Calculation and analysis of the steady-state line ampacity weather sensitivity coefficients

Electric Power Systems Research 181 (2020) 106181

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Calculation and analysis of the steady-state line ampacity weather sensitivity coefficients

T

Alen Pavlinića,*, Vitomir Komenb a b

HEP ODS d.o.o., DP Elektroistra, Vergerijeva 6, Pula, Croatia HEP ODS d.o.o., DP Elektroprimorje, Viktora Cara Emina 2, Rijeka, Croatia

ARTICLE INFO

ABSTRACT

Keywords: Dynamical line rating systems Line Line ampacity Sensitivity analysis Power system computing

The continuous increase in electrical power demand entails the need for larger ratings of overhead lines. For spatial, economic and ecological reasons, contemporary solutions are focused on incrementing the ratings of existing lines. One of these solutions is the real-time determination of line ampacity according to the measured or predicted weather and/or line parameters. The paper is focused on a novel calculation procedure and analysis of the steady-state line ampacity (SSLA) weather sensitivity coefficients (WSCs). The calculation of WSCs is based on the sensitivity analysis of the methodology for the SSLA given in Cigre N° 601 Technical Brochure. On the one hand these WSCs allow a fast estimate of the potential in the introduction of real-time ampacity calculation concept. On the other hand they give a good basis on the requirements for the measurement equipment installed in contemporary dynamical line rating systems (DLRSs). The interrelations between the WSCs and the SSLA are investigated using correlation and regression analysis. Finally, according to the typical design parameters of overhead lines the WSCs and their interrelations with the SSLA are calculated and analysed for typical aluminium steel-reinforced conductors (ACSRs) present in the Croatian transmission system.

1. Introduction The line ampacity is a limitation in the duration and value of line current with the goal to restrict the conductor temperature under the maximum allowable limit (defined by the conductor manufacture). Conductor temperature is restricted in order to limit clearance between the conductor and ground, clearance to other conductors, and for the protection of overhead conductor tensile strength loss or permanent conductor damage caused by heat. Furthermore, the term line ampacity could refer to the steady-state or short-term (dynamical) line ampacity. The steady-state line ampacity (SSLA) is calculated from the steadystate heat balance equation under the assumption of thermal equilibrium [1–12]. Differently, the short-term line ampacity is calculated from the conductor non-steady heat equation under the assumption of no thermal equilibrium [13–16]. The methodologies for the calculation of line ampacity are based on standard methodologies [1–16] or on complex numerical modelling [17–20]. Additionally, the main differences in the methodologies for calculation of line ampacity are present in the calculation of alternating current (AC) resistance [1,2,21,22], convective cooling [1,2,23,25], and in the neglection of the radial and axial conductor temperature distribution [18,26,27]. Moreover, the majority of the methods neglect ⁎

the heating due to corona and evaporative cooling in the calculation of the SSLA. The corona heating can be significant at times of high humidity and high wind speeds, but it is normally irrelevant for SSLA calculation because at that times the convective effects are much more important. The cooling due to evaporation is usually neglected due to the fact that is unlikely for the entire line to be wet and the difficulty of assessment [1,11,28,31]. In this paper the SSLA is calculated using the standard methodology given in Cigre Technical Brochure N° 601 (Cigre 601) [1]. This method, along with the IEEE 738-2012 standard methodology [2], is widely used in contemporary DLRSs for the line ampacity (thermal rating) calculation [32–35]. For the calculation of the alternating current (AC) resistance the methodology given in Appendix A of Cigre Technical Brochure N° 345 [22] is used. In this paper the radial and axial conductor temperature distributions are ignored. The main focus of this paper is the analysis of the SSLA WSCs. This concept represents a form of sensitivity analysis of the SSLA. The sensitivity analysis of the line ampacity is discussed in Refs. [1–3,7,29–31], although without the introduction of the WSCs. The WSCs are a novel concept in the analysis, and are calculated as partial derivatives of the SSLA. Four different WSCs are calculated to analyse the sensitivity of the SSLA on: wind speed (v), wind angle (δ), ambient temperature (Ta)

Corresponding author. E-mail address: [email protected] (A. Pavlinić).

https://doi.org/10.1016/j.epsr.2019.106181 Received 24 March 2019; Received in revised form 7 December 2019; Accepted 26 December 2019 Available online 06 January 2020 0378-7796/ © 2019 Elsevier B.V. All rights reserved.

Electric Power Systems Research 181 (2020) 106181

A. Pavlinić and V. Komen

and global radiation intensity (S). To analyse the interrelations between the WSCs and the SSLA regression and correlation analysis are used. The correlation between the WSCs and the SSLA is studied by the Pearson correlation coefficients (Spearman coefficients gave similar results) [36]. To find the dependency of the SSLA on the WSCs or on the WSCs and D nonlinear regression using polynomial approximations of the WSCs is used [37]. The correlation and regression analysis were performed in three different regions depending on the valid convective cooling formula of calculation (determined mainly by the value of v). Finally, the paper for common design parameters of overhead lines explores on three ACSRs of cross section 240/40 mm2, 367/57 mm2 and 490/65 mm2 the values of the WSCs. The results are discussed in terms of the numerical values of the WSCs, and on the introduced errors of the equipment in weather DLRSs. Moreover, after obtaining the numerical values of the WSCs the interrelations between the WSCs and the SSLA are investigated by determining the numerical values of Pearson correlation coefficients, and coefficients of the approximated nonlinear regression function which defines the dependency of the SSLA on the WSCs. Moreover, numerical values of the extended regression function coefficients that define the dependency of the SSLA on the WSCs and outer conductor diameter (D) are also calculated.

Thus, the solar heating is dependent on D, the solar absorptivity of conductor surface (αs), and S. The parameter αs changes with aging and upon the environment of the conductor. The value of αs varies from around 0.2 for a bright new conductor to around 0.9 for a weathered conductor in an industrial environment. Moreover, αs could be measured on real overhead lines [1,39,40]. 3) Convective cooling (Pc) Convective cooling is the most important cooling element of overhead line. Generally, two convective cooling mechanisms are present [1,2,23–25]. The first one is natural convective cooling which occurs when no wind speed is present. This type of cooling occurs due to the follow of air caused by a higher temperature of the conductor in respect to Ta (valid when the line is energized). The second one is forced convection which is caused by a flow of wind of certain speed and angle. Natural convection is dominant at low wind speeds, while forced for higher values. In this paper the calculation of convective cooling is done according to Cigre 601, and is based on two Nusselt numbers for natural and forced convection. The following equations are used for the calculation of convective cooling [1]:

Pc1 =

Pc 2 =

2. Mathematical model for the calculation of the SSLA

Pc

(1)

Pr = 0

In Eq. (1) the balance between the heat gained (Joule, magnetic and Solar heating) and lost (convective and the radiative cooling) is assumed, and all the elements are calculated per unit length due to the neglection of the axial conductor temperature distribution. The components of Eq. (1) could be described as follows: 1) Conductor alternating current (AC) resistance (Rac) In case of ACSRs Rac is dependent on the direct current (DC) resistance (Rdc), the conductor temperature (Tc), skin effect, core losses due to hysteresis and eddy currents and the transformer effect [1,21,22]. The proximity effect is commonly ignored for ACSRs due to large distances of overhead line conductors [38]. In this paper Rac is calculated as follows [1,22]:

Pr =

Ta) A (Gr Pra )m (1

(Tc

Ren1

Ta) B1

1.76 10

(A1 + B2 sin(

6

2.5)

)m1)

(5) (6)

D [(Tc + 273) 4

(7)

(Ta + 273) 4]

The radiative cooling is dependent on Tc, Ta, D and on the conductor emissivity (ε). The parameter ε changes over time and rages from 0.2 to 0.9. This parameter could also be determined by measurements [41]. The SSLA is calculated from Eq. (1) assuming Tc equal to the maximum allowable conductor temperature (Tmax). In case of ACSR with even number of aluminium layers the SSLA may be calculated by the following equation:

(2)

R ac= (1 + k s+ km + kred ) RdcTc

f

(Tc

The natural convection is calculated according to Eq. (5) and is dependent on D, Ta, conductor temperature (Tc), elevation above ground (y), and angle of conductor inclination on the horizontal (β). The forced convection is calculated according to Eq. (6) and is apart from the parameters mentioned for the natural convection (except β) dependent on v, δ and diameter of aluminium layers (d). 4) Radiative cooling (Pr) The radiative cooling is the total heat of the conductor radiated to the sky and to the ground and surroundings. According to the Stefan–Boltzmann law and assuming the surrounding with infinite surface this element may be calculated as [1–12]:

In all methodologies which ignore the radial and axial conductor temperature distribution the SSLA is calculated from the steady-state heat balance equation defined as follows [1–12]:

g = I 2 Rac + Ps

f

In case of ACSRs with even number of aluminium layers due to small values of core losses and transformer effect (coefficients km and kred) only the skin effect (ks) is taken into account [1,21,22]. For ACSRs with uneven number of aluminium layers due to the transformer effect Rac is dependent on the line current (I). This dependency (Rac(I)) is nonlinear and calculated according to Eq. (2) for different values of I. In this paper to obtain the WSCs a spline approximation of Rac(I) on a small interval [Il, Ih) is used. This approximation is expressed as follows:

In case of ACSRs with uneven number of aluminium layers due to the nonlinear dependence Rac(I) the SSLA is obtained by an iterative procedure on Eq. (1). In this paper the Newton–Raphson method was used [42]. This method calculates the value of the line current in each iteration by the following equation:

R ac= Rac (Ih)

I I+1 = I i +

Ih

Rac (Il ) Il

(I

Il ) + Rac (Il ), Rac

[Il , Ih)

IR =

(3)

2) Solar heat gain per unit length (Ps) The sun emits radiation which heats the overhead line conductor. Two different methodologies for the calculation of this element are present. One based on the reflective, diffusive and direct solar radiation [1,11]. The other one is simplified and based only on S [1]. In DLRSs due to cost effectiveness the measurement of S with pyranometers is extensively used. Moreover, the simplified method based on S is adequately accurate, and some constants in the complex method are hard to determine for real overhead lines (like reflectance of the ground) [1,2,39,40]. For all of these reasons the method based on S is used, and this element is calculated as follows:

Ps =

s

DS

Pc (Tmax , v, , Ta) + Pr (Tmax , Ta) Rac (Tmax )

Ps (S )

g (I i ) g (I i )

(8)

(9)

In Eq. (9) g′ represent the partial derivative of g in respect to I. At the end of the numerical procedure the value of IR is obtained. The whole process of calculation is presented in Fig. 1. In Fig. 1 Nu1 and Nu2 represent the Nusselt numbers for natural and forced convection, which also define two forms of Eq. (1) according to the type of convection (g1 and g2). The other introduced parameters are needed for the iterative procedure of calculation (the indicated values were used for the calculations in the case analysis). 3. Mathematical model for the calculation of SSLA WSCs

(4)

The base for the determination of the SSLA WSCs is the differential 2

Electric Power Systems Research 181 (2020) 106181

A. Pavlinić and V. Komen

wind speed from 1.4 m/s to 10 m/s. Thus, for very low wind speeds (less than 0.1 m/s) the natural convection is always dominant, while for very high wind speeds (above 2.2 m/s) the forced convection with B1 = 0.048, n1 = 0.8 is valid. Between these two limiting wind speeds (between 0.1 m/s and 2.2 m/s) there is a mixture of natural and the two defined types of forced convection depending on the value of Tmax, δ, Ta, β, D, y, d, and v. To study the strength of the relationships between the WSCs and SSLA the Pearson’s product-moment coefficients are explored. Pearson’s product-moment coefficient is the measurement of correlation and ranges (depending on the correlation) between +1 and −1. Positive correlation is seen if one variable increases simultaneously with the other, while for a negative correlation one variable decreases when the other increases. Pearson’s product-moment coefficient between two variables x and y is calculated according to Ref. [36]: Fig. 1. Process of IR calculation for ACSRs with uneven number of aluminium layers.

sensitivity analysis. In terms of sensitivity analysis the sensitivity coefficient is basically the ratio of change of the parameter output according to one parameter input while all the other parameters remain constant. If the partial derivative can be analytically calculated the sensitivity coefficient is represented as the partial derivative of the dependent variable (output) in respect to the independent variable (one of the input). This is possible only if an explicit algebraic equation which describes the relationship between the independent variables and the dependent variable is known [43,44]. Assuming for an ACSR that the parameters D, d, β, αs, ε, y, Rac(Tmax) and Tmax are constant, the SSLA is only dependent on Ta, v, δ and S. Thus, in this situation the four WSCs are calculated as follows:

K v= K =

KT = K S=

IR v IR

IR Ta IR S

nt j=1

pxy =

nt j=1

(x j

(x j

xav ) (zj

xav

z av )

nt j=1

)2

z av )2

(z j

(14)

The Pearson’s product-moment coefficients, and the correlation matrix were calculated using the Matlab function corr. [45]. To find the dependency of the SSLA on the WSCs the nonlinear regression analysis was used [37]. Nonlinear regression is a form of regression in which the output (SSLA) is modelled as a function which is a nonlinear combination of the model parameters (WSCs). For the first region (natural convection dominant) the selected function for nonlinear regression is of form: 4

IR

1

+

2

Ks +

j+2

KTj

(15)

j=1

According to Eq. (15) in the first region the SSLA approximation function is modelled with a linear dependency on Ks, and a fourth order polynomial dependency on KT. This approximation function is selected due to minimal errors between the actual results and the results obtained by the regression function for the analysed cases of WPs and ACSRs. In the first region the dependency of SSLA on Kv and Kδ is neglected due to the fact that these coefficients are 0 (natural convection is independent on v and δ). For the second and third region (forced convection dominant) the selected function for nonlinear regression is more complex and of form:

(10) (11) (12) (13)

In case of ACSR with even number of aluminium layers the WSCs are obtained as partial derivatives of Eq. (8). In case of ACSR with uneven numbers the procedure for obtaining WSCs is more complex. In this paper as the dependency of Rac(I) is represented by a linear spline approximation (according to Eq. (3)) the function g defined in Eq. (1) is of cubic polynomial order. Thus, an analytic solution for IR is also possible to obtain, and the WSCs are calculated by applying Eqs. (10)–(13). The WSCs in this paper were calculated using the Matlab function diff. [45].

2

IR

1

+ j=1

2 j+1

2

K vj + j=1

j+3

Kj+ j=1

2 j+5

KTj + j=1

j +7

KSj +

10

KS (16)

KT

According to Eq. (16) in the second and third region the SSLA approximation function is modelled with a second order polynomial dependency on Kv, Kδ, KT, Ks. Additionally, the term KT·Ks is found to be very important for the correct approximation. This type of approximation function is selected to minimize the errors. To extend the regression analysis on ACRSs with different values of D the nonlinear regression which relates IR on the WSCs and D is performed according to the following expressions:

4. Correlation and regression analysis of the SSLA and WSCs In this paper the correlation and regression analysis of the SSLA and WSCs are explored in three regions. The first region includes all the cases in which the natural convective cooling is dominant (according to Eq. (5)), while the second and third region include all the cases in which the forced convection is dominant (according to Eq. (6)). The difference between the second and third region is in the value of B1 and n1 in Eq. (6). The second region has B1 = 0.641, n1 = 0.471, while the third B1 = 0.048, n1 = 0.8 (if the roughness coefficient is below 0.05 than B1 = 0.178, n1 = 0.633). For the analysed cases and ACSRs the first region includes some cases between wind speeds from 0 m/s to 1.3 m/s. The second region is valid for some cases with wind speeds from 0.1 m/s to 2.2 m/s. Finally, the third region is valid for some cases with

4

IR

D

1

+

2

Ks +

j+2

KTj +

7

D+

8

D2

(17)

j=1 2

IR

D

1

+ j=1

K S KT +

2 j+1

11

K vj +

D+

j=1

12

D2

2 j+3

Kj+ j=1

2 j+5

KTj + j=1

j +7

KSj +

10

(18)

Eq. (17) is valid for the first region and is very similar to Eq. (15) the differences are that D multiplies the right-hand side of Eq. (15), and the 3

Electric Power Systems Research 181 (2020) 106181

A. Pavlinić and V. Komen

addition of the second order polynomial dependency on D. Similarly, Eq. (18) is valid for the second and third region and is obtained by multiplying the right-hand side of Eq. (16) with D, and by adding the second order polynomial dependency on D. These changes in the approximation functions are done to minimize the errors introduced by the regression analysis, and to take into account the changes in D. The coefficients of the nonlinear regression (β coefficients with appropriate index) were calculated using the Matlab function nlinfit. [45].

pyranometers is also important. Apart from the weather stations the equipment of weather DLRSs includes: current transformers (if the current measurements are not acquired from the supervisory control and data acquisition system), auxiliary power supply, data loggers and a server [33,50]. The auxiliary power supply usually consist of a battery, voltage regulator and photovoltaic modules. The data loggers are located in the control boxes mounted on the line poles and communicate using wireless technology or wired connections with the weather measurement sensors. They also serve to partially store and process the sensor data. Finally, the data is sent to the server with a communication device using radio or mobile networks. The server except the hardware is equipped with proprietary software to configure the data loggers and to calculate the SSLA (based on worst measured WPs and external WP date). Apart from the SSLA the software may also calculate SSLA predictions, sags, and clearance violations [35,39,40,48].

5. Equipment of weather DLRS Different types of DLRSs are used for the real-time determination or prediction of the SSLA. A good review of existing DLRSs is given in Refs. [33–35]. The weather DLRSs calculate the SSLA from Eq. (1) according to the real-time measurements or predictions of weather parameters (WPs) and are based on standard methodologies for SSLA calculation [1–12]. At first these systems used local measurements from weather stations mounted on the overhead line poles. In this manner a point measurement of WPs is obtained. In order to take into account the axial variations of WPs multiple weather station are installed along the line. The installation of meteorological stations in all overhead line spans is impractical and uneconomic, and thus the concept of critical span is introduced. In the context of critical spans weather stations are placed on poles in sheltered regions where low wind speeds values and/or parallel wind speed direction are expected. Additionally, in some cases, the basic critical span setting criteria is the risk of clearance violations and some practical limitations (available communications links, buildings etc.) [39,40,46,47]. With the advancement of technology, the concept of solely local measurement was extended. On the one hand from point measurements with correlation techniques the axial distribution of WPs between weather stations is obtained. On the other hand based on external WPs data the axial WPs variations are obtained by the use of numerical weather predictions within the Limited area region. The main focus of contemporary weather DLRSs is in the use of external data [47]. For the measurements in the weather stations the following is valid: The wind speed is measured by cup type or propeller type or ultrasonic anemometers. In case of propeller type anemometers the wind direction is measured by the integral wind wane, while for the other types a separate wind wane is needed. The general accuracy of wind speed measurements with anemometers used for DLRSs is of up to ± 1–2%, with a resolution of 0.01–0.1 m/s and range 0−75 m/s. The accuracy of wind angle measurements is of ± 0.5–3° with a resolution of 0.1°–1°. The stated accuracies may even be worse for different wind and ambient parameters [35,40,48]. As v and δ have a high impact on SSLA and can change abruptly in a short time interval along the overhead line conductor (possibility of wind vortex of up to 12 m/s) different methods are used to reduce the variability and accuracy of SSLA. These methods include averaging v and δ on some time intervals (typically 10 min), assuming a fixed δ of less than 25°, and multiplying v with a coefficient which takes into account the measurement errors and the distance of the measurement equipment from the line conductor [39,40,47]. The ambient temperature is measured by temperature sensors. The most commonly used temperature sensors in DLRSs applications are thermistors, thermocouples, and resistance thermometers. These sensors for common Ta values have accuracies of ± 0.1–1 °C, and a resolution of temperature measurement less than 0.1 °C. The accuracy of the temperature sensor is also temperature dependent [35,48].The ambient temperature is fairly constant along the overhead line (differences less then ± 1° within the ruling span section), and instantaneous values of Ta are used for the calculation of SSLA [40]. The global solar radiation intensity is measured by pyranometers. These sensors have a total accuracy estimated to 2–5% [35,49]. As S could change rapidly (cloud appearance) the time response of

6. Case study and results 6.1. Input data and remarks on the performed calculations The input parameters for ACSR 240/40 mm2, 360/57 mm2 and 490/65 mm2 geometry, and for the AC resistance calculation were taken from references [22,51]. The frequency is assumed as 50 Hz. The other parameters which are constant in all the calculations have the following values: y = 0 m, αs = ε = 0.9, β = 0° and Tmax = 80 °C. The static SSLAs (ISR) of the considered ACSRs are: 605 A, 780 A and 950 A. These values of line current are permanently allowed to flow into the considered ACSRs, and calculated assuming a set of worst-case WPs according to Ref. [39]. In the calculation of the SSLAs and WSCs a total of 21,304,738 cases were analysed. To achieve this the WPs were changed in the following manner: v from 0 m/s to 10 m/s with a step of 0.1 m/s, Ta from −20 °C to +40 °C with a step of 1 °C, δ from 0° to 90° with a step of 1°, and S from 0 W/m2 to 1480 W/m2 with a step of 40 W/m2. These changes of WPs practically cover all the situations in the design range of the overhead lines (an exception is the limitation of v to 10 m/s due to the calculation of extremely large values of SSLA). In the numerical Newton–Raphson method for the SSLA calculation of ACSRs with three aluminium layers the allowed error was of 0.001. All the calculations were performed in Matlab. 6.2. Results of SSLA and SSLA potential In Fig. 2 the calculated values of the SSLAs are presented. The analysed cases were sorted to arrange the values of the SSLAs for ACSR 240/40 mm2 from minimum to maximum values. According to Fig. 2

Fig. 2. Calculated values of SSLA (IR). 4

Electric Power Systems Research 181 (2020) 106181

A. Pavlinić and V. Komen

Table 1 Statistic values of KT. ACSR

Type of value

KT (A/°C) Ta

2

240/40 mm

360/57 mm2 490/65 mm2

Minimum Maximum Average Minimum Maximum Average Minimum Maximum Average

−20–0 °C

0–20 °C

20–40 °C

−14.04 −3.58 −8.7 −18.22 −4.62 −11.28 −22.13 −5.62 −13.7

−15.98 −4.08 −9.82 −20.74 −5.29 −12.74 −25.18 −6.45 −15.48

−19.68 −4.83 −11.79 −25.58 −6.28 −15.32 −31.03 −7.67 −18.64

Fig. 3. Percentage of cases in percentual intervals of ISR. Table 2 Statistic values of KS.

for an ACSR 240/40 mm2 the range of SSLAs is of 292 A–2368 A. Similarly, for an ACSR 360/57 mm2 this range is of 370 A–3072 A, while for an ACSR 490/65 mm2 of 452 A–3749 A. To observe the potential of implementing weather DLRSs all the cases in Fig. 2 were divided in percentual intervals of the static SSLA (ISR) and the percentage of cases in each interval was calculated. The results are shown in Fig. 3. From Fig. 3 it is possible to conclude that only about 3% of cases have smaller values of SSLA in respect to ISR. In 52–55% cases the calculated SSLA is in the interval of 100–120% of ISR. In other intervals, the percentage of cases varies from 5% to 10%. Therefore, assuming that each case has equal probability, it is most likely to increase the SSLA up to 20% in respect to ISR. These results are in good accordance with the practical measurements given in Refs. [52,53].

ACSR

Type of values

KS (A/((W/m2)) S

240/40 mm2 360/57 mm2 490/65 mm2

Minimum Maximum Average Minimum Maximum Average Minimum Maximum Average

0–520 W/ m2

520–1000 W/ m2

1000–1480 W/ m2

−0.142 −0.028 −0.056 −0.191 −0.038 −0.076 −0.240 −0.047 −0.095

−0.169 −0.028 −0.058 −0.228 −0.038 −0.079 −0.289 −0.047 −0.098

−0.227 −0.028 −0.06 −0.312 −0.038 −0.082 −0.403 −0.048 −0.103

each WSC are presented in Tables 1–4. The calculated values of KT shows that an increase of Ta by 1 °C results on average in a decrease of the SSLA from 8.7 A to 18.64 A. The negative values of KT are higher for greater conductor cross sections and higher values of Ta. The maximum negative values of this

6.3. Results of WSCs To analyse the impact of the WPs on the SSLA the WSCs for each case were calculated according to Eqs. (10)–(13). The calculated values of the WSCs are presented in Fig. 4. The calculated statistics values of

Fig. 4. Calculated values of the WSCs in the analysed cases. 5

Electric Power Systems Research 181 (2020) 106181

A. Pavlinić and V. Komen

a maximum error from approximately 0 A to 82.43 A is expected for the wind angle measurement. In percentage of the SSLA this error is from 0% to 5.87%. The measurement error increases for larger conductor cross sections and lower values of δ. The sensitivity coefficient of wind speed shows that an increase of v by 1 m/s results on average in an increase of SSLA from 69.95 A to 244.92 A. The values of this coefficient are higher for lower values of v and greater conductor cross sections. At very low wind speeds (0.1–0.2 m/s) there is a discontinuity of the partial derivative due to the change of natural to forced convection. In this interval extremely high values of the coefficient are noted. In the range of v from 0.1 m/s to 2.2 m/s some discontinuities are also present due to the change from natural to different types of forced convection. The maximum changes of this coefficient ranged from 104 A/(m/s) to 1041.41 A/(m/s). These values are extremely high compared to other WSCs. In reality the wind speed may suddenly change along the line causing a substantial change of the SSLA. For example if Kv of 230 A/(m/s) is selected then for a change of v from 4 to 2 m/s an approximate decrease of 460 A in the SSLA is obtained. Thus, in weather DLRSs the right measurement or estimation of wind speed is of crucial importance especially for lower values of v. Assuming an error of wind speed measurement of 2% a maximum measurement error from approximately 0 A to 28.88 A is expected. In percentage of the SSLA this error is from 0% to 1.12%. This error increases for larger conductor cross sections and lower values of v.

Table 3 Statistic values of Kδ. ACSR

Type of value

Kδ (A/°) δ

2

240/40 mm

360/57 mm2 490/65 mm2

Minimum Maximum Average Minimum Maximum Average Minimum Maximum Average

0–30°

30–60°

60–90°

0 17.51 8.57 0 22.67 11.02 0 27.48 13.3

0 11.32 4.84 0 14.67 6.23 0 17.79 7.53

0 5.5 1.58 0 7.12 2.04 0 8.65 2.46

coefficient ranged from 14.04 A/°C to 31.03 A/°C. Assuming a ambient temperature change from 20 °C to 10 °C along a long line with KT = −9/°C this would approximately increase the SSLA by 90 A. Regarding the measurements in DLRSs assuming the worst-case temperature sensors accuracy of 1 °C, a maximum error from 3.58 A to 31.3 A is expected due to the Ta measurement. In percentage of the calculated SSLA this is from 0.41% to 4.73%. This error increases for larger conductor cross sections and higher values of Ta. The calculated values of KS show that an increase of S by 1 W/m2 results on average in a decrease of the SSLA from 0.056 A to 0.103 A. The negative values of this coefficient increase with S, and the conductor cross section. The maximum negative changes of this coefficient ranged from 0.142 A/(W/m2) to 0.403 A/(W/m2). At first the impact of S on SSLA seems low. Although, in reality S could change in a wide range of values, that could lead to a substantial decrease of SSLA. For example if S is increased from 520 W/m2 to 1000 W/m2 with KS = −0.1 A/(W/m2) this approximately reduces the SSLA by 48 A. Thus, the correct measurement of S is very important. Assuming a pyranometer accuracy of 5% the maximum error introduced in the calculation of SSLA is from approximately 0 A to 29.89 A. In percentage of the SSLA this error is from 0% to 6.6%, and increases for larger conductor cross sections and higher values of S. The maximum percentage error is quite high because at the maximum calculated error of 28.89 A the value of the SSLA was low. The sensitivity coefficient of wind angle shows that an increase of δ by 1° results on average in an increase of SSLA from 1.58 A to 13.3 A. The values of this coefficient are higher for lower values of δ and greater conductor cross sections. The maximum changes of this coefficient ranged from 5.49 A/° to 27.48 A/°. The minimum values are zero due to the fact that natural convection is independent on δ and v. In reality the wind angle may suddenly change along the line which can substantially affect the SSLA. For example if Kδ of 13.3 A/° is selected, then for a change of wind angle from 30° to 0° an approximate decrease of 399 A in the SSLA is obtained. Thus, in weather DLRSs the right measurement or estimation of wind angle is important especially for low values of δ. Assuming a maximum error of angle measurement of 3°

6.4. Results of correlation and regression analysis For the first region, where the natural convection is dominant the correlation matrix is given in Table 5. According to the results strong positive Pearson’s correlation coefficients from 0.965 to 0.969 are calculated for the relationship between IR, KT and IR, KS. Moreover, a strong positive correlation of 1 is present between KT, KS. Thus, in this region the SSLA rises by increasing the values of KT and KS. For the second and third region, where the forced convection is dominant the correlation matrix is given in Table 6. According to the results strong positive Pearson’s correlation coefficients from 0.945 to 0.962 are calculated for the relationship between IR, KS. Between IR and the other WSCs the correlation is weak. This fact is particularly correct for Kv (coefficients from −0.097 to −0.242) and Kδ (coefficients from −0.075 to −0.181). Moreover, between IR and KT in the second region the correlation is positive (coefficients from 0.27 to 0.513) while in the third region the same correlation is negative (coefficients from −0.301 to −0.363). Between the WSCs only a medium correlation is observed between Kv, Kδ (coefficients from −0.45 to −0.406), Kv, KT (coefficients from 0.319 to 0.617), and KS, KT (coefficients from −0.366 to 0.532). Thus, the results of the correlation analysis are completely different compared to the ones calculated for the first region. The calculated approximation regression function coefficients according to Eq. (15) are given in Table 7 (expressed in terms of the corresponding ACSR D in m). Notable differences between the coefficients are present depending on the ACSR cross section. These approximation coefficients lead in the regression analysis to the best adjusted root mean error squared (R-Squared) of 1. This approximation function could be used for a fast estimate of the SSLA or SSLA potential. For example assuming for an ACSR 240/40 mm2 the value of KT = −6.83 A/°C and KS = −0.12 W/m2 the calculated value of the SSLA with the approximation function is equal to 544.12 A, while the actual value of the SSLA was 544.44 A. The calculated approximation regression function coefficients for the second and third region according to Eq. (16) are given in Table 8 (expressed in terms of the corresponding ACSR D in m). According to Table 8 high differences in value between the coefficients are present depending on the region and the ACSR cross section. These approximation coefficients lead in the second region to the

Table 4 Statistic values of Kv. ACSR

Type of value Kv (A/(m/s)) v 0–2 m/s 2–4 m/s 4–6 m/s 6–8 m/s 8–10 m/s

240/40 mm2 Minimum Maximum Average 2 360/57 mm Minimum Maximum Average 490/65 mm2 Minimum Maximum Average

0 1141.48 155.4 0 959.98 200.90 0 1100.01 244.92

43.53 230.28 130.56 71.14 298.19 171.61 86.35 361.03 208.08

44.87 155.63 98.5 57.93 201.60 127.5 70.28 244.29 154.47

38.55 123.08 80.99 49.81 159.45 104.87 60.43 193.36 127.05

34.18 104.04 69.95 44.18 134.79 90.59 53.61 163.58 109.77

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Table 5 Correlation matrix for first region.

Table 7 Values of regression function coefficients for first region according to Eq. (15).

ACSR

WSC

KT

KS

IR

240/40 mm2

KT KS IR KT KS IR KT KS IR

1.000 1.000 0.968 1.000 1.000 0.967 1.000 1.000 0.966

1.000 1.000 0.969 1.000 1.000 0.967 1.000 1.000 0.965

0.968 0.969 1.000 0.967 0.967 1.000 0.966 0.965 1.000

360/57 mm2 490/65 mm2

Coeff.

ACSR 240/40 mm2 (D = 21.9 × 10−3 m)

ACSR 360/57 mm2 (D = 26.6 × 10−3 m)

ACSR 490/65 mm2 (D = 30.6 × 10−3 m)

β1 β2 β3 β4 β5 β6

123477.4·D 223475.6·D 30649.3·D 4690.9·D 307.5·D 7.8·D

128376.8·D 183659.9·D 23683.3·D 2765.9·D 137.1·D 2.6·D

131344.6·D 278537.5·D 16596.9·D 1776.3·D 70.1·D 1.1·D

absolute percentage errors ranged from 3.08% to 9% for all the approximations. For the purpose of analysis it is worth mentioning that the first region included (depending on the ACSR cross section) from 2.85% to 3.57% of all the analysed cases. In the second region this percentage was from 10.73% to 15.73%, while in the third from 68.53% to 78.5%. Thus, the largest part of the cases are covered by the third region. Additionally, to use the defined approximation functions the values of the WSCs in each region should be known. The calculated statistics values of the WSCs in each region are given in Table 11. Analysing Tables 11 and 4 it could be concluded that the results for Kv from approximately 0 m/s–2 m/s (in Table 4) are divided between the first two regions and the third region statistically covers the other cases. For the other WSCs it is hard to asses for which WPs value a certain WSC is in the defined region (in each region all the combination of other WPs except v are present). To define a value of the WSC according to the corresponding WP one possibility is the use of Fig. 4. Although, the right selection of the approximation function for the SSLA calculation is still a problem. One way to overcome this problem is to check for the desired combination of WP and input data in which region the convective cooling lays. This problem of the adequate function selection is only present if the wind speed is between 0.1 m/s and 2.2 m/s. In other cases the approximation function for the first region (below 0.1 m/s) or third region (above 2.2 m/s) should be selected. Moreover, when calculating the SSLA according the approximated functions the minimal and maximal values of WSCs in the defined region according to Table 11 should not be exceeded.

adjusted R-Squared from 0.992 to 0.993, while in the third region the adjusted R-Squared was from 0.986 to 0.99. The presented approximation functions allow to calculate the SSLA and SSLA potential basing the calculations on the WSCs. For example assuming for an ACSR 240/ 40 mm2 in the second region the value of Kv = 207.71 A/(m/s), Kδ = 6.38 A/º, KT = −4.42 A/ºC and KS = −0.092 A/(W/m2) the calculated value of the SSLA with the approximation function is equal to 721.06 A, while the actual SSLA was 710.9 A. The calculated approximation regression function coefficients for the first, second and third region according to Eqs. (17) and (18) are given in Table 9. According to Table 9 high differences in value between the coefficients are present depending on the region. For these approximations the values of the adjusted R-Squared had the values of 0.997, 0.987, and 0.984. The presented approximation functions allow to calculate the SSLA and SSLA potential basing the calculations on the WSCs and D. For example assuming for an ACSR 240/40 mm2 (D = 21.9 × 10−3 m) in the second region the value of Kv = 207.71 A/(m/s), Kδ = 6.38 A/°, KT = −4.42 A/°C and KS = −0.092 A/(W/m2) the calculated value of the SSLA with the approximation function is equal to 731.67 A, while the actual SSLA was 710.9 A. To evaluate the errors introduced by the regression analysis in Table 10 the absolute percentage maximum, minimum and average errors between the actual and approximated values of the SSLA are given. According to the results given in Table 10 in the first and third region the absolute maximum and average percentage errors are higher for ACSRs with greater cross section, while in the second region these errors are lower for the same ACSRs. The minimum percentage values of the absolute error are approximately 0% for all cases. Moreover, analysing the results for the approximations given by Eqs. (17) and (18) it could be concluded that these approximations lead to higher maximum percentage error values compared to the ones given for individual ACRSs (according to Eqs. (15) and (16)). The maximum

7. Discussion of results The calculation of WSCs have clearly shown that the SSLA is from the aspect of numerical impact mostly dependent on: v, Ta, δ and S. In reality this ranking could be changed depending on the possible variations of the WPs and the line route. Moreover, due to the nonlinear

Table 6 Correlation matrix for second and third region. Region

Second region

ACSR 2

240/40 mm

360/57 mm2

490/65 mm2

Third region

WSC

Kv

Kδ

KT

KS

IR

Kv

Kδ

KT

KS

IR

Kv Kδ KT KS IR Kv Kδ KT KS IR Kv Kδ KT KS IR

1.000 −0.406 0.398 −0.220 −0.210 1.000 −0.433 0.358 −0.164 −0.155 1.000 −0.450 0.319 −0.134 −0.120

−0.406 1.000 0.288 −0.145 −0.175 −0.433 1.000 0.264 −0.152 −0.176 −0.450 1.000 0.234 −0.162 −0.181

0.398 0.288 1.000 0.285 0.270 0.358 0.264 1.000 0.423 0.405 0.319 0.234 1.000 0.532 0.513

−0.220 −0.145 0.285 1.000 0.961 −0.164 −0.152 0.423 1.000 0.962 −0.134 −0.162 0.532 1.000 0.962

−0.210 −0.175 0.270 0.961 1.000 −0.155 −0.176 0.405 0.962 1.000 −0.120 −0.181 0.513 0.962 1.000

1.000 −0.425 0.567 −0.083 −0.097 1.000 −0.430 0.598 −0.180 −0.185 1.000 −0.432 0.617 −0.244 −0.242

−0.425 1.000 0.340 −0.107 −0.137 −0.430 1.000 0.295 −0.063 −0.099 −0.432 1.000 0.267 −0.036 −0.075

0.567 0.340 1.000 −0.304 −0.301 0.598 0.295 1.000 −0.345 −0.341 0.617 0.267 1.000 −0.366 −0.363

−0.083 −0.107 −0.304 1.000 0.953 −0.180 −0.063 −0.345 1.000 0.949 −0.244 −0.036 −0.366 1.000 0.945

−0.097 −0.137 −0.301 0.953 1.000 −0.185 −0.099 −0.341 0.949 1.000 −0.242 −0.075 −0.363 0.945 1.000

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Table 8 Values of regression function coefficients for second and third region according to Eq. (16). Region

Second region

Third region 2

2

2

Coeff.

ACSR 240/40 mm (D = 21.9 × 10−3 m)

ACSR 360/57 mm (D = 26.6 × 10−3 m)

ACSR 490/65 mm (D = 30.6 × 10−3 m)

ACSR 240/40 mm2 (D = 21.9 × 10−3 m)

ACSR 360/57 mm2 (D = 26.6 × 10−3 m)

ACSR 490/65 mm2 (D = 30.6 × 10−3 m)

β1 β2 β3 β4 β5 β6 β7 β8 β9 β10

99768.3·D −4.1·D 3.98·10−3·D −59.9·D −8.5·D −317.8·D −45.3·D 1047427.1·D 3425894.8·D 2348.3·D

101897.8·D −2.97·D 2.2·10−3·D −58.8·D −4·D −151.8·D −30.6·D 756469.6·D 1717411.6·D 2537.5·D

103782.7·D −2.28·D 1.3·10−3·D −59.7·D −2.1·D −36.4·D −22.9·D 582911.4·D 985673·D 2444.6·D

158067.3·D −9.40·D 8.34·10−4·D 27.6·D −8.0·D −1518.3·D −39.6·D 2802345.7·D 16212420.8·D −13998.7·D

158156.6·D −10.78·D 2.14·10−3·D 25.2·D −5.3·D −1765.1·D −34.3·D 2020683.7·D 8507225.4·D −11961.5·D

157240.5·D −8.16·D 1.68·10−3·D 27.0·D −3.9·D −1968.2·D −31.1·D 1574956.6·D 5260801.4·D −10403.3·D

be considered. For this purposes the average WSCs could give a reasonable good insight of the potential. Extreme calculated values of the WSCs should be ignored. For example from the results of Table 4 for an ACSR of 240/40 mm2 the average increase of IR is in the region of v from 0 to 2 m/s 155.4 A per m/s, while in the region of v from 8 to 10 m/s the same increase is only of 104.04 A. Thus, with the WSCs the nature of the SSLA ampacity potential is numerically expressed in relation to the corresponding WP interval. If a direct calculation of the SSLA according to the calculated values of WSCs is wanted than the approximation functions from regression analysis must be calculated for the analysed ACSR. As beforehand defined the type of approximation function depends on the convective cooling and is calculated in three regions. For an example of the approximation function use, assume that the value of the SSLA is calculated for a 240/40 mm2 ACSR, and for v = 6 m/s, Ta = 20 °C, S = 1000 W/m2 and δ = 0° (other input data are according to subchapter 6.1). From Fig. 4 the defined values of WPs correspond to the following values of the WSCs: KT = −9.11 A/ºC, KS = −0.069 A/(W/m2), Kδ = 0 A/º and Kv = 59.86 A/(m/s). Now, as v is greater than 2.2 m/s the analysed case is in the third region. Calculating the SSLA according to Eq. (16) and using the coefficients from Table 8 leads to the value of the SSLA of 943.93 A. The actual SSLA calculated according to (2) is of 960.53 A. Thus, the result is within the maximum absolute error limit of 4.47% (for ACSR 240/40 mm2 and region 3). Similarly, using Eq. (18) and the coefficients from Table 9 the calculated SSLA is of 936.88 A. This result is also within the maximum error limit of 9%. From these results the WSCs could be efficiently used for a fast estimate of the SSLA. Moreover, the WSCs give a very good insight on how each WP affects the SSLA. Regarding the measurements the WSCs proven that in the weather DLRSs the most important is the right measurement of δ and v (especially δ) at their low values. For example looking at the values of Kδ in Table 3 it is obvious that for lower values of δ due to higher values of Kδ (if a constant accuracy of the δ is assumed) the errors introduced by anemometers are higher. Additionally, the highest percentage measurement error was calculated for the measurement of S. This high percentage is the consequence of high negative values of Ks in the

Table 9 Values of regression function coefficients for all regions according to Eqs. (17) and (18). Coeff.

Region

β1 β2 β3 β4 β5 β6 β7 β8 β9 β10 β11 β12

First region

Second region

Third region

−366.8 1501739.1 −12285.1 1162.9 43.0 0.65 6274.6 3459003.8 / / / /

−37352.4 −1.8 3.396 × 10−4 −94.7 −2.5 −329.03 −43.0 615390.5 1104979.4 2861.6 55561.6 2819853.2

−35634.3 5.0 −4.468 × 10-2 0.6 −5.4 −754.1 −5.0 1657383.7 5647795.1 −9228.1 57279.7 4923030.6

dependency of the WSCs and the corresponding WP, a recommendation is in grouping the WSCs statistic values into intervals of WPs. In all the analysed cases for a certain ACSR the values of Tmax, ε, αs, β and y were held constant. The parameter αs was taken with the worstcase value, while the value of ε was selected for a weathered conductor. Lowering αs will decrease only KS, while lowering the value of ε increases KT, Kδ, Kv, and decreases KS. Increasing the value of β would increase the value of KT and decrease the value of KS for the cases in which the natural convection is dominant (low v and δ). Increasing Tmax would increase the value of all WSCs. To verify the changes of WSCs at higher y values all the simulations were repeated with y = 2000. In case of y = 2000 the average values were for: KT from −7.28% to −8.46% lower, KS from 7.4% to 8.77% higher, Kδ from −9.25% to −9.78% lower and Kv from −6.55% to −21.39% lower. Thus, y is also an important parameter to be considered for the analysed line and point of measurement. The WSCs itself could serve as a fast estimate of real-time SSLA introduction potential. Although, in this use the intervals of WPs must

Table 10 Statistical values of the absolute percentage error between the actual and approximate SSLA values. ACSR

Absolute percentage error between the actual and approximate SSLA values First region Min. (%) 2

240/40 mm 360/57 mm2 490/65 mm2 Alla a

1.2 2.9 3.4 5.4

× × × ×

Second region Max. (%)

−4

10 10−5 10−5 10−5

3.08 3.54 3.84 5.05

Mean (%) 0.08 0.09 0.1 1.2

Min. (%)

Third region Max. (%)

−7

2.7 × 10 1.9 × 10−8 2.1 × 10−6 3 × 10−7

5.5 4.8 4.5 7.82

Approximations according to Eqs. (17) and (18). 8

Mean (%) 1.3 1.22 1.21 2.25

Min. (%) 5.4 1.4 2.5 4.5

× × × ×

−8

10 10−7 10−8 10−7

Max. (%)

Mean (%)

4.47 6.5 8.1 9

1.8 2.03 2.26 3.2

Electric Power Systems Research 181 (2020) 106181

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Table 11 Statistical values of sensitivity coefficients in first, second and third region. Region

First region

ACSR 2

240/40 mm

360/57 mm2

490/65 mm2

Second region

Third region

WSC

Min.

Max.

Mean

Min.

Max.

Mean

Min.

Max.

Mean

Kv (A/(m/s)) Kδ (A/°) KT (A/°C) KS (A/(W/m2)) Kv (A/(m/s)) Kδ (A/°) KT (A/°C) KS (A/(W/m2)) Kv (A/(m/s)) Kδ (A/°) KT (A/°C) KS (A/(W/m2))

0 0 −12.64 −0.227 0 0 −16.9 −0.312 0 0 −21.42 −0.403

0 0 −4.39 −0.078 0 0 −5.67 −0.104 0 0 −6.88 −0.13

0 0 −6.06 −0.108 0 0 −7.87 −0.145 0 0 −9.62 −0.181

43.53 0 −12.18 −0.227 61.38 0 −15.26 −0.31 78.1 0 −18.74 −0.401

1141.48 8.7 −3.58 −0.051 959.98 10.36 −4.63 −0.073 1100.01 11.65 −5.63 −0.096

180.12 2.88 −6.74 −0.083 245.64 3.36 −8.42 −0.118 308.47 3.78 −10.07 −0.152

34.18 0 −19.68 −0.146 44.18 0 −25.79 −0.219 53.61 0 −31.03 −0.296

236.9 17.51 −4.4 −0.028 349 22.68 −5.25 −0.038 457.6 27.48 −6.09 −0.47

95.37 5.64 −11.08 −0.05 127.65 7.17 −14.19 −0.069 158.34 8.6 −17.09 −0.087

region where the SSLA is the lowest. This fact could be also proven by analysing Table 11 where the minimal values for Ks are more negative for region one and two (low values of SSLA) compared with the region three. Thus, taking into account the real accuracies of the measurement equipment the introduced errors for SSLA calculation are mostly due to anemometers and pyranometers and less due to temperature sensors. In this field further research is possible taking into account the real accuracy curves of the sensors, thus taking into account the dependency of the accuracy on the corresponding WP. The calculated values in the results part assume a constant maximal accuracy for all the sensor equipment and all cases of WP. The correlation analysis has proven only a strong correlation between IR, KT in the first region, while in all the regions only a very strong correlation is present between IR, KS. This fact is due to the nonlinear dependency of IR on the other WSCs. The nonlinear regression approximation is considered to be effectively applied on the problem of connecting the value of IR to the WSCs or even more extensively on the WSCs and D. More components in Eqs. (15) and (16) or in Eqs. (17) and (18) are possible to add, but a small increase in the accuracy is obtained. If a higher accuracy in the second and third region is desired than the forced convective cooling could be divided into more than two regions which are not only dependent on B1, n1 but also on A1, B2, m1 (according to Eq. (6)). This analysis is outside the scope of this paper. It is worth to emphasize that before using Eqs. (17) and (18) the correct grouping of the ACSRs in respect to the roughness of conductor (d/(2·(D-d))) is needed. The ACSRs should be divided into two groups. The first group of ACSRs includes all those with a roughness of conductor below or equal to 0.05, while the second includes the ACSRs with a roughness of conductor above 0.05. This grouping of ACSRs is important due to the fact that the Cigre 601 methodology changes the coefficients B1 and n1 depending on the value of conductor roughness [1].

impact. Moreover, a nonlinearity of the WSCs is present for all of the coefficients, and higher positive (Kv, Kδ) or negative (KS, KT) values of these coefficients are calculated for greater conductor cross sections. According to the correlation analysis a strong correlation is present only between IR, KT in the first region, and Ks, IR in all regions. In other situations due to the nonlinear nature of the WSCs little or no correlation between IR and the WSCs (except KS) is present. Regression analysis has shown that in the cases where natural convection is dominant the dependency of the SSLA on the WSCs can be effectively modelled with a linear dependency on Ks, and fourth order polynomial dependency on KT. In the regions where forced convection is dominant the mention dependency could be modelled by a second order polynomial dependency on Kv, Kδ, KT, Ks, and an additional term KT·Ks. These approximate functions could be used for a fast estimation of the SSLA value basing the calculation on the WSCs. A very similar but more extensive model which relates the SSLA on the WSCs and D can also be effectively used for the fast estimate of the SSLA or SSLA potential. Although, this model has a slightly lower accuracy in respect to the individual regression approach. For the analysed cases and ACSRs it was clearly theoretically determined that in the majority of cases the SSLA increment is up to 20% in respect to the static SSLA. The calculated average WSCs for v ranged from 69.95 A/(m/s) to 262 A/(m/s), for Ta from −8.7 A/°C to −18.64 A/°C, for δ from 1.58 A/° to 13.3 A/°, and for S from −0.056 A/(W/m2) to −0.103 A/(W/m2). The maximum expected measurement errors expressed as percentage of the SSLA are approximately for v from 0% to 1.12%, for δ from 0% to 5.87%, for Ta from 0.41% to 4.73%, and for S from 0% to 6.6%. The absolute percentage errors between the actual and approximate SSLA values for the defined approximation regression functions ranged from nearly 0% to 9%. These errors are dependent on the analysed region and ACSR cross section. Authors’ contributions section

8. Conclusion

Alen Pavlinić and Vitomir Komen invented the idea of the paper. Alen Pavlinić developed the theory, performed all the computations and presented the results. Vitomir Komen verified the analytical methods, and the results of the paper. Finally, the authors discussed the results and contributed to the final manuscript version.

The paper introduced a novel concept of the SSLA WSCs. These WSCs are mathematically calculated as partial derivatives of the SSLA in respect to v, Ta, δ, S, and assuming constant all the other parameters required for the SSLA calculation. The WSCs give a good insight of the potential in the introduction of the real-time ampacity concept, because they numerically express the rise of IR on the corresponding WP. Additionally, the WSCs could be effectively used for the determination of the errors introduced by the measurement equipment used in the weather DLRSs. The sensitivity analysis clearly shown that v is the most influential WP on the SSLA. The other WPs (Ta, δ and S) have a lower

Conflict of interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Electric Power Systems Research 181 (2020) 106181

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Appendix A Table A1 Table A1 List of symbols. Symbol

Name

Measurement unit

A, m αs B1, n1, A1, B2, m1 β

Constants of natural convective cooling Conductor surface absorptivity coefficient Constants of forced convective cooling

/ / /

Angle of conductor inclination on the horizontal Regression function coefficients Diameter of outer layers Outer conductor diameter Wind angle Error in the Newton-Raphson method emissivity coefficient of conductor surface Steady-state line ampacity heat balance equation Grashof number i-th iteration Line current Static line ampacity Steady-state line ampacity High value of current for spline approximation Low value of current for spline approximation j-th power, index of summation Thermal conductivity of the air Coefficient of core losses due to hysteresis and eddy currents Skin effect coefficient Redistribution effect coefficient Sensitivity coefficient of wind angle Sensitivity coefficient of global solar radiation Sensitivity coefficient of ambient temperature Sensitivity coefficient of wind speed Maximum number of iteration Sample size Convective heat lost per unit length Prandtl number Radiative heat lost per unit length Solar heat gain per unit length Conductor AC resistance per unit length Conductor DC resistance per unit length at 20 °C Conductor DC resistance per unit length at conductor temperature Stefan–Boltzmann constant Global solar radiation intensity Ambient temperature Conductor temperature Maximum allowable conductor temperature Wind speed Value of first variable Average value of first variable Elevation of conductor above sea level Value of second variable Average value of second variable

[°]

β1,…,β10 d D δ err ε g, g1, g2 Gr i I ISR IR Ih Il j λf km ks kred Kδ KS KT Kv maxiter nt Pc Pra Pr Ps Rac Rdc RdcTc σ S Ta Tc Tmax v x xav y z zav

10

/ [m] [m] [°] / / / / / [A] [A] [A] [A] [A] / [W/(K m)] / / / [A/(°)] [A/(W/m2)] [A/(°C)] [A/(m/s)] / / [W/m] / [W/m] [W/m] [Ω/m] [Ω/m] [Ω/m] [W m−2 K−4] [W/m2] [°C] [°C] [°C] [m/s] / / [m] / /

Electric Power Systems Research 181 (2020) 106181

A. Pavlinić and V. Komen

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