US20070038396A1

US20070038396A1 - Parameter estimation for and use of a thermal model of a power line

Info

Publication number: US20070038396A1
Application number: US11/503,164
Authority: US
Inventors: Marek Zima; Petr Korba; Albert Leirbukt; Mats Larsson
Original assignee: ABB Schweiz AG
Current assignee: Hitachi Energy Switzerland AG
Priority date: 2001-12-21
Filing date: 2006-08-14
Publication date: 2007-02-15
Also published as: DE60119555D1; CA2470630A1; AU2002366845A1; US7107162B2; WO2003055028A1; EP1324454A1; DE60119555T2; ATE326073T1; US20050222808A1; BR0215234A; EP1324454B1; CA2470630C

Abstract

A relationship between a temperature T<SUB>l </SUB>of a power line or power transmission conductor 10, an electrical quantity of the power line such as a current l or power flow P through the power line, as well as meteorological quantities or ambient conditions of the power line such as wind speed W, wind direction, humidity, solar radiation S and ambient temperature T<SUB>a</SUB>, is established in the form of a thermal model of the power line. Values of the aforementioned quantities or variables are continuously measured, and the collected values of the quantities are evaluated in order to update model parameters of the thermal model during operation of the power line. In an exemplary embodiment, an average temperature representative of the entire line is determined via two phasor measurement units (PMU) 11, 11' providing synchronized phasor values from two ends of the power line. An ohmic resistance of the power line can be computed from the phasor values, from which in turn the average line temperature can be derived.

Description

CROSS-REFERENCE TO RELATED PATENT APPLICATIONS

This application is a continuation-in-part application under 35 U.S.C. §120 which claims the benefit of the filing date of allowed U.S. patent application Ser. No. 10/499,701, fled May 3, 2005 as a 35 U.S.C. §371 application of PCT/CH02/00682 filed Dec. 11, 2002, and which in return claims priority under 35 U.S.C. §119 to European Patent Application No. 1811254.0, filed on Dec. 21, 2001 in the European Patent Office, the disclosures of which are all incorporated herein in their entireties by reference.

BACKGROUND

An operation of electric power transmission lines is disclosed.
As a consequence of the electric utility industry deregulation and liberalization of electricity markets, the amount of electric power exchanged between remote regions and trading activities between different countries are steadily increasing. In addition, due to the emerging desire to optimize assets, substantially increased amounts of power are transmitted through the existing networks, occasionally causing congestion, transmission bottlenecks and/or oscillations of parts of the power transmission systems. In particular, thermal constraints can impose limitations on power flow in critical power flow paths or power transmission corridors interconnecting distinct areas. Exemplary reasons for these thermal constraints are an annealing of and/or a permanent damage to the conductors caused by severe overloads as well as an increase in conductor length with the temperature of the power line conductor. The latter may lead to unsatisfactory ground clearance due to line sag and possible flash-over to nearby trees or other line conductors, with subsequent trip by the protection system as a result.
A number of symptoms or effects relate to an elevated line temperature and therefore can influence the maximum allowable temperature of a specific electric power transmission line. Among the former are a degradation of mechanical properties of the conductors and connectors (loss of mechanical strength and integrity as well as accelerated component aging), an increase in conductor sag, an increase in resistive losses, and a potential damage to devices or equipment attached to the conductors (e.g. for power line communication).
Both the maximum allowable conductor temperature and the worst-case weather conditions used in calculating line ratings are selected by the individual network owners or independent Transmission System Operators (TSO). The operational temperature for a specific overhead power lines generally varies between, for example, 50 and 100° C.
Since the line temperature is not measurable in a straight forward way, an alternative limit in terms of maximum allowable power transfer or maximum allowable current can be derived based on worst-case scenario assumptions. This limit is usually referred to as the “ampacity” of the line. The additional assumptions made can be subjective, and/or the resulting thermal limits in terms of power transfer or current can be made on a somewhat ad-hoc basis. Also, since they are based on a worst-case scenario, they can be unnecessarily conservative. Consequentially, direct monitoring of the thermal limits in terms of temperature instead of power transfer or current can be provided using an on-line measurement of the line temperature in order to evaluate, during operation, whether a line is loaded close to its operational temperature limit or not.
A number of techniques have been proposed and several products are available to measure or infer the temperature of power line conductors during operation. These comprise the use of infrared cameras, mechanical tension measurements, direct sag measurements, predictive meteorological methods, or the use of phasor measurement data.
Infrared cameras may be used to take a digital picture of a power line, the color information of which is subsequently analyzed in a signal processing step in order to derive the temperature of the conductors. This technique can perform monitoring of the temperature of particular hotspots that are known a-priori.
Mechanical tension measurements between the tower and the isolator in combination with solar radiation and ambient temperature measurements can be based on the fact that the tension of the line conductor is approximately inversely proportional to its length. From the relationship between tension and length of the conductor, the line sag of a single span and the conductor temperature can be inferred. Likewise, line sag monitors directly measure the line sag of a single span through for example GPS (global positioning system) or laser measurement techniques.
Predictive meteorological methods and products based on the IEEE 738-1993 “Standard for Calculating the Current-Temperature of Bare Overhead Conductors”, the disclosure of which is incorporated herein by reference in its entirety, have been proposed to model the dependency between the line ampacity and various operational and ambient properties. These methods involve a number of meteorological measurements such as air temperature, wind speed, angle between wind and conductor and the elevation above sea level. The IEEE 738-1993 standard then specifies a computational procedure that can be used to estimate a steady-state conductor temperature from the meteorological measurements alone, i.e. without reverting to an independent measurement of the line temperature. The standard is based on a purely static model which does not account for the time-dependent behaviour of the line temperature and which is difficult to tune since various input parameter data may be assumed and detailed meteorological data is required.
The patent application EP 1324454, the contents of which are hereby incorporated herein by reference, describes a way of determining an actual average conductor temperature, via a calculated series resistance, from on-line phasor measurements. The average line temperature is largely independent of assumptions regarding any line parameters, such as the inductance, reactance or susceptance of the power line conductor. The method includes determining time-stamped current phasor information and voltage phasor information for a first end and a second end of the line, computing an ohmic resistance of the line from the phasor information, and computing an average line temperature from the ohmic resistance.
A state or condition of an electric power system at one specific point in time can be obtained from a plurality of synchronized phasor measurements or snapshots collected across the electric power system or power transmission network. Phasors are time-stamped, complex values such as amplitude and phase, of local electric quantities such as currents, voltages and load flows, and can be provided by means of Phasor Measurement Units (PMU). These units involve a very accurate global time reference, obtained e.g. by using the Global Positioning Satellite (GPS) system or any other comparable means, and allowing synchronization of the time-stamped values from different locations. The phasors are sampled at, for example, a rate of 20 to 60 Hz with a temporal resolution of less than 1 millisecond, and thus can provide a rather dynamic view on transient states that goes beyond the rather static view as provided by scalar values such as RMS values of voltages or currents and relied upon by SCADA/EMS systems.
Accordingly, parameters of an electric power network may be estimated by repeatedly measuring, at a plurality of network locations, synchronized values of electrical network variables; and identifying there from, during network operation, currently valid parameters of a mathematical model of the power network. In particular and by way of example, the Patent Application EP-A 1 489 714, the contents of which are incorporated herein by reference, discloses an adaptive detection of electromechanical oscillations in electric power systems based on a linear time-varying model. A system quantity or signal such as e.g. the amplitude or angle of the voltage or current at a selected node of the network is sampled, and the parameters of the linear model representing the behaviour of the power system are estimated by means of Kalman filtering techniques. This process can be carried out in a recursive manner, i.e. every time a new value of the system quantity is measured the parameters of the model are updated. Finally, from the estimated parameters of the model, the parameters of the oscillatory modes, such as frequency and damping, are deduced and presented to an operator. This adaptive identification process can provide a real-time analysis of the present state of the power system.

SUMMARY

A method and system are disclosed that can, at any time during operation, provide a reliable forecast of a power line conductor temperature. An exemplary method of estimating model parameters of a thermal model of a power line as well as a use of the thermal model are disclosed.

BRIEF DESCRIPTION OF THE DRAWINGS

The subject matter of the invention will be explained in more detail in the following text with reference to preferred exemplary embodiments which are illustrated in the attached drawings, of which:
FIG. 1 schematically shows a power line with a number of measurement devices,
FIG. 2 depicts a process of model parameter estimation, and
FIG. 3 depicts measured values of a number of quantities recorded for a period of 6 h.
The reference symbols used in the drawings, and their meanings, are listed in summary form in the list of reference symbols. In principle, identical parts are provided with the same reference symbols in the figures.

DETAILED DESCRIPTION

In accordance with exemplary embodiments, a relationship between a temperature of a power line or power transmission conductor, an electrical quantity of the power line such as a current or power flow through the power line, as well as meteorological quantities or ambient conditions of the power line such as wind speed, wind direction, humidity, solar radiation and ambient temperature, can be established in the form of a thermal model of the power line and repeatedly calculated or updated during operation of the power line. To this end, values of the aforementioned quantities or variables can be continuously or periodically measured or sampled, and the collected values of the quantities evaluated in order to update or tune model parameters of the thermal model. Including the temperature of the power line as a variable of the thermal model allows using, without diminishing its validity, a simple model or even a black box model with a limited number of model parameters. The latter may be updated without excessive computational efforts as frequently as desired, which ultimately increases the reliability, at any time during operation, of a forthcoming line temperature prediction.
In an exemplary embodiment disclosed herein, an average temperature representative of the entire line is determined via two Phasor Measurement Units (PMU) providing synchronized phasor values from two ends of the power line. An ohmic resistance of the power line is computed from the phasor values, from which in turn the average line temperature can be derived. As the PMUs are primarily provided for other purposes, e.g. for determining electrical quantities, such a double use avoids the need for any dedicated line temperature sensing device. In addition, as the PMUs are generally mounted indoors in a protected environment, they are less exposed to environmental stress than any other line temperature sensing device.
On the other hand, it can be advantageous to calibrate the conversion from the resistance to the temperature of the power line by means of such a dedicated line temperature sensor. As the latter may only be temporarily needed for the specific purpose of calibration, it may be expensive or otherwise cumbersome without impairing the subsequent operation of the power line.
In an exemplary variant, the meteorological data is obtained by subscription and imported from an external source such as a meteorological institute which is distinct from the Transmission System Operator (TSO). Relying on the data from a specialist can, if desired, avoid dedicated measurement units located on or close to the line conductor and operated by the TSO. Due to a relatively slow change in ambient conditions and their small geographical gradients, any potential temporal or geographical offset between the meteorological data and electrical data is of a lesser concern and can be ignored.
In an advantageous variant, an adaptive method or algorithm is based on a recursive calculation of the model parameters for each time-step, based on new values of the measured quantities and the old values of the model parameters. As opposed to the collection of data over a time window and then performing the parameter identification at once, any change in the power system can thus be detected much faster. In this context, the thermal model can be a linear autoregressive model of finite order, and an adaptive Kalman Filter can be used to estimate its model parameters.
The thermal model may be a nonlinear parametric model based on a heat balance equation. Such a physically inspired model can offer higher confidence when used for simulation, prediction or extrapolation and can be used in place of, e.g. a linear parametric model with no physical meaning, such as an Auto-Regressive Moving Average model (ARMA). The ARMA model on the other hand has the advantage that, except for the model order, no a-priori assumptions on model structure and parameters have to be made.
In a further aspect of an exemplary embodiment, the thermal model can be used to calculate a power line temperature given actual or forecasted values of electrical and meteorological quantities as e.g. provided by load predictions or weather forecasts. By comparing this predicted temperature with a temperature limit for the power line, a maximum amount of current or electrical power flow that can be transported on the line without violating the line temperature or sag limits may be derived, for example, by means of a simulation or inversion of the thermal model. Determination of a maximum flow that will result in a certain conductor temperature is particularly useful in order to determine the actual power flow limits to be used in a balance market clearing process. Since these limits are less conservative than a-priori known limits, less expensive balance power can be scheduled resulting in an economical gain for the TSO.
A computer program product is disclosed which includes computer program code means for controlling one or more processors of a model parameter estimator, a line temperature predictor or a Power Flow Control device connected to the power line. Such a computer program product can include a computer readable medium containing therein the computer program code means.
FIG. 1 shows a power line 10 that is part of a power system comprising a plurality of power generating means and power consumers interconnected by a transmission network. At two ends of the power line, two synchronized Phasor Measurement Units (PMU) 11, 11′ are provided and can be mounted in respective substation control buildings or any desired, suitable location. Sensing devices for measuring meteorological quantities at one or more locations in the vicinity of the power line 10 are collectively referred to as weather station 12. Means 13 for measuring electrical quantities of the power line are depicted schematically, some of their components such as instrument transformers or process busses may also be used by the PMUs. An independent line temperature measurement device 14 is likewise shown. The weather station 12, the means 13 and the device 14 collect and provide a number of equidistantly sampled measurements of ambient conditions (wind speed W, wind direction, solar radiation S, ambient temperature T_a, humidity), electrical quantities (line current l, power flow P) and line temperature (T_l) to a processor 15 of a model parameter estimator, a line temperature predictor or a Power Flow Control device.
FIG. 2 depicts an exemplary process of model parameter estimation according to an exemplary embodiment. The measured values of the meteorological and electrical quantities, collectively denoted as input variables u, are fed to the processor 15 for estimating or tuning, based on parameter identification or fitting techniques such as Kalman filters, Maximum Likelihood or Least-Squares, values of model parameters A, B, . . . ; C_l, . . . of a thermal model of the power line. Based on the momentary values of the electrical and meteorological quantities as well as of the model parameters, a line temperature prediction T_l ^fis produced as an output variable, and any difference between the latter and a corresponding independent measurement of the line temperature T_l, is fed back for further evaluation.
The thermal model can be a standard linear black box model in transfer function or state-space form. For dealing with several input variables u the discrete-time state-space form:
x(kT+T)=Ax(kT)+Bu(kT)+Ke(kT) (a)
y(kT)=Cx(kT)+Du(kT)+e(kT) (b)
x(0)=x0 (c)
is the most convenient one. Here, x denotes the dynamic state of the model, u the driving input variables, y the output of the system that model should reproduce and e Gaussian white noise, whereas A, B, . . . are model parameters. Linear models are attractive because of the simple parameter estimation techniques that are available and because of the fact that virtually no a-priori knowledge needs to be given, except for which measurements to use. On the other hand, such linear models can only be used to predict the line behaviour with rather small variations in the input variables, since non-linear contributions between the conductor temperature and the measured quantities could be quite substantial when the variations in the conductor temperature and/or measurement quantities are large. Accordingly, a linear model is suitable for short-term predictions on the order of minutes. Particularly a prediction interval of some 5-30 minutes (or lesser or greater) can be used to compute dynamic ratings of power lines based on a 5-30 minute forecast (or lesser or greater) and based on the identified dependencies between the conductor temperature and the electrical and meteorological measurements. This rating can be most beneficially used in the market clearing for the balance market which usually takes place on a similar time scale.
Alternatively, a physically inspired thermal model based on a heat-balance equation can have the advantage that model parameters which are known with enough certainty can be fixed a-priori, wherein such a thermal model could be valid also with quite large variations in the operating points. However, such a model would be non-linear and can involve more complicated parameter estimation techniques than the linear black box models. The extended Kalman filter has been shown to perform well in the estimation of parameters in non-linear models, although also other options are available. An exemplary heat balance equation has the form $C_{l} \frac{ⅆ T_{l}^{i}}{ⅆ t} = q_{i n} (I_{i}^{i}, T_{a}^{i}, {Solar}^{i}, \dots) - q_{out} (T_{a}^{i}, {Wind}^{i}, \dots),$
where C_lis a model parameter reflecting a characteristic thermal time or thermal capacity of the line, and where q_inrepresents the incoming heat flow to the conductor with main contributions from the sun's radiation and the heat produced by resistive losses in the conductor, and where q_outis the total heat loss of the conductor. The heat loss depends on many factors, for example the radiation and conduction to the surrounding air which in turn depends on factors like the wind speed and direction and the air humidity. The two heat transfer terms can involve a number of further model parameters.
For prediction in the longer term a higher model order and long data sets can be used so that the daily and even weekly or monthly variations can be modelled. Predictions based on such models could be used for example in the computation dynamic ratings of lines in the day ahead market, which typically are executed 24 hours ahead with update intervals of one hour.
FIG. 3 shows some measurement data from a field test, in which the weather and electrical phasor measurements were recorded during an observation window of six hours following the connection of a 380 kV transmission line at midday. The measured quantities are (top left plot) the conductor temperature (continuous line) and the ambient temperature (dashed line), the line current (top right plot) the humidity (middle left plot), the wind speed (middle right plot) and the solar radiation (bottom plot) at a specific location along the line. The recorded measurement samples were evaluated for the identification of the parameters A, B, . . . of a first order discrete time state-space model as indicated above, with the aim of accurately reproducing the line temperature. On the basis of the identified model parameters, the effect of a 100 A increase in line current and of a one degree change in ambient temperature have been simulated, yielding an increase in the line temperature of about 2.5° C. and 1° C. respectively.
The phasor data v₁, i₁; v₂, i₂can be collected from phasor measurement units that are distributed over a large geographical area, i.e. over tens to hundreds of kilometres. Since the phasor data from these disparate sources are analysed in conjunction, they refer to a common phase reference. Therefore, the different phasor measurement units have local clocks that are synchronised with each other to within a given precision. Such a synchronisation of the phasor measurement units can be achieved with a known time distribution system, for example the global positioning (GPS) system. In a typical implementation, the phasor data is determined, for example, at least every 200, or every 100, or preferably every 20 milliseconds, with a temporal resolution, for example, less than 1 millisecond. In an exemplary embodiment, the temporal resolution is less than 10 microseconds, which corresponds to a phase error of 0.2 degrees. Each measurement is associated with a time stamp derived from the synchronised local clock. The phasor data therefore comprises time stamp data.
According to an exemplary variant, the temperature of the line is determined in the following way: The electric line parameters, or at least the ohmic resistance R_lof the line, i.e. the real part R_lof the line impedance Z=R_l+jX_lare determined from measured or computed phasor information representing some or all of the voltage and current phasors at the two ends of line.
In a first variant, it is assumed that the shunt capacitance jX_cremains essentially constant (e.g., ±10 percent or lesser or greater) during power line operation and is known from other measurements, design parameters or calculations. Then the two voltage phasors v₁and v₂are determined at either end of the line and one of the current phasors i₁or i₂. Let i₁be measured. Then the impedance Z is $Z = \frac{v_{1} - v_{2}}{i_{1} - v_{1} \cdot {jX}_{C}} .$
In a second variant, no assumption on shunt impedances is made, and the two voltage phasors v₁and v₂and the two current phasors i₁or i₂are measured or determined from measurements. Determining the actual electrical line parameters R_l, X_l, X_cfrom these measurements is common knowledge. Since resulting equations for the electrical line parameters are non-linear, numerical methods such as Newton-Raphson approximation are used for determining actual parameter values. The resulting line parameters are actual values wherein they are determined online and represent the actual state of the power line. The average line temperature T₁is computed from the ohmic resistance R_lby modelling a relationship between temperature and resistance as linear, i.e.
R _l =R ₀(1+α₀(T _l −T ₀)),
where R₀is a known material property specified by the power line conductor manufacturer, i.e. a reference resistance dependent on the construction of the line, and where α₀is a material constant for the line cable and wherein T₀is, for example, a reference (e.g., ambient) temperature of the line. The linear relationship is typical for common conductor materials such as copper or aluminium. As an example, the parameter values are such that for a line temperature change of ΔT=30° C. the resistance changes by about ΔR_l=12%. The equation for the chosen relationship is solved for T₁, which gives the desired average line temperature.
It will be appreciated by those skilled in the art that the present invention can be embodied in other specific forms without departing from the spirit or essential characteristics thereof. The presently disclosed embodiments are therefore considered in all respects to be illustrative and not restricted. The scope of the invention is indicated by the appended claims rather than the foregoing description and all changes that come within the meaning and range and equivalence thereof are intended to be embraced therein.
List of Designations

10 power line
11, 11′ Phasor Measurement Units
12 weather station
13 electrical quantity measurement means
14 line temperature measurement device
15 processor

Claims

1. A method of estimating model parameters (A, B, . . . ; C_l, . . . ) of a thermal model of a power line, comprising:

measuring values of an electrical quantity (l, P) of the power line and of meteorological quantities (T_a, S, W) representing ambient conditions of the power line;

calculating values of the model parameters from said measured values;

measuring, repeatedly during operation of the power line, momentary values (lⁱ, T_a ⁱ, Sⁱ, Wⁱ) of the electrical and meteorological quantities;

measuring concurrently a momentary value (T_l ⁱ) of a temperature (T_l) of the power line; and

calculating, repeatedly during operation of the power line, the values (Aⁱ, Bⁱ; . . . ; C_l ⁱ, . . . ) of the model parameters from said measured values.

2. The method according to claim 1, wherein measuring a momentary value (T_l ⁱ) of the temperature of the power line comprises:

measuring, by means of two Phasor Measurement Units, synchronized phasor data (v₁ ⁱ, i₁ ⁱ; v₂ ⁱ, i₂ ⁱ) at two ends of the power line;

computing a value (R_l ⁱ) of an electrical resistance (R_l) of the power line from the phasor data; and

computing, from the value (R_l ⁱ) of the electrical resistance of the power line, an average line temperature as the momentary value (T_l ⁱ) of the temperature of the power line.

3. The method according to claim 2, wherein computing the average line temperature involves an analytical expression with parameters (R₀, α₀) suitably calibrated by means of an independent line temperature measuring device.

4. The method according to claim 1, wherein measuring momentary values (T_a ⁱ, Sⁱ, Wⁱ) of the meteorological quantities involves measuring the values by a provider of meteorological data other than an operator of the power line.

5. The method according to claim 1, comprising:

generating a series of measured values (T_l ¹, T_l ², . . . ; u¹, u², . . . ) of the temperature of the power line and the electrical and meteorological quantities; and

adaptively calculating updated values (A^k, B^k; . . . ; C_l ^k, . . . ) of said model parameters every time a new value (T_l ^k, u^k) of the temperature of the power line or the electrical and meteorological quantities is measured.

6. The method according to claim 1, wherein the thermal model is a nonlinear parametric model based on a heat balance equation.

7. A method for predicting a value (T_l ^f) of the line temperature of the power line, using a thermal model with momentary model parameter values (Aⁱ, Bⁱ; . . . ; C_l ⁱ, . . . ) estimated according to claim 1, comprising:

providing forecasted values (u^f) of the electrical and meteorological quantities; and

calculating a power line temperature forecast (T_l ^f) based on the momentary model parameters and the forecasted values of the electrical and meteorological quantities.

8. The method according to claim 7, comprising:

comparing the power line temperature forecast (T_l ^f) with a power line temperature limit; and

calculating a maximum allowable value of the electrical quantity (l, P) there from.

9. The use according to claim 8, comprising:

using a linear thermal model for the power line; and

providing the maximum allowable value of the electrical quantity (l, P) to a balance market clearing process.

10. The method according to claim 2, comprising

11. A method for predicting a value (T_l ^f) of the line temperature of the power line, using a thermal model with momentary model parameter values (Aⁱ, Bⁱ; . . . ; C_l ⁱ, . . . ) estimated according to claim 2, comprising:

12. A method for predicting a value (T_l ^f) of the line temperature of the power line, using a thermal model with momentary model parameter values (Aⁱ, Bⁱ; . . . ; C_l ⁱ, . . . ) estimated according to claim 3, comprising:

13. A method for predicting a value (T_l ^f) of the line temperature of the power line, using a thermal model with momentary model parameter values (Aⁱ, Bⁱ; . . . ; C_l ⁱ, . . . ) estimated according to claim 4, comprising:

14. A method for predicting a value (T_l ^f) of the line temperature of the power line, using a thermal model with momentary model parameter values (Aⁱ, Bⁱ; . . . ; C_l ⁱ, . . . ) estimated according to claim 5, comprising:

providing forecasted values (u^f) of the electrical and meteorological quantities;

15. A method for predicting a value (T_l ^f) of the line temperature of the power line, using a thermal model with momentary model parameter values (Aⁱ, Bⁱ; . . . ; C_l ⁱ, . . . ) estimated according to claim 6, comprising:

16. A method for predicting a value (T_l ^f) of the line temperature of the power line, using a thermal model with momentary model parameter values (Aⁱ, Bⁱ; . . . ; C_l ⁱ, . . . ) estimated according to claim 10, comprising:

17. A method of estimating a parameter of a thermal model of a power line, comprising:

measuring a value of an electrical quantity of the power line and at least one meteorological quantity representing an ambient condition of the power line;

calculating a value of the parameter of the thermal model from said electrical quantity and said meteorological quantity;

measuring, during operation of the power line, a momentary value of each the electrical quantity an the meteorological quantity;

measuring a momentary value of a temperature of the power line; and

calculating, during operation of the power line, a value of the parameter from said measured values of the electrical quantity, the meteorological quantity and the temperature.