Coping with the Ice Accumulation Problems on Power Transmission Lines P.N. Shivakumar 1, J.F.Peters 2, R.Thulasiram 3, and S.H.Lui 1 1 Department of Mathematics 2 Department of Electrical & Computer Engineering 3 Department of Computer Science (http://home.cc.umanitoba.ca/~shivaku/) University of Manitoba Winnipeg, Manitoba, Canada Dec.7-9, 2002 ICIWIM 2002 1
Outline! Problem statement! Related work! Definitions! Parallel implementation! Measurement of ice thickness! Results and discussions! Ice Storm 98 (Some pictures) Dec.7-9, 2002 ICIWIM 2002 2
Problem Statement During winter ice forms on electrical transmission lines, and due to the wind and increase in its weight, the lines with the electrical poles are likely to be pulled down causing heavy casualties. Every winter, power lines and communication towers collapse in freezing rain storms. " The glaze ice load on trees,wires,and structural members,combined with wind acting on the increased projected area, causes these failures. " Repair costs after a severe storm can be hundreds of millions of dollars, and electrical outages may deprive residents and businesses of heat,water,and power for extended periods. Dec.7-9, 2002 ICIWIM 2002 3
Problem Statement (Contd.) This was one of the major problems in the Ice Storm 98 of Montreal and other eastern parts of Canada. " 4 million people in freezing dark without heat and light because of destroyed electrical network " Other losses were millions of trees, 120,000 km of power lines, 130 major transmission towers and 30,000 electrical poles " To avoid such causalities, the ice formed should be melted as soon as it forms. A part of the ice melts because of the heat from transmission itself. Dec.7-9, 2002 ICIWIM 2002 4
Objective To develop mathematical model of Ice melting problem and approaches to measuring ice-accumulation. The design includes various factors by analyzing the boundaries between the different material regions and the heat transfer equations that govern the melting process. Ice melting is due to an applied fault current was developed. Dec.7-9, 2002 ICIWIM 2002 5
Ice Storm 98 Dec.7-9, 2002 ICIWIM 2002 6
Ice Storm 98 Ice Rolling Techniques Dec.7-9, 2002 ICIWIM 2002 7
Model! This model identifies four material regions around power lines and heat transfer equations were developed with appropriate Boundary Conditions in Cartesian coordinates.! Finally, a 2 dimensional cross-sectional model for melting ice on an electrical transmission line due to applied fault current was developed. Dec.7-9, 2002 ICIWIM 2002 8
Wire (1) Ice (3) Water (2) Dec.7-9, 2002 ICIWIM 2002 9
Q = i 2 r q = α i T c i T = div t q Γ 12 1 2 T (1) α 1 = α n α α T (1) n T (3) = 3 µ ( T env T (3 ) n α T (2 ) 2 T (2) n n = λ v 1 2 Γ 23 ) Γ 34 Γ 23 3 Dec.7-9, 2002 ICIWIM 2002 10
g Dec.7-9, 2002 ICIWIM 2002 11
Ice weight-forced convection Buogancy-driven Convection g Dec.7-9, 2002 ICIWIM 2002 12
Ice Metal Water Dec.7-9, 2002 ICIWIM 2002 13
Complete Model To account for the convection need to know " the velocity field of macroscopic motion of the liquid " the velocity of the ice as a whole with respect to resting wire (see Figure). The equations for velocities are obtained from the momentum balance. Momentum of a small volume of the liquid leads to the Navier-Stokes equations T( 2)! c ( ) 2 + v, T ( 2 ) = α 2 T ( 2 ). t Dec.7-9, 2002 ICIWIM 2002 14
While the global momentum of the ice brings in an additional integro-differential equation.! Ω dv ice 1 ( ) ( ) 3 3 g p x, y n x, y ds dt = Ω! +! ρ Γ 23 Dec.7-9, 2002 ICIWIM 2002 15
Initial Implementation Model! We do not consider the effects of gravity, convection and also the exterior climatic conditions.! Hence, the regions are shown in concentric circles rather than in ellipses. Dec.7-9, 2002 ICIWIM 2002 16
Governing Equations The governing are equations are: T() i Ci = α i T() i + Qi,... i = 1, 2,3 t Q 0 1 The Boundary Conditions are: Wire-Water Boundary Γ 12 a) T and heat flux are continuous T T b) (1) (2) α 1 = α 2 = 0 n n Water-Ice Γ 23 a) T1 = T2 = 0 (melting temperature) T(2) T(3) b) Stefan condition: α 2 α3 ( va( x), n) n = λ n Dec.7-9, 2002 ICIWIM 2002 17 Here Ω 1 Ω 2 Ω 3 Ω 4 -- metal (electric wire) -- water melted off ice -- ice -- atmosphere
Governing Equations cont d v A where is the velocity of ice melting in the normal direction γ ( B) given by v x = n A( ) n where γ ( B) is a position of a point on water-ice boundary Dec.7-9, 2002 ICIWIM 2002 18
Finite-Difference Equations The above equations are used to calculate the values of temperature in different regions Dec.7-9, 2002 ICIWIM 2002 19
Variables Used in the Equations 1-wire 2-water 3-ice! C1, C2 and C3 are the specific heat values.! T is the Temperature! n =Number of time steps! t = 0.0001! α = 0.007297+27! Grid size=( x) = ( y) = 0.002! i, j are the axes, with i pointing upwards! Q1 = internal heat source in the wire; a constant! Q2 = Q3 = 0 Dec.7-9, 2002 ICIWIM 2002 20
Implementation Details! Based on the mathematical model, Parabolic PDEs were subject to central differencing and transformed into Finite-Difference Equations (FDEs).! Cartesian coordinates were used and grid size of 1000 x 1000 was super-imposed onto the problem domain, such that the boundary of wire does not change with respect to time and each node of the grid is considered to be a computational node.! The values of temperature for different regions are calculated by solving these FDEs at each processor, until steady state is reached as t # infinity;! for example in the current project the time steps n #10 6. Dec.7-9, 2002 ICIWIM 2002 21
Implementation Details! Since the current model is a set of concentric circles, only one quadrant is considered for computation and analysis due to symmetricity.! This quadrant is distributed among four processors, each having 250 computational grid points in i and j direction individually.! Each processor computes the values of temperature in a SPMD (Single Program Multiple Data) manner.! Since there is a constant heat source in the wire (Q1), wire equations are not computed explicitly. Dec.7-9, 2002 ICIWIM 2002 22
Our Model P1 P2 P3 P4 4 x 4 grid Processors (P) Dec.7-9, 2002 ICIWIM 2002 23
Computational Issues! One other focus of the overall project is addressing computational issues such as partitioning, load balancing and communication/ synchronization! Decomposition or Domain Decomposition is the process of integrating the equations on rectangular domains, such that data can be decomposed uniformly by assigning rectangular sub-domains to each processor.! Load Balancing is the process of ensuring equal load among all processors, so that each processor has enough work and they finish execution of a task at the same time.! Communication and Synchronization between the processors are necessary as each processor performs only a segment of the main task, individually. Dec.7-9, 2002 ICIWIM 2002 24
Computation Platform Beowulf is a class of parallel workstations, developed to evaluate and characterize the design space of single user dedicated systems in price and performance. Cluster of Workstations (COW) Beowulf Workstation Model Name Intel Pentium III CPU Model Name Pentium III (Katmai) CPU MHz 1000.065 CPU MHz 501.146 Cache Size 256 KB Cache Size 512 KB Memory 526 MB Memory 526 MB Dec.7-9, 2002 ICIWIM 2002 25
Several steps in the current study $Fluid Dynamics/Partial Differential Equations $ Basic ideas of fluid mechanics and differential equations $Physics of heat transfer issues and boundary conditions $ Implications of changing a boundary condition is quite large. Importance of BC as well as how these BC affect the physics of the problem is addressed in this step $Finite-Difference method $ Though simple this technique has its own intricacies for implementation to capture the physics. $Implementation issues $ The machine characteristics and assigning the evenly distributed load on individual processors were addressed. Dec.7-9, 2002 ICIWIM 2002 26
Discussions! The Finite Difference code is producing acceptable results in the coarser grid, which is computationally less challenging. Physics is not fully captured in the coarser grid.! A finer computational grid takes more than one million time steps, for steady state solutions.! Experiments are being performed on a Beowulf cluster (parallel computing system).! The overall execution time is expected to reduce significantly, when compared to the sequential code.! More results are being generated! Fine tuning the code to capture the physics is going to take much longer time. Dec.7-9, 2002 ICIWIM 2002 27
Parallel Implementation of the Ice Melting problem Dec.7-9, 2002 ICIWIM 2002 28
Estimation of Ice Thickness on Wires Goal: Estimation of ice thickness covering transmission lines Available information: Digital images Problem One: Wires are not parallel to image borders Problem Two: Low image resolution Solution: Digital Image Processing Point of reference: The wire itself Task: Measure wire thickness Tool: δ-mesh Dec.7-9, 2002 ICIWIM 2002 29
Robot navigates along wire type 1 obstacle direction of robot movement a1 a2 a3 array of 3 "forward" proximity sensors a4 sensor to measure vertical distance % Sensors look for obstacles % Robot crawls around each detected obstacle % Line-crawling robot moves like a caterpillar Dec.7-9, 2002 ICIWIM 2002 30
Proposed Ice Measurement System %Digital Image Processing Issues %Measuring Frame of Ref. %System Calibration/Testing %Implementation %Experiments %Performance %Concluding Remarks Rough Inclusion δ-mesh Linear Interpolation Angle Detection Observed Wire Thickness Nearest Neighborhood Gouraud Shading Nearest Triangle Gouraud Shading Dec.7-9, 2002 ICIWIM 2002 31
Digital Image Processing, 1 Distance from an object when taking a picture Zoom used Magnitude Image resolution Location of an object Location Scene light (i.e., lighting conditions at the time that a picture is taken) Film speed Aperture Exposure time Aperture (i.e., variable opening by which light enters a camera) Dec.7-9, 2002 ICIWIM 2002 32
Digital Image Processing, 2 SF = f (aperture, location, magnitude ) A need for the frame of reference New problem formulation 1. Take a picture of a wire (partly bare and partly ice-covered). 2. Measure OWT(Observed Wire Thickness) and OIT(Observed Ice Thickness). 3. Given wire thickness, find out Scale Factor from equation OWT = SF * wire thickness 4. Find out ice thickness from equation OIT = SF * ice thickness Dec.7-9, 2002 ICIWIM 2002 33
Measuring Frame of Reference. Rough Inclusion y µ W x µ :!! [0,1] W wire Dec.7-9, 2002 ICIWIM 2002 34 sky
Measuring Frame of Reference. δ-mesh Rough Membership Fn. : µ Indistinguishability Relation Ind A : B y δ, δ ρ( X [ y] B ) ( X ) = δ ρ([ y] ) B Ing δ B = x x U a B a x δ = a x δ 2 A, ( ) = {(, '). ( )/ δ = ( ')/ δ } [ u] = equivalence classes of Ing ( B), u U B Each equivalence class of Ing A,δ (B) corresponds to one cell in the δ-mesh Dec.7-9, 2002 ICIWIM 2002 35 A, δ
Measuring Frame of Reference. δ-mesh Dec.7-9, 2002 ICIWIM 2002 36
Measuring Wire Thickness Image of the wire high variability Rotated image very small variability Dec.7-9, 2002 ICIWIM 2002 37
Measuring Wire Thickness Search for the best δ in each dimension to preserve only important information Dec.7-9, 2002 ICIWIM 2002 38
Contribution of this Work What is new: discovery of method of eliminating signal noise using δ-mesh cells. Significance: elements of a δ-mesh cells are computationally equivalent. Dec.7-9, 2002 ICIWIM 2002 39
Conclusion Mathematical model of internal heating of ice-clad power system transmission line Parallel computation approach to ice-melting suggested. Idea of δ mesh presented Approach to measuring thickness of the ice on a power transmission line given Digital image processing used in study of ice accumulation. Dec.7-9, 2002 ICIWIM 2002 40
Acknowledgements Manitoba Hydro Natural Sciences and Engineering Research Council (NSERC) of Canada Institute of Mathematics, Rzeszów University, Rzeszów, Poland University of Information Technology and Management, Rzeszów, Poland Dec.7-9, 2002 ICIWIM 2002 41
Thank You Dec.7-9, 2002 ICIWIM 2002 42