Ampacity Calculation of Long Line Overhead Cables

Model descriptionSolution of the dinamyc model

Parameter determinationData filtering

Conclusions and future work

Ampacity Calculation of Long Line Overhead Cables

Julen Alvarez

Joint work with

Enrique Zuazua and Inaki Garabieta

BCAM seminar

17/01/2011

Julen Alvarez Aramberri - 17/01/2011 Ampacity Calculation of Long Line Overhead Cables




Motivation





Goals

General Objectives

Compute the ampacity of a long line cable.

Determine some parameters of the model.

Establish whether it is necessary or not to filter the measured data.

Decisions based on:

To get a mean square error as small as possible.

Definition: Mean Square Error

The mean square error (MSE) between two series (of length N) x and y isdefined by:

MSE =1

N

N∑k=1

(xk − y k

)2





Outline

1 Model descriptionMathematical ModelsDefinition of the variables

2 Solution of the dinamyc modelDeduction of the modelNumerical resolution of the ODE

3 Parameter determinationProblem FormulationHarmony SearchNumerical resultsSensitivity analysis

4 Data filteringData filteringResults

5 Conclusions and future workConclusions





Mathematical ModelsDefinition of the variables

Outline











Mathematical Models

CIGRE Working Group 22.12, (Thermal behaviour of overheadconductors), Study Committee 22-Working Group 12, August 2002

Dynamic Model

dT (t)

dt=

1

mc(Pj + Ps − Pc − Pr )

Static Model

Pj + Ps = Pc + Pr

Heats

Pj : Generated heat due to Joule effect.Ps : Generated heat due to solar radiation.Pc : Convective cooling.Pr : Cooling due to radiation.






Definition of the variables

Heat definitions

Pj = I 2 R [1 + α (Tc − 20)]

Ps = αs S D

Pc = π λf (Tc − Ta) Nu

Pr = πDεσB

[(Tc + 273)4 − (Ta + 273)4

]Definition of the constants

R : Resistance at 20ºC per unit length.α : Temperature coefficient of resistance per degree Kelvin.D : Diameter.σB : Stefan-Boltzmann constant.






Definition of the variables

More definitions

Reynolds nº: Re =ρrvvD

νfρr : relative air density

vv : wind velocity(m/s)

νf : kinematic viscosity(m2/s)

Nusselt nº: Nu = Nu90 [A + B (sinδv )]

Nu90 = C (Re)n

Empirics Equations

Tf = 0.5(Tc + Ta)

νf = 1.32 · 10−5 + 9.5 · 10−8Tf

λf = 2.42 · 10−2 + 7.2 · 10−5Tf

ρr = exp(−1.16 · 10−4 y), y: the height above sea level(m)






Measured variables and parameters to determine.

Heat definition

Pj = I 2 R [1 + α (Tc − 20)]

Ps = αs S D

Pc = π λf (Tc − Ta) C

(ρrvvD

νf

)n

[A + B (sinδv )]

Pr = πDεσB

[(Tc + 273)4 − (Ta + 273)4

]Measured variables Notation Parameters Notation

Wind direction δv Absorptivity αs

Radiation S Emissivity εRoom temperature Ta A A

Wind velocity vv B BIntensity I Limit velocity limit velocity

Conductor temperature Tc






Measured variables and parameters to determine.

Heat Definitions

Pj = I 2 R [1 + α (Tc − 20)]

Ps = αs S D

Pc = π λf (Tc − Ta) C

(ρrvvD

νf

)n

[A + B (sinδv )]

Pr = πDεσB

[(Tc + 273)4 − (Ta + 273)4

]Measured variables Notation Parameters Notation

Wind direction δv Absorptivity αs

Radiaton S Emissivity εRoom temperature Ta A A

Wind velocity vv B BIntensity I Limit velocity limit velocity

Conductor temperature Tc






Notation

Notation

x t(a1, ..., a5, ut1, ..., u

t5) : Solution to the ODE in t.

(a1, ..., a5) : Set of the five parameters to determine.(ut

1, ..., ut5) : Measurements of the physical variables in t.

MSE in our problem

MSE =1

N

∑t

(x t(a1, ..., a5, u

t1, ..., u

t5)− θt

)2

θt is the measured value for the conductor temperature in t, and N is the totalnumber of measurements.





Deduction of the modelNumerical resolution of the ODE

Outline











Deduction of the model

Heat equation

∂T

∂t=

λ

γc

(∂2T

∂r 2+

1

r

∂T

∂r+

1

r 2

∂2T

∂ϕ2+∂2T

∂z2

)+

q (T , ϕ, z , r , t)

γc

c : Specific heat capacity. q : Power by unit of volume.γ : Mass density. λ : Thermal conductivity.

Cylindrical simetry and semi-infinite length

∂T

∂t=

λ

γc

(∂2T

∂r 2+

1

r

∂T

∂r

)+

q (T , r , t)

γc

Constant radial distribution of the temperature and m = γA and q = PA

, whereA is the normal section of the cilindre and P is the power by unit of length.

dT

dt=

1

mc(Pj + Ps − Pc − Pr )






Tried methods for solving the ODE

Tried methods

ode45: Based on an explicit Runge-Kutta (4,5) formula, theDormand-Prince pair.

ode23: An implementation of an explicit Runge-Kutta (2,3) pair ofBogacki and Shampine.

ode113: A variable order Adams-Bashforth-Moulton PECE solver.

ode15s: A variable order solver based on the numerical differentiationformulas (NDFs).

ode23s: Based on a modified Rosenbrock formula of order 2.

ode23tb: An implementation of TR-BDF2, an implicit Runge-Kuttaformula with a first stage that is a trapezoidal rule step and a secondstage that is a backward differentiation formula of order two.

ode23t: An implementation of the trapezoidal rule using a ”free”interpolant.






Results for the diferents methods.

Set of 800 observations. Values of the parameters: absorptivity = 0.84,emissivity = 0.68, A = 0.78, B = 0.29 y limit velocity = 0.72.

Method MSE Time(s)

ode45 0,4574 19,9ode23 0,4566 15,7

ode113 0,4570 15,4ode15s 0,4543 17,4ode23s 0,4594 24,2

ode23tb 0,4562 19,22ode23t 0,4566 13,6

MSE and execution time.






Comparation between dinamyc and static models

MSE

Parameters Dynamic StaticSet 1 0,3879 0,4476Set 2 0,4116 0,4201Set 3 0,4601 0,5138Set 4 0,5522 0,5747Set 5 0,3366 0,4041Set 6 0,4929 0,5195Set 7 0,3529 0,3929

Comparation between dinamyc (ode23t) and static models.





Problem FormulationHarmony SearchNumerical resultsSensitivity analysis

Outline











Problem Formulation

Problem Formulation

Function to minimize:

mina1,...,a5

1

N

∑t

(x t(a1, ..., a5, u

t1, ..., u

t5)− θt

)2

Physical considerations

Range of existence:

absorptivity , emissivity , A, B ∈ (−0.05 1.25)

limit velocity ∈ (0.5 ∞)






Harmony Search: Definition of the algorithm

Inspired in

Imitates the music improvisation process applied by musicians. Each musicianimprovises the pitches of his/her instrument to obtain a better state ofharmony.

Some definitions

HM : Harmony MemoryHMS : Harmony Memory SizeHMCR : Harmony Memory Consideration RatePAR : Pitch Adjustment Ratebw : Distance to do the local searchNI : Number of Improvisations







x : HARMONY : (a1, a2, a3, a4, a5)

Inicialization of the Harmony Memory

HM =

a1

1 a12 a1

3 a14 a1

5

a21 a1

2 a23 a2

4 a25

... ... ... ...

aHMS1 aHMS

2 aHMS3 aHMS

4 aHMS5







Improvisation of new harmony

Once the HM is built, a new harmony is generated based on the HM or on analeatory factor: xnew = (xnew

1 , ..., xnewn )

Improvisation steps

Experience Consideration

Random Selection

Pitch Adjustment

Update of Harmony Memory







1 Step 1 : I n i t i a l i z e t h e HM.23 w h i l e ( s t o p c r i t e r i a i s not s a t i s f i e d ( c o n v e r g e n c e or NI ) )45 Step 2 : I m p r o v i s e a new harmony .67 f o r i =1:N % N i s the number o f d e c i s i o n v a r i a b l e s .8 i f U( 0 , 1 ) < HMCR % Memory c o n s i d e r a t i o n .9 x new ( i ) = x , % x i n the i−s t column o f HM.

10 i f U( 0 , 1 ) < PAR % Pi t ch Adjustment .11 x new ( i ) = x new ( i ) + (U(0 ,1)−0.5)∗bw ;12 end13 e l s e % Random s e l e c t i o n .14 x new ( i ) = U( 0 , 1 ) % I f the v a r i a b l e e x i s t s i n ( 0 , 1 )15 end16 end1718 Step 3 : Update t h e HM.1920 i f x new b e t t e r than x w o r s t ( i n HM )21 s u s t i t u t e x w o r s t by x new ( i n HM)22 end2324 end ( w h i l e )






Numerical results

Parameters of the model

Set of 2223 observations. Value of the parameters of HS:HMS = 12,HMCR = 0.9,PAR = 0.2, bw = 0.1,NI = 5000.

Parameter Value

Absorptivity 0.9531Emissivity 0.8655

A 0.5863B 0.0053

limit velocity 0.8675MSE 9.5454

Results of the optimization process.






Graphic results

Comparation between the measured and the calculated temperatures after theoptimization process






Sensitivity analysis

Linearizationx ′(t) = F

((x t(a1, ..., a5), a1, ..., a5, u

t1, ..., u

t5

)x(0) = x0

Analysis in: a1 = a∗1 , a2 = a∗2 , ... , a5 = a∗5 .

yj(t) = ∂aj xt(a1, ..., a5).

y ′j (t) = Fx(x t(a∗1 , a

∗2 , ..., a

∗5 ))yj(t) + Faj (x(a∗1 , a

∗2 , ..., a

∗5 ))

yj(0) = 0

Finite difference

yj =x t(aj + ε)− x t(aj)

ε






Results

Types∫ T

0

|y(s)|2ds∫ T

0

|y(s)|ds

maxs(|y(s)|)






Results

Square Absolut Max

absorptivity 35.67 77.82 0.80emissivity 5.89 ·104 6.02 ·103 21.01

A 9.74 ·105 2.56 ·105 76.27B 1.02 ·105 7.73 ·103 35.83

Linealization.

Square Absolut Max

absorptivity 40.31 101.79 0.87emissivity 4.19 ·104 5.07 ·103 18.52

A 7.08 ·105 2.17 ·104 65.21B 1.28 ·105 6.19 ·103 37.45

Finite difference.





Data filteringResults

Outline











Definition of the filters

Centered Moving Average (CMA)

Let Yt be a temporal series of data. CMA filter transforms this original serie as:

CMAN (Yt) =1

N

r= N2∑

r=− N2

wrYt−r ,

where

wr =

0.5 , r = ±N

2

1 , otherwise.






Definition of the filters

Kalman filter: Univariate local level

yt : Measurements vector. αt : Phenomenon which is described by a vector ofmagnitudes (state), that is unobservable in principle.

yt = αt + εt , εt ∼ N(0, σε)

αt+1 = αt + ηt , ηt ∼ N(0, ση)

So, with

at : E(αt |Yt−1)

Pt : Var(αt |Yt−1)

The Kalman filter updates at and Pt as:

at+1 = at + Kt(yt − at)

Pt+1 = Pt(1− Kt) + Qt

where Kt = Pt(Pt + Ht)−1 is the gain matrix.






Numerical results

Filter MSE

Without filter 0.664Kalman 1.2806CMA(1) 0.6788CMA(2) 0.6924CMA(3) 0.7114CMA(4) 0.7368CMA(5) 0.7502CMA(6) 0.7764CMA(7) 0.8022CMA(8) 0.8282CMA(9) 0.8674

CMA(10) 0.9054

MSE for differents filters.





Conclusions

Outline










Conclusions

Conclusions

Conclusions

The dynamic model for the evolution of the temperature has been solved.

This model is physicaly more realisitic than the static one.The MSE in the dynamic model is always smaller.

Implementation of an efficient method to determine the paramaters of themodel.

If we assume that the temperature of the conductor is measured goodenough, is not necessary to filter the measured data.





Conclusions

Ampacity Calculation of Long Line Overhead Cables

Julen Alvarez

Joint work with

Enrique Zuazua and Inaki Garabieta

BCAM seminar

17/01/2011


Ampacity Calculation of Long Line Overhead Cables

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