varies according to the temperature
Thermal resistances (Ti ) and capacitances (Qi ) for various of the cable. This relationship between the cable temperature
layers of the cable and its surrounding need to be accurately and resistance is defined by the (1). To rectify this, Wc at time
computed. For known internal and external diameters Di and ti is modified and written as:
De of a layer, its thermal resistance is [1]:
  Wc (ti ) = I(ti )2 (R0 (1 + α(θc (ti ) − θc (t0 ))) (32)
ρi Dei
Ti = ln (27)
2π Dii where R0 and θc0 are the cable conductor resistance and
where, ρthi is the thermal resistivity of the material of layer temperature estimated by the temperature monitoring method
i. For each sub-layer of the soil, thermal resistivity Tsi was and used as the initial conditions for (31) at time t0 .
calculated as [4]: The modified system (31) can be written using the state-
  space notation:
ρi Dei ln(2) 0
Tsi = ln + (28) x = Ax + Bu (33)
2π Dii N
where, N is the number of soil sub-layers. Thermal capacitance where the state vector x is [θc θscr θj θs1 ... θsN ]T and
of any layer can be calculated as [1]: conductor and ambient temperatures (θc and θa ) are known.
π The driving function (B) for a given time period can be
Qi = (De2i − Di2i )Ci (29) determined using the forecasts of the generation and load units.
4
The thermal response of the cable over the given period of time
where, Ci is the volumetric specific heat of the respective cable can be obtained by solving the system of differential equations.
layer or soil sub-layer. The time domain solution of (33) is the superposition of
A transient is considered long when it lasts longer than natural and forced response of the system and for a given
1
3 ΣT.ΣQ, where ΣT and ΣQ are the internal thermal resis- period (t0 -t1 ) can be given as:
tance and capacitance of the cable. Short duration transients Z t1
for different cable types last anywhere between 10 minutes to Λt −1
x(t) = P e P x(t0 ) + P e B Λt
e−Λt dt (34)
1 hour. This paper focuses on transients lasting longer than 1 t0
hour that is the long transients. For long duration transients,
van Wormer coefficient p to divide the insulation is given by where Λ is the diagonal matrix made of the eigenvalues of
[1]: the matrix A and P is the left eigenvector.
1 1 As discussed, the initial value of the state vector (x(t0 ))
p=  − 2 (30)
Deins Deins can be calculated during the steady-state conditions using the
2ln −1 available real-time estimates of the conductor temperature.
Diins Diins
Using the steady-state condition x0 = 0, and substituting the
where, Deins and Diins are the external and internal diameter value of conductor temperature (θc ) and the known ambient
of the insulator. temperature (θa ), (31) can be rewritten as a system of lin-
ear equations of form (21). Initial Values of unknown state
B. Transient Thermal Analysis variables are estimated using the solution given by (22) and
utilized in (34).
The state variables of interest are the temperature of the
The solution of the complete TEE model of a cable and the
conductor, screen, jacket and the multiple soil layers. The rate
surrounding soil was verified by comparing it to the solution
of change of the state variables can be described by the set of
given by a FEM based model created in the commercial
equations:
software Comsol Multipysics 5.4. For demonstration purpose
1 θc − θscr


0 a single-phase 10 kV cable was modelled with four layers. A

 θc = Wc + Wd1 −


 Q1 T1 copper conductor, an XLPE insulation, a Lead alloy sheath as

 0 1 θ c − θscr θs − θj
screen and jacket made up of PVC. The cross section area of
θs = Ws + Wd2 + −


Q3 T1 T3 the conductor is 330 mm2 . The properties of other cable layers



 θ0 = 1 θscr − θj − θj − θs1




 j
and the surroundings are presented in the Table IV and can be
Qs1 T3 Ts1 (31) found in detail in [1]. The cable is buried at a depth of 1 m.
0 1 θ − θ θ − θs2


 θ s1 =
j s1
−
s 1
Ambient soil temperature was chosen to be 15 ◦ C. In the TEE
Qs2 Ts1 Ts2



 model the soil layer was divided into 100 equal thickness sub-

 .
.



 . layers. In the Comsol model, the soil is modelled as a rectangle

 0 1 θsN −1 − θj θs − θa
of width of 20 m and a depth of 20 m. The boundaries of the
 θsN = − N


QsN TsN −1 TsN rectangle have fixed ambient temperature.

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JOURNAL OF LATEX CLASS FILES, VOL. , NO. , MONTH YEAR 9

1.5
TABLE IV Predicted current
S PECIFICATIONS OF THE C ABLE USED IN THE CABLE MODELLING AND Rated currrent
1

Current (p.u.)

Current (kA)
THERMAL SIMULATION PROCESS . 1

0.5 0.5
Layer Dex (mm) ρth (Km/W) C (MJ/m3 K)

Insulation 30.1 3.5 2.0 0 0

0 5 10 15 20 25
Screen 31.4 - 1.47
Jacket 35.8 5.0 1.7
Soil - 1.0 2.0 Constant resistance
Varying resistance
80

Temperature (°C)
Thermal limit

60

The static rating of the single cable according to IEC 60287 40

was calculated to be 947 A. A step of rated current was given 20

0 5 10 15 20 25
for 24 hours and the conductor temperatures from each method Time (hour)
were recorded. A 24 hours step was for day-ahead planning
and assessment of thermal limits. The results are presented Fig. 7. Top: Predicted current profile in the cable. Bottom: Predicted cable
conductor temperature based on the power profile and the thermal response
in the Fig. 6. It was observed that the TEE method gives for the used TEE model
a reasonably accurate solution with a maximum deviation of
around 1 ◦ C.
Next, thermal response to a multi-step driving function section present in the Dutch MV network. The results showed
was calculated. After a steady state condition, a 28 hour that the method in the current setup was capable of monitoring
forecasted power flow as shown in the top half of the Fig. the cable temperature up to an accuracy of ±5◦ C. In this work,
7 was simulated. The initial temperatures of the screen, the skin and proximity effect were not included in estimation
jacket and other soil layers were calculated based on the problem. Their inclusion in either the temperature estimation
steady-state temperature of the conductor and the ambient or uncertainty calculation could be a task for the future.
soil temperature. The thermal response was calculated using Application of the real-time cable temperature monitoring
(34) and Fig. 7 presents two solutions: original state-space method in assessment of thermal state of the cable for forested
model with constant resistance and modified model with loading scenario was shown. Utilization of the TEE model was
varying conductor resistance. The rated current and maximum shown to match the FEM results. For the modelled cable the
conductor temperature are marked as constants in the plots. It uncertainty in the predicted thermal response of the cable lied
is observed that using real-time resistance based temperature within 2 ◦ C. However to simplify the demonstration process,
updates, thermal profile of cables can be predicted to allow TEE model of a single cable was used. More complex TEE
the assessment of the dynamic thermal sate of the conductor models of multiple cables laid in different formations is a task
for predicted loading scenarios. for the future. The thermal resistivity of the whole soil layer
was assumed to be the same. However different soil types
VII. C ONCLUSION and humidity levels at different layers could require multiple
This paper presented a new method for monitoring of cable thermal resistivity values. More work is also required to study
temperature using PMU data. The method is based on an additional cases where the cables cross multiple soil types.
algorithm which can estimate accurate resistance of a 3-phase Application of the presented thermal assessment method to
cable system in real-time to calculate the temperature of the facilitate DLR is most suitable for cable sections connected
cable conductor. Application of this temperature monitoring to varying wind or solar power parks where the goal is to
method was demonstrated utilizing PMU data from a cable maximize the absorption and delivery of renewable power
using the available cable infrastructure. The grid operators
could in advance analyze the thermal response of the critical
80 10
cable infrastructure and request production curtailment/storage
8
70 actions if necessary. This would help them to optimize the
6

60
capacity planning of cables.
4
Temperature (oC)

2
Difference

50
0
40
-2 ACKNOWLEDGMENT
30 -4

-6 This work has received funding from the European Unions

20 TEE Including Soil
COMSOL -8 Horizon 2020 research and innovation programme MEAN4SG
Difference
10
0 5 10 15 20 25
-10 under the Marie Skodowska-Curie grant agreement 676042.
Time (hours)
The authors would like to thank Arjen Jongepier for the
Fig. 6. Comparison of the thermal response of the combined TEE model with
PMU data obtained in the Enduris 50 kV grid. The authors
the response from the FEM based model simulated in Comsol. The driving would also like to thank Dr. Helko van den Brom and Dr. Gert
step function is the rated current for the cable. Rietveld from VSL for their support.

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Transactions on Power Delivery
JOURNAL OF LATEX CLASS FILES, VOL. , NO. , MONTH YEAR 10

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