➢ Τ∆𝑇c ∆𝑡 = Τ𝐼2𝑅 𝑇𝑐 + 𝑄𝑠 − 𝑄𝑐 𝑇𝑐 − 𝑄𝑟 𝑇𝑐 𝐻 (2)

➢ captures the actual variation in conductor temperature 𝑇𝑐.

Enhanced Weather Modelling for Dynamic Line Rating

Fulin Fan (e-mail: f.fan@strath.ac.uk), Prof. Keith Bell, and Prof. David InfieldDepartment of Electronic and Electrical Engineering, University of Strathclyde, Glasgow, United Kingdom

Sponsors: Energy Technology Partnership, Scottish Power Energy Networksand National Grid Electricity Transmission

Start Date: Apr. 2013

2. Objective• The project aims to develop a weather-based model that provides dynamic line

rating (DLR) forecasts in the form of percentiles (i.e. probabilistic forecasting):

➢ describing the probability of a particular DLR forecast being exceeded;

➢ system operators have time to exploit additional power transfer headroom;

➢ system operators can make an informed judgement about risk.

• This poster is to describe a fast-computational approach to probabilistic forecasting of transient-state or short-term DLR that yields the maximum allowable conductor temperature within a specified time period.

1. Background• The ampacity of an overhead line (OHL) is conventionally limited to a static line

rating that is derived from a maximum allowable conductor temperature and conservative weather conditions (e.g. low wind speed) for a particular season.

• Dynamic Line Rating is the maximum permissible level of power flow that can pass through an OHL safely and reliably under prevailing weather conditions.

• The additional headroom of an OHL’s ampacity exploited by dynamic line rating can help network operators accommodate growth in power flow and facilitate connections of distributed generation.

The ratios of 10-minute and 30-minute transient-state DLR forecasts and weather observation based DLRs to the SLRs on 27/3/13 for a particular 132kV span.

• Steady-state calculation:

➢ assumes that the conductoris in thermal equilibrium.

➢ 𝐼2𝑅 𝑇𝑐 + 𝑄𝑠 − 𝑄𝑐 𝑇𝑐 − 𝑄𝑟 𝑇𝑐 = 0 (1)

• Transient-state calculation:

➢ considers the heat capacity 𝐻of the conductor (thermal inertial);

Solar heat gain 𝑄𝑠

Joule heat gain 𝐼2𝑅

3. Steady-State vs Transient-State

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Transient-state

Steady-state and transient-state 𝑇𝑐 estimated from the measured weather data and line currents on 24/2/2013 for a particular 132kV overhead span. Distributions of differences between transient-state and steady-state DLRs at a particular

span, showing that transient-state DLRs are greater than steady-state DLRs for most of the time while the enhancement decreases with increases with forecast look ahead time.

5. An Enhanced Analytical Method• The IEEE analytical method for transient-state 𝑇𝑐 calculation is enhanced to:

➢ additionally consider changes in weather variables;

➢ fulfil the requirement of the conductor being in thermal equilibrium prior to the step change.

• Approximating equation (2) to be a first-order linear time-invariant system,

𝑇𝑐 𝑡 = 𝑇𝑐𝑖 + 𝑇𝑠𝑠,𝑐𝑓 − 𝑇𝑐𝑖 ∙ 1 − 𝑒 Τ−𝑡 𝜏 (3)

• where 𝜏 is the thermal time constant at which the change of 𝑇𝑐 reaches 63.2% of the difference between the initial conductor temperature 𝑇𝑐𝑖 and steady-state final conductor temperature 𝑇𝑠𝑠,𝑐𝑓 that is derived from line current 𝐼𝑓and weather conditions 𝑤𝑐 after step changes by the secant method,

𝜏 = ൗ𝑇𝑠𝑠,𝑐𝑓 − 𝑇𝑐𝑖 ∙ 𝐻 𝐼𝑓2 − 𝐼𝑖,𝑒𝑞

2 ∙ 𝑅 𝑇𝑐 (4)

• where an equivalent steady-state initial line current 𝐼𝑖,𝑒𝑞 is inferred from 𝑇𝑐𝑖and 𝑤𝑐 based on equation (1),

𝐼𝑖,𝑒𝑞2 = Τ𝑄𝑐 𝑇𝑐𝑖 ,𝑤𝑐 + 𝑄𝑟 𝑇𝑐𝑖 , 𝑤𝑐 − 𝑄𝑠 𝑤𝑐 𝑅 𝑇𝑐𝑖 (5)

Less attention has been given to transient-state DLRs due to that:

1) not necessary when sampling intervals of variables are long enough (e.g. 1 hour) for a conductor to fully respond to step changes…

✓ 10-minute data is provided in this work.

2) computation time cost to estimate transient-state DLRs is much greater…

✓ reduce computation time via enhancement of an analytical method presented in IEEE Std. 738 and use of the secant method, i.e. a fast root-finding algorithm.

3) transient ratings can be maintained within the specified time period only…

✓ but it will provide higher additional ampacity headroom than steady-state DLRs.

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Transient-State Rating - Steady-State Rating (Amps)

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Convection heat loss 𝑄𝑐

Radiation heat loss 𝑄𝑟

6. The Secant Method

➢ using a succession of roots of secant lines to better approximate the root of a non-linear function;

➢ having a faster convergence process than the bisection method.

• The secant method is used in this work to:

➢ estimate 𝑇𝑠𝑠,𝑐𝑓 from 𝑤𝑐 and 𝐼𝑓;

➢ estimate the transient-state DLR that yields the maximum allowable conductor temperature in the specified time period.

• The secant method is a fast root-finding algorithm,

𝑥𝑘 =𝑥𝑘−2𝑓 𝑥𝑘−1 − 𝑥𝑘−1𝑓 𝑥𝑘−2

𝑓 𝑥𝑘−1 − 𝑓 𝑥𝑘−2

7. Process and Results of Transient Rating Forecasting

4. Practicability of Applying Transient Rating Forecasts

(b) Predictive distributions of weather variables for up to half hour (3 steps) ahead

(c) 104 sets of correlated weather samples

Monte Carlo sampling

Thermal model for transient-state DLRs on the basis of the enhanced analytical method and the secant method

(a) Historic weather data (air temperature, solar radiation, wind speed and direction)

Conditionally heteroscedastic auto-regressive forecasting models

A rank correlation based pairing method

(d) 104 sampled values for 10-minute, 20-minute and 30-minute transient-state DLRs

Kernel Density Estimation

(e) Percentiles of 10-minute, 20-minute and 30-minute transient-state DLR forecasts

Key Results:

• The fast-computational approach requires about 2 seconds to calculate 10-minute, 20-minute and 30-minute transient ratings for 2 × 104 scenarios;

• The approach developed here gives greater than 10% improvement over persistence forecasting in root mean square errors of point forecasts;

• The positive correlations among the paired random weather samples of the same parameters at different future moments expand the distributions of 20-minute and 30-minute transient-state DLR forecasts, improving the calibrationat the lower percentiles.

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