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Preliminary model comparison for Dynamic

Thermal Rating estimation

Emanuele Ogliari, Alfredo Nespoli, Roberto Faranda

Department of Energy

Politecnico di Milano

Milan, Italy

{emanuelegiovanni.ogliari;alfredo.nespoli;roberto.faranda}@polimi.it

Davide Poli

DESTEC

University of Pisa

Pisa, Italy

davide.poli@unipi.it

Fabio Bassi

Terna Rete Italia S.p.A.

National Dispatcing

Italy

fabio.bassi@terna.it

Abstract—The large-scale diffusion of renewable power gener-

ators contributes to trigger a crisis in power system components

that started to operate closest to their thermal limits, thus

increasing the risk of network congestions. To manage with these

contingencies induced by Renewable Energy Sources generators,

the Transmission System Operators have to implement proper

corrective actions. Transmission System Operators usually adopts

well-known physical methods, based on weather forecasts, to

estimate the temperature of the lines in order to avoid an

expensive installation of monitoring devices on their network.

However, this estimation often results in inaccurate forecasts

of the conductors’ temperature, due to many complexities and

parameters which should be considered in the physical model of

the line or to not precise weather forecasts. This paper proposes

an innovative method based on Artiﬁcial Neural Network to eval-

uate the conductor’s temperature and consequently the Dynamic

Thermal Rating in a given overhead line. The results based on

real case studies and measures, clearly show the effectiveness and

the potential of the proposed method.

Index Terms—DTR, Thermal estimation, CIGRE thermal

model, ANN, Overhead line

I. INTRODUCTION

The large-scale diffusion of renewable power generators all

over the transmission and distribution networks is leading a

global transition from a centralized to a distributed generation.

In particular, the inherent unpredictability of the generation

of wind and photovoltaic power systems has required the

system operators to schedule more reserve resources for the

instantaneous balancing between the power produced by the

generators and the one requested by the loads, including

system losses.

The increasing penetration of renewable energy sources

(RES) in power systems, together with the decentralization

of the energy production, is requiring increasing capacity

of the electrical grid in order to integrate such a variable

power generation. Therefore, in order to avoid exceeding the

degradation of the mechanical properties of the conductors

and to respect phase-to-ground clearances, maximum current

in the lines must be limited [1] [2] [3] [4].

It is well known that reliable OverHead Line (OHL) op-

eration requires, amongst other things, the rigorous satisfac-

tion of proper thermal constraints, which limit the conductor

temperature in order to keep the line sag in ﬁxed allowable

ranges, avoiding the risk of ground faults. The transmission

capacity of an OHL is the current (ampacity) that corresponds

to the maximum acceptable temperature of conductors. Hence

the maximum acceptable temperature is limited by the thermal

performance of the material. Therefore it is extremely useful

to give an accurate evaluation of the temperature of any line in

the grid, namely Dynamic Thermal Rating (DTR) of the line.

DTR, also referred to as Dynamic Line Rating (DLR), can be

done by directly monitoring the temperatures of conductors,

or uses weather and load forecasting to estimate their future

trend, in order to calculate the actual capacity of a transmission

line [5] [6]. Therefore, this can be achieved either by means

of direct measurements or by an effective estimation method.

The former implies a very expensive solution which is not

adopted, while the latter is cheaper but often can be not very

accurate. Indeed, the adopted method has to consider several

parameters affecting the temperature of the line and, at the

same time, should be not too complicated to be used. The

temperature of the conductors depends on the current in the

line and on the local weather conditions [7] [8] [9] [10].

The beneﬁts deriving by the application of DTR techniques

in existing power systems have been assessed in several papers,

which proposed different solution methodologies, including

those based on ﬁrst-order components thermal models, or

using distributed sensors for the direct measurement of the

conductor temperature [11].

These techniques, if integrated in advanced optimization

frameworks, known as Weather Condition-based Optimal

Power Flow (W-OPF) or Electro-Thermal OPF (ET-OPF),

could reliably improve the components loadability, enhancing

the congestion management ﬂexibility, and maximizing the

RES generators exploitation [12] [13] [14] [15].

The goal of this work is to compare the performances

of different methods for the forecast of the temperature of

OHL. The comparison is made between a physical model

(CIGRE) [1] [16], and a statistical method, based on Artiﬁcial

Neural Networks (ANN) [17]. Such comparison is of particular

interest, especially when the forecasted wind speed is lower

than 2 m/s, which is proven to be the range where physical

models provide a worse result [18].

978-1-7281-0653-3/19/$31.00 © 2019 IEEE

II. DTR ES TI MATI ON M OD EL S

In the scientiﬁc literature it is possible to ﬁnd several DTR

estimation models, in the following the physical CIGRE model

and a novel technique based on ANN, are presented.

A. Physical CIGRE model

In deterministic weather-based DTR procedures, the temper-

ature of conductors is estimated using a proper thermal model

of the transmission line, usually the one developed by CIGRE

[7] or by IEEE [8].

Conductor’s temperature depends on various factors such as

conductor material, diameter, superﬁcial characteristics, load

current, and weather conditions insisting on it. Once these

latter are known, the above mentioned standard allows to cal-

culate the conductor’s temperature given the line current or the

capability, given the maximum feasible operating temperature.

In physical models like CIGRE or IEEE, the thermal behavior

of the conductor is assessed based on the following ﬁrst order

dynamic equation:

dTc

dt =1

mcp

·[qs+qj−qc−qr](1)

Fig. 1. Thermal behavior of OHL Conductor [7]

where:

mcpis the conductor heat capacity per unit length, deﬁned

as the product of its speciﬁc heat capacity, cp, and the mass

per unit of length, m. If the conductor consists of different

materials, its heat capacity is the sum of heat capacities of all

the ith strands, as mcp=Pimi·cp,i.

qcis the convected heat loss rate per unit of length. Its

calculation requires a different expression depending on the

wind speed.

When the wind speed is different from zero, forced convection

occurs and the convective heat losses can be computed through

equation 2 and 3.

qc1=Kangle·1.01 + 1.35·Re0.52·kf·(Tc−Ta)(2)

qc2=Kangle·0.754·Re0.6·kf·(Tc−Ta)(3)

where Re is the Reynolds number, Tathe air temperature

and Tcthe conductor core temperature. While qc1applies for

low wind speed, qc2is for high wind speed. To properly choose

the correct one, for any wind speed both of them have to be

computed and the largest one has to be used [8].

The term Kangle accounts for the dependance of the con-

vective heat loss rate on the wind direction and it is calculated

as:

Kangle = 1.194 −cos(φ)+0.194·cos(2φ)+0.368·sin(2φ)(4)

where φrepresents the angle between the wind direction and

the conductor axis.

When the wind speed is zero, natural convection occurs and

the rate of heat loss can be computed as following:

qcn = 3.645·ρ0.5

f·D0.75

0·(Tc−Ta)1.25 (5)

In order to compute the air density, air viscosity and air

thermal conductivity are estimated at the temperature Tfilm =

0.5·(Tc+Ta)through the polynomial expressions:

µf=1.458 ·10−6·(Tfilm + 273)1.5

Tfilm + 383.4(6)

ρf=1.293 −1.525 ·10−4·He+ 6.379 ·10−9·H2

e

1+0.00367·Tfilm

(7)

kf= 2.424 ·10−2+ 7.477 ·10−5·Tfilm −4.407 ·10−9·T2

film

(8)

qris the radiated heat loss rate per unit length, calculated

as:

qr= 17.8·D0·ε·"Tc+ 273

100 4

−Ta+ 273

100 4#(9)

where εis the conductor emissivity, which depends on the

age of the conductor, and ranges between 0.27 to 0.95, with

a value of 0.5 proposed for example by IEEE, and D0is the

external conductor diameter.

qsis the solar heat gain rate per unit length and is given by

the expression:

qs=α·Qse·sin(θ)·D0(10)

θ=arcos[cos(Hc)·cos(Zc−Zl)] (11)

and Hcis the solar altitude, Zcthe solar azimuth, and Zlthe

azimuth of line. Corrected rate of solar heat gain as a function

of the altitude, Qse, equals Ksolar ·Qs, where Ksolar =A+

B·He+C·H2

e, and:

A= 1

B= 1.148 ·10−4

C=−1.108 ·10−8

(12)

Qsis given, for a clear or industrial atmosphere, by a

polynomial expression in Hc:

Qs=A+B·Hc+C·H2

c+D·H3

c+E·H4

c+F·H5

c+G·H6

c(13)

where the constants A,B,C,D,E,F,G,Hare shown

in Table I

TABLE I

POLYNOMIAL COEFFICIENTS FOR Qs

Clear atmosphere Industrial atmosphere

A−42.2391 A53.1821

B63.8044 B14.2110

C1.9220 C6.6138 ·10−1

D3.46921 ·10−2D−3.1658 ·10−2

E−3.61118 ·10−4E5.4654 ·10−4

F1.94318 ·10−6F−4.3446 ·10−6

G−4.07608 ·10−9G1.3236 ·10−8

Finally, qjare joule heat losses, i.e. the contribution of

thermal energy loss for Joule effect. This term depends on the

electric resistance of the conductor at the temperature Tavg

and on the square of line current I:

qj=I2·R(Tc)(14)

The conductor resistance R(Tc)depends on the system

frequency, the line current and the temperature. In the CIGRE

and in the IEEE standards, the electric resistance is assumed

to change only with respect to temperature Tc. Given a value

of electric resistance for high temperatures, Th, and one for

low temperatures, Tl, it is possible to derive a linear model

for electric resistance as a function of conductor temperature:

R(Tc) = R(Th)−R(Tl)

Th−Tl·(Tc−Tl) + R(Tl)(15)

Tc

R

R(Tl)

R(Th)

TlTh

Fig. 2. Resistance-Temperature of the conductor trend

Actually, the resistance of the conductor increases with

temperature somewhat faster than the linear model as shown

in ﬁgure 2. This means that, if one calculates the resistance

for a value of Tavg between Tland Th, the value of resistance

calculated with the linear model will be greater than the real

one, and so it is conservative for the conductor temperature

calculation. Instead, if we calculate the resistance for Tavg

larger than Th, the calculated value of resistance will be lower

than the real one, whence an estimation of a non-conservative

conductor temperature value. For this reason equation 15 is

commonly adopted for a conductor temperature lower than

175 oC. Resistance value has to include variations due to Skin

Effect, Magnetic Core Effect and Radial Temperature Gradient

in the conductor.

B. Statistical ANN based model

Artiﬁcial Intelligence (AI) has been long used by the

scientiﬁc community in several branches. Among the different

forecasting techniques, Artiﬁcial Neural Network (ANN) is

one of the most used [19] [20]. The basic structural and

functional unit of ANN is the “neuron”. The inputs of each kth

neuron xiare multiplied by some properly tuned parameters

called weights wk,i then summed with the neuron bias bk, as

shown in Figure 3. Finally, the output ykis produced through

an activation function fby means of the following equation:

yk=f X

i

(xi·wk,i +bk)!(16)

Fig. 3. Neuron model

Given its ability to generalize an arbitrary complex function,

a two-layer Feed Forward Neural Network (FFNN) is adopted

in this work. For this type of architecture, neurons are orga-

nized in layers and each layer receives in input the output of

the previous layer and provides its output to the following one

as in ﬁgure 4. To properly choose its size in terms of number

of neurons, a sensitivity analysis must be performed [21].

The weights among the layers are initially randomly chosen,

then they have to be optimized using a procedure called

training; as explained and proven in [21], to reduce the error,

it is best to adopt an ensemble forecast. Hence, the algorithm

is run independently several times (trials). Each trial requires

then a newly initialized FFNN, trained randomly allocating the

available samples between training and validation set, keeping

only their numerosity constant. Finally, the obtained results

are averaged. The number of trials is then a parameter to be

optimized. A trade off between the increase of the performance

and the computational burden is necessary.

III. FORECASTING METHODS APPLICATION

The two previously described methods were implemented

and tested to estimate the conductor’s temperature of an Italian

Fig. 4. Example of a generic Feed Forward Neural Network architecture

Transmission OHL. Further details on the line such as voltage

and current are intentionally not provided on TSO request,

without loosing the generality and the validity of the proposed

comparison.

The CIGRE model is able to forecast the conductor tempera-

ture providing the weather parameters and the OHL variables

together. The coefﬁcients of the thermal model are available

in literature and do not require to be furtherly tuned. The

main drawback of this physicalmodel is that with wind speed

lower than 2 m/s, the accuracy of the model is highly

jeopardized. On the other hand, ANNs learn main trends and

correlations among the different parameters, therefore they are

able to adapt to varying conditions and to capture correlations

not explicated in the physical model, provided that they are

properly trained with historical data.

The experimental data employed in this analysis refer to the

time period of one month in which the following weather

forecast were available:

•Day of the Year

•Hour of the Day

•Ambient Temperature (K)

•Wind Speed (m/s)

•Wind Direction (◦)

•Global Solar Radiation (W/m2)

•Atmospheric Pressure (bar)

•Relative Humidity (%)

In the same time frame, the temperature of the OHL was

measured every 5 minutes.

For a coherent sampling of all the considered parameters, the

hourly mean values were calculated. In the following section,

the comparison has been performed only for the 64 samples

with wind speed below 2 m/s. These parameters represent

the input of both methods compared in this paper for the

temperature of the conductor estimation.

As CIGRE method is a physical model, the temperature esti-

mation for each sample can be obtained directly by following

the procedure described in II-A, while for the ANN method a

tuning of the statistical model parameters must be performed

in advance, as previously described in II-B. In particular,

a sensitivity analysis has been carried out to set the ANN

parameters such as, number of neurons in the hidden layers,

number of trials in the ensemble forecast and samples amount

in the training and validation set respectively. In this work

90% of the available samples are assigned to the training set,

while the remaining 10% were devoted to validate the model

similarly to what is performed in [22]; 40 trials were adopted

in the ensemble forecast of an ANN with 8 neurons in each

of the two hidden layers.

IV. PRELIMINARY RESULTS

In this section, 64 hourly samples with a wind speed lower

than 2 m/s are used for the comparison of the two estimation

methods.

The absolute errors made by the two forecasting methods,

expressed by equation 17, are compared in Figure 5:

|∆Th|=|Tm,h −Tf,h |(17)

where Tm,h is the measured temperature of the conductor

at time hand Tf,h is the forecast temperature, obtained either

with the physical or the statistical method, at the same time

h.

10 20 30 40 50 60

Selected hourly samples

0

1

2

3

4

5

6

7

8

9

10

| Th|

CIGRE

ANN

Fig. 5. Comparison of the absolute errors committed by the analysed methods

In Figure 5 it is possible to see that the ANN method

provides a very good estimation performance with respect to

the CIGRE method, giving a maximum absolute error equal

to 4.55 oC. Generally speaking, this is largely below the one

made by the physical model. In Figure 6 the outcomes of

the simulations are shown. To better highlight the comparison

between the two methods, results are here presented in terms

of the ratio between the statistical and the physical method.

On the x-axis the number of samples are reported, while

the y-axis shows on a logarithmic scale the corresponding

ratio (Rh) between the errors made by ANN (∆TANN,h) and

(∆TCI GRE,h ), as in equation 18

Rh=∆TANN ,h

∆TCI GRE,h

(18)

The blu line in Figure 6 is drawn as a reference, being the

case in which the two methods make the same forecasting

error.

As it is possible to notice observing the orange line in the

graph, the ANN outperforms the physical model, returning a

more accurate result in 60 cases out of 64.

Furthermore, as far as the four remaining cases are concerned,

10 20 30 40 50 60

Selected hourly samples

10-3

10-2

10-1

100

101

102

Rh

Rh

Fig. 6. Absolute errors of the Statistical method compared to the Physical as

the reference

which correspond to the two positive spikes in Figure 6 the

absolute error made by ANN is lower than 2oC: in those

speciﬁc cases, the error made by the physical method is very

low, so the ratio is increased. In these cases, it must be pointed

out that both estimations are very close to the measured values.

TABLE II

NUM ERI CA L RES ULTS S HOW N IN FIG UR E 6

Indicator |∆TANN (oC)| |∆TCI GRE (oC)|Rh

Min 0.01 0.08 0.14

Max 4.55 9.63 0.47

Mean 1.06 4.50 0.24

Std dev 0.93 1.90 0.49

In Table II, some aggregate statistical values are provided to

better compare the two methods. As it is possible to see, the

temperature forecast performed with the ANN is more reliable

within the described wind range, being both the average error

committed and its standard deviation lower, as well as the

maximum and minimum values.

V. CONCLUSIONS

In this work two methods for the estimation of the tem-

perature of the conductor of a given OHL are compared.

In literature the CIGRE physical model is well known and

adopted, however it shows limited accuracy for peculiar wind

speed conditions. For this reason an ANN has been trained

and the preliminary results of the two forecasting method have

been compared with wind speed less than 2 m/s.

In this condition ANN has shown a better forecasting accuracy

outperforming the physical method on 60 over the 64 hourly

samples that which have been considered, with an absolute

mean error of 1.06 oCinstead of 4.5oC. These preliminary

results encourage future works and a deeper analysis will be

extended to different transmission line typologies increasing

the number of cases under investigation.

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