Model description Solution of the dinamyc model Parameter determination Data filtering Conclusions and future work Ampacity Calculation of Long Line Overhead Cables Julen Alvarez Joint work with Enrique Zuazua and I˜ naki Garabieta BCAM seminar 17/01/2011 Julen Alvarez Aramberri - 17/01/2011 Ampacity Calculation of Long Line Overhead Cables
34
Embed
Ampacity Calculation of Long Line Overhead Cables - · PDF fileModel description Solution of the dinamyc model Parameter determination Data ltering Conclusions and future work Ampacity
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
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
Model descriptionSolution of the dinamyc model
Parameter determinationData filtering
Conclusions and future work
Motivation
Julen Alvarez Aramberri - 17/01/2011 Ampacity Calculation of Long Line Overhead Cables
Model descriptionSolution of the dinamyc model
Parameter determinationData filtering
Conclusions and future work
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
Julen Alvarez Aramberri - 17/01/2011 Ampacity Calculation of Long Line Overhead Cables
Model descriptionSolution of the dinamyc model
Parameter determinationData filtering
Conclusions and future work
Outline
1 Model descriptionMathematical ModelsDefinition of the variables
2 Solution of the dinamyc modelDeduction of the modelNumerical resolution of the ODE
Julen Alvarez Aramberri - 17/01/2011 Ampacity Calculation of Long Line Overhead Cables
Model descriptionSolution of the dinamyc model
Parameter determinationData filtering
Conclusions and future work
Deduction of the modelNumerical resolution of the ODE
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 )
Julen Alvarez Aramberri - 17/01/2011 Ampacity Calculation of Long Line Overhead Cables
Model descriptionSolution of the dinamyc model
Parameter determinationData filtering
Conclusions and future work
Deduction of the modelNumerical resolution of the ODE
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.
Julen Alvarez Aramberri - 17/01/2011 Ampacity Calculation of Long Line Overhead Cables
Model descriptionSolution of the dinamyc model
Parameter determinationData filtering
Conclusions and future work
Deduction of the modelNumerical resolution of the ODE
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.
Julen Alvarez Aramberri - 17/01/2011 Ampacity Calculation of Long Line Overhead Cables
Model descriptionSolution of the dinamyc model
Parameter determinationData filtering
Conclusions and future work
Problem FormulationHarmony SearchNumerical resultsSensitivity analysis
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 ∞)
Julen Alvarez Aramberri - 17/01/2011 Ampacity Calculation of Long Line Overhead Cables
Model descriptionSolution of the dinamyc model
Parameter determinationData filtering
Conclusions and future work
Problem FormulationHarmony SearchNumerical resultsSensitivity analysis
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
Julen Alvarez Aramberri - 17/01/2011 Ampacity Calculation of Long Line Overhead Cables
Model descriptionSolution of the dinamyc model
Parameter determinationData filtering
Conclusions and future work
Problem FormulationHarmony SearchNumerical resultsSensitivity analysis
Harmony Search: Definition of the algorithm
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
Julen Alvarez Aramberri - 17/01/2011 Ampacity Calculation of Long Line Overhead Cables
Model descriptionSolution of the dinamyc model
Parameter determinationData filtering
Conclusions and future work
Problem FormulationHarmony SearchNumerical resultsSensitivity analysis
Harmony Search: Definition of the algorithm
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
Julen Alvarez Aramberri - 17/01/2011 Ampacity Calculation of Long Line Overhead Cables
Model descriptionSolution of the dinamyc model
Parameter determinationData filtering
Conclusions and future work
Problem FormulationHarmony SearchNumerical resultsSensitivity analysis
Harmony Search: Definition of the algorithm
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 )
Julen Alvarez Aramberri - 17/01/2011 Ampacity Calculation of Long Line Overhead Cables
Model descriptionSolution of the dinamyc model
Parameter determinationData filtering
Conclusions and future work
Problem FormulationHarmony SearchNumerical resultsSensitivity analysis
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.
Julen Alvarez Aramberri - 17/01/2011 Ampacity Calculation of Long Line Overhead Cables
Model descriptionSolution of the dinamyc model
Parameter determinationData filtering
Conclusions and future work
Problem FormulationHarmony SearchNumerical resultsSensitivity analysis
Graphic results
Comparation between the measured and the calculated temperatures after theoptimization process
Julen Alvarez Aramberri - 17/01/2011 Ampacity Calculation of Long Line Overhead Cables
Model descriptionSolution of the dinamyc model
Parameter determinationData filtering
Conclusions and future work
Problem FormulationHarmony SearchNumerical resultsSensitivity analysis