Study of Galloping Mitigation:

Tug Boats and Power Lines

November 11, 2015

By Jay Quint, PE, PEng

Learning Objectives

► Myths

► Weather Cases

► Mitigation

► Events

► Mechanics

► Tug boats?

► Metric

► CIGRE 322

2

► Aeolian Vibration

► Wake Induced Oscillation

► Galloping

Wind Induced Conductor

Motions

3

Galloping Myth Busting ► Galloping is a Force

Equilibrium

► Galloping is a Single Span

problem

► Flashovers are the primary

problem

𝑓𝐿

𝐺𝑀𝑊𝑖𝑟𝑒

𝑓𝐷

4

Observed Galloping on

Operating Lines

► Single loop

galloping

observed

on spans

greater

than 200 m

(600-700

ft)

5

Wind During Galloping

on Operating Lines

► Minimum wind speed = typically 5-7 m/s (~20-25 mph)

6

Ice and Wind During

Galloping on Operating

Lines

7

Galloping Mitigation

Options ► Increase the phase

spacing

► Interphase spacers

► Aerodynamic devices

► Tuning devices

► Ice removal “The Ice

Buster”

8

Mitigation: Increased

Phase Spacing per RUS

Voltage 115 kV 138 kV 230 kV 345 kV 500 kV

Phase-Phase 0.46 m

(1.5 ft)

0.46 m

(1.5 ft)

0.76 m

(2.5 ft)

1.07 m

(3.5 ft)

1.83 m

(6.0 ft)

Phase-Ground 0.30 m

(1.0 ft)

0.30 m

(1.0 ft)

.61 m

(2.0 ft)

.76 m

(2.5 ft)

1.22 m

(4.0 ft)

𝑀(𝑓𝑡) = 1.25 𝑆𝑎𝑔 .5𝑖𝑛 𝑖𝑐𝑒/32 𝐷𝑒𝑔 𝐹 + 1.0

9

► Single Loop Galloping

Mitigation:

Interphase Spacers ► Most widely

used device

on all

voltages

and bundles

► Prevents

flashovers

► Galloping /

Dynamic

loading

persist

10

Mitigation:

Interphase Spacers ► Typical Interphase Spacer installations

11

Mitigation:

Aerodynamic Devices ► T2/VR2 Conductor

• One Full Twist over ~3m (8-10 ft) for T2 Drake (2 x 795 KCMIL 26/7

ACSR)

• See ASTM B911: Standard Specification for ACSR Twisted Pair

Conductor

• Vertical bundle of controls galloping

12

Mitigation:

Aerodynamic Devices ► Air flow spoilers

• Limits: Single conductors

• 25% percent of the span is wrapped in two groups

• Galloping prevention / amplitude reduction

13

Galloping Mitigation:

Aerodynamic Devices ► AR Twister:

• Eccentric weight mounted about the conductor

• Creates a smooth ice profile

• Two per span (minimum)

14

Galloping Mitigation:

Tuning Devices ► Detuning Pendulums

► Torsional Damper and Detuner

(TDD)

15

Galloping Mitigation:

Summary of Options

16

Galloping Mitigation:

Other Options

17

Galloping Events and

the Results

► 345 kV, Tubular Steel

Line Retrofit with

Interphase Spacers, MN

Winter 2014

► 345 kV, Lattice Tower

Failures, SD Winter

2010

► 345 kV, H-Frame Tubular

Steel Cascade Failure,

IA Winter 1990

ICE

WIND

Flashover &

Structural Damage

18

Galloping Mechanics:

Modes of Failure

► 1 Degree of Freedom:

Vertical

► Cable System: Damped

Simple Harmonic

Motion

► 2 Degrees of Freedom:

Vertical and

Torsional

► Dynamic Analysis of a

Cable Section

19

Analysis of Vertical

Conductor Motion ► The motion will be (+) or (-) from a neutral

position

► Likewise the conductor velocity will also

change signs

► Analysis of a conductor in motion

Y=-MAX, ⅆ𝑦/ ⅆt =0

Y=0, (+/-) Max = ⅆ𝑦/ ⅆt

Y=+MAX, ⅆ𝑦/ ⅆt =0

20

Physics: NASA Drag

Coefficients

WIND (UO)

Drag Force (𝑓𝑑) =1

2𝜌𝐴𝐶𝑑𝑈𝑂

2

𝐶𝑑 −𝑠𝑝ℎ𝑒𝑟𝑒 = .07 − .5

𝐶𝑑 − 𝑓𝑙𝑎𝑡 𝑝𝑙𝑎𝑡𝑒 = 1.28

𝐶𝑑 − 𝑏𝑢𝑙𝑙𝑒𝑡 = 0.295

21

Physics: Aerodynamic

Forces

► Asymmetrical Cable Shapes and the Lift Force

WIND (UO)

+𝑓𝐿

Lift (𝑓𝐿) =1

2𝜌𝑎𝑖𝑟ⅆ𝑣𝑟

2𝐶𝐿 𝜑 ANGLE OF ATTACK (𝜑)

(- 𝜑)

(+𝜑) −𝑓𝐿

22

Aerodynamic Forces on

an Iced Conductor

► Lift and Drag Coefficient vs Angle of Attack

+LIFT 𝑓𝐿

+DRAG 𝑓𝐷

(+ 𝜑) (- 𝜑)

-π π π/2 -π/2

-LIFT 𝑓𝐿

+DRAG 𝑓𝐷

90 45

-45 90

23

Physics: Simple

Harmonic Motion (SHM) ► Simple Harmonic Motion describes Galloping

24

Analysis of Vertical

Conductor Motion

JJQ

► Change in Angle of Attack due to conductor

motion

► Relative Wind at Neutral Position

(+) ⅆ𝑦/ ⅆt

− ⅆ𝑦/ ⅆt

WIND (UO)

WIND (UO) -α

(+ 𝜑) ANGLE OF ATTACK

(- 𝜑) ANGLE OF ATTACK

α

25

► Determine if the Aerodynamic Forces are

damping or destabIilizing the vertical

motion

► Case 1: Initial Angle of Attack = 90 deg, π/2

Analysis of Vertical

Conductor Motion

dy/dt =(+)

Δ𝑓𝐿=(-) dy/dt =(-)

Δ𝑓𝐿=(+)

26

𝑓𝐿 = (ⅆ𝑓𝐿ⅆ𝜑)𝛼

Review of Lift and

Drag: Damped ► For a small ∆𝜑 (𝑤ℎ𝑖𝑐ℎ 𝑖𝑠 𝛼), the lift force is linear:

+LIFT 𝑓𝐿

+DRAG 𝑓𝐷

(+ 𝜑) (- 𝜑)

-π π π/2 -π/2

-LIFT 𝑓𝐿

+DRAG 𝑓𝐷

27

► Determine if the Aerodynamic Forces are

damping or destabIilizing the vertical

motion

► Case 1: Initial Angle of Attack = 0 deg, 0

Analysis of Vertical

Conductor Motion

dy/dt =(+) Δ𝑓𝐿=(+)

dy/dt =(-) Δ𝑓𝐿=(-)

28

Review of Lift and

Drag: Destabilized ► For a small ∆𝜑 (𝑤ℎ𝑖𝑐ℎ 𝑖𝑠 𝛼), the lift force is linear:

+LIFT 𝑓𝐿

+DRAG 𝑓𝐷

(+ 𝜑) (- 𝜑)

-π π π/2

-LIFT 𝑓𝐿

+DRAG 𝑓𝐷

𝑓𝐿 = (ⅆ𝑓𝐿ⅆ𝜑)𝛼

29

Galloping: An

Aerodynamic

Instability

𝑓𝐿 cos 𝛼 > 𝑓𝐷 sin 𝛼, SYSTEM UNSTABLE

tan−1 𝛼 = (−ⅆ𝑦/ⅆ𝑡)/𝑈𝑂

− ⅆ𝑦/ ⅆt

WIND (UO)

α

For small 𝛼, in Radians

𝛼 = (−ⅆ𝑦/ⅆ𝑡)/𝑈𝑂

sin 𝛼 = 𝛼

cos 𝛼 = 1

𝑓𝐿 1 > 𝑓𝐷(α)

𝑑𝑓𝐿

𝑑𝜑 𝛼 > 𝑓𝐷(α)

𝑓𝐷

𝑓𝐿 = (ⅆ𝑓𝐿ⅆ𝜑)𝛼

𝑓𝐿 cos 𝛼 < 𝑓𝐷 sin 𝛼, SYSTEM DAMPED

𝑑𝑓𝐿

𝑑𝜑 > 𝑓𝐷, SYSTEM UNSTABLE

30

Idealized Structure &

Cable System ► Cable system with fixed ends, sag (mass)

► Tension enters the system as a restorative

force (opposite to motion)

𝐹𝑟𝑒𝑠𝑡𝑜𝑟𝑒 = −𝑘𝑦 𝑡 + 𝑚𝑔 − 𝑘(𝑌𝑖𝑛𝑖𝑡𝑖𝑎𝑙)

𝐹𝑜𝑟𝑐𝑒𝑠 =ma=m(ⅆ2𝑦/ⅆ𝑡2)

𝑚 (ⅆ2𝑦/ⅆ𝑡2) + 𝑘𝑦(𝑡) = 0

DIFFERENTIAL EQUATION:

𝑚𝑔 − 𝑘(𝑌𝑖𝑛𝑖𝑡𝑖𝑎𝑙) = 0 WHERE:

31

Cable System: Simple

Harmonic Motion ► Hooke’s Law: cable spring constant

𝜎 = 𝐸(𝜀)

GIVEN: 𝜎 = Δ𝑇/𝐴

𝜀 = Δ𝑙/𝐿

𝑅𝑒𝑠𝑡𝑜𝑟𝑎𝑡𝑖𝑣𝑒 𝑓𝑜𝑟𝑐𝑒 𝑘𝑥 = Δ𝑇 =𝐸𝐴

𝐿(Δ𝑙)

THEREFORE: 𝑘 =𝐸𝐴

𝐿

32

Physics: Sum of the

Forces in SHM ► Key Relationships and Solution

y(t) = 𝑦𝑚𝑎𝑥 cos(𝜔𝑡)

DIFFERENTIAL EQUATION:

WHERE:

𝑚(ⅆ2𝑦/ⅆ𝑡2) = −𝑘𝑦(𝑡)

ⅆ2𝑦/ⅆ𝑡2 = −𝜔2𝑦𝑀𝐴𝑋 cos(𝜔𝑡)

THEREFORE: −𝑚𝜔2𝑦𝑚𝑎𝑥 cos(𝜔𝑡) = 𝑚ⅆ2𝑦/ⅆ𝑡2 = −𝑘𝑦(𝑡)

THEREFORE THESE RELATIONSHIPS EXIST: 𝑚𝜔2 = (2𝜋𝜈)2 = 𝑘

𝑁𝑎𝑡𝑢𝑟𝑎𝑙 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 (𝜈𝑁𝐴𝑇) =1

2𝜋

𝑘

𝑚

SOLUTION DESCRIBING LOCATION

VALUE DESCRIBING SYSTEM:

MECHANICAL ENERGY: 𝐸𝑛𝑒𝑟𝑔𝑦 = .5𝑘 𝑦 𝑡 2+ .5𝑚ⅆ𝑦

ⅆ𝑡

2

y(t) = 𝑦𝑚𝑎𝑥 cos(𝜔𝑡)

33

Physics: Damped

Harmonic Motion ► Damping vane in a viscous fluid attached to

an oscillating block

𝐹𝑜𝑟𝑐𝑒𝑠 = −𝑘𝑦 − 𝑏𝑣 + (𝑚𝑔 − 𝑘(𝑌𝑖𝑛𝑖𝑡𝑖𝑎𝑙)) =ma

NOW THE SUM OF THE FORCES:

DAMPING FORCE:

𝐹𝐷𝐴𝑀𝑃𝐼𝑁𝐺 = −𝑏 𝑉𝐸𝐿𝑂𝐶𝐼𝑇𝑌 = −𝑏(ⅆ𝑦

ⅆ𝑡)

𝐹𝑜𝑟𝑐𝑒𝑠 = −𝐹𝑟𝑒𝑠𝑡𝑜𝑟𝑒 − 𝐹𝑑𝑎𝑚𝑝𝑖𝑛𝑔 + (𝐹𝑔𝑟𝑎𝑣𝑖𝑡𝑦 − 𝐹𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑠𝑝𝑟𝑖𝑛𝑔 𝑙𝑜𝑎𝑑) =ma

𝑚𝑔 − 𝑘(𝑌𝑖𝑛𝑖𝑡𝑖𝑎𝑙) = 0

m(ⅆ2𝑦

ⅆ𝑡2) + 𝑏

ⅆ𝑦

ⅆ𝑡+ 𝑘𝑦(𝑡) = 0

WHERE:

NEW DIFFERENTIAL EQUATION:

34

Physics: Damped

Harmonic Motion

𝐶𝑟𝑖𝑡𝑖𝑐𝑎𝑙 𝐷𝑎𝑚𝑝𝑖𝑛𝑔 (𝑐𝑐) = 2 𝑘𝑚

𝐷𝑎𝑚𝑝𝑖𝑛𝑔 𝑅𝑎𝑡𝑖𝑜 𝜍 =𝐴𝑐𝑡𝑢𝑎𝑙 𝐷𝑎𝑚𝑝𝑖𝑛𝑔 𝑐

𝐶𝑟𝑖𝑡𝑖𝑐𝑎𝑙 𝐷𝑎𝑚𝑝𝑖𝑛𝑔 𝑐𝑐=

𝑏

2 𝑘𝑚

𝐷𝑎𝑚𝑝𝑒ⅆ 𝑁𝑎𝑡𝑢𝑟𝑎𝑙 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦(𝜈𝐷𝐴𝑀𝑃) = 𝜈𝑁𝐴𝑇 1 − 𝜍2

𝐴𝑛𝑔𝑢𝑙𝑎𝑟 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 (𝜔) = (𝑘

𝑚−

𝑏

4𝑚2)

y(t) = 𝑦𝑚𝑎𝑥 𝑒−(𝑏𝑡2𝑚)cos(𝜔𝑡)

NEW DIFFERENTIAL EQUATION:

SOLUTION DESCRIBING LOCATION

m(ⅆ2𝑦

ⅆ𝑡2) + 𝑏

ⅆ𝑦

ⅆ𝑡+ 𝑘𝑦(𝑡) = 0

VALUE DESCRIBING SYSTEM:

𝐸𝑛𝑒𝑟𝑔𝑦 𝐹𝐼𝑁𝐴𝐿 + 𝐸𝑛𝑒𝑟𝑔𝑦𝐷𝐼𝑆𝑆𝐼𝑃𝐴𝑇𝐸𝐷 = 𝐸𝑛𝑒𝑟𝑔𝑦 𝐼𝑁𝐼𝑇𝐼𝐴𝐿 MECHANICAL ENERGY:

35

Physics: Damped

Harmonic Motion ► For a transmission line single or double

bundle 𝜍 = .01

36

Normal Case: Cable

Motion is Underdamped

SHM

► Review the force equation:

► Air Resistance (Drag) Acts as a Viscous

Damper

► Results: Aerodynamic Stability

𝐹𝑜𝑟𝑐𝑒𝑠 = −𝐹𝑟𝑒𝑠𝑡𝑜𝑟𝑒 − 𝐹𝑑𝑎𝑚𝑝𝑖𝑛𝑔 + (𝐹𝑔𝑟𝑎𝑣𝑖𝑡𝑦 − 𝐹𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑠𝑝𝑟𝑖𝑛𝑔 𝑙𝑜𝑎𝑑) =ma

𝑓𝐿 cos 𝛼 < 𝑓𝐷 sin 𝛼, SYSTEM DAMPED

dy/dt =(+)

Δ𝑓𝐿=(-) dy/dt =(-)

Δ𝑓𝐿=(+)

37

► Review the force equation:

► Aerodynamic Lift Over Comes Drag: Excitation

► Results: Aerodynamic Destabilized, but

Requires Wind Energy

𝐹𝑜𝑟𝑐𝑒𝑠 = −𝐹𝑟𝑒𝑠𝑡𝑜𝑟𝑒 + 𝐹𝑒𝑥𝑐𝑖𝑡𝑖𝑎𝑡𝑖𝑜𝑛 + (𝐹𝑔𝑟𝑎𝑣𝑖𝑡𝑦 − 𝐹𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑠𝑝𝑟𝑖𝑛𝑔 𝑙𝑜𝑎𝑑) =ma

Galloping Case: Cable

Motion is Excited SHM

𝑓𝐿 cos 𝛼 > 𝑓𝐷 sin 𝛼, SYSTEM UNSTABLE

dy/dt =(+) Δ𝑓𝐿=(+)

dy/dt =(-) Δ𝑓𝐿=(-)

38

Torsion: The Second

Degree of Freedom ► Pitching Moment creates the twisting motion

WIND (UO) (+ 𝜑) ANGLE OF ATTACK

-α

+𝜃𝑖𝑐𝑒

+θ

𝜑 = −𝛼 + 𝜃 + 𝜃𝑖𝑐𝑒 PITCHING MOMENT =𝑀𝑊= 1

2𝜌𝑎𝑖𝑟ⅆ

2𝑣𝑟2𝐶𝑀 𝜑

Mw

39

Vertical / Torsional

Motion Phasing ► Determine impacts of motion out of phase

► Exaggerates the Aerodynamic Instability

► Case 1: Max Twist at Y=0

40

Vertical / Torsional

Motion Phasing ► Determine impacts of phased oppositely

(Aeroelastic)

► Case 2: Max Twist not at Y=0, “Coupled Flutter”

Occurs

41

Structural Components:

3 Centers of an Iced

Cable ► 2 degrees of freedom complicated by

structural data

AERODYNAMIC CENTER

CENTER OF GRAVITY

CENTER OF CABLE

42

Coupled Oscillation:

Multi-span Systems ► Mechanism of tension transfer between spans

43

Galloping Standing

Waves in a Section ► Creates Dynamic Loading:

• Dynamic Loads on a Tangent: Vertical Force 2 times the static load

(CIGRE 322)

• Dynamic Loads on a Deadend: Tension Force 2.5 times the static load

(CIGRE 322)

44

Tug Boat Drive Train:

Voith Schneider

Propeller

► Voith Schneider Propoller: Tug Boat Power

45

Conclusions ► Galloping weather: ice plus high winds

► Galloping mitigation exists

► Single cable due to aerodynamic

instability

► Multi-cable due to aero-elastic

instability

46

Questions

47

Advanced topics of

Galloping Analysis ► Discussion of Cigre, Alberta ESO, and RUS galloping

design

► The dynamic loading

• Effects on the structures due to allowing the cable to act as a spring

• Structure has one fundamental frequency and wire section has a different

frequency

► Multi-Span analysis for fundamental natural

frequency

► Vertical and torsion motion, 2 total degrees of

freedom system: forces and stiffness

► Vertical, torsion, and horizontal motion, 3 total

degrees of freedom: forces and stiffness

► Weather cases, ice shapes, aerodynamic coefficients

► Amplitude of motion (energy balance wind as

excitation)

► Influence of hardware configuration

48