Upload
others
View
11
Download
1
Embed Size (px)
Citation preview
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
3 Parameter determinationProblem FormulationHarmony SearchNumerical resultsSensitivity analysis
4 Data filteringData filteringResults
5 Conclusions and future workConclusions
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
Mathematical ModelsDefinition of the variables
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
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
Mathematical ModelsDefinition of the variables
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.
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
Mathematical ModelsDefinition of the variables
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.
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
Mathematical ModelsDefinition of the variables
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)
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
Mathematical ModelsDefinition of the variables
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
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
Mathematical ModelsDefinition of the variables
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
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
Mathematical ModelsDefinition of the variables
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.
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
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
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.
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.
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
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.
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
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
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
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
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
Results
Types∫ T
0
|y(s)|2ds∫ T
0
|y(s)|ds
maxs(|y(s)|)
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
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.
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
Data filteringResults
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
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
Data filteringResults
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.
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
Data filteringResults
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.
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
Data filteringResults
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.
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
Conclusions
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
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
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.
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
Conclusions
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