Transcript lecture2

Stochastic Optimization
Uncertainty and Reliability
Analysis
Water Resources Planning and Management: M6L2
D Nagesh Kumar, IISc
Objectives
 To learn the major elements of planning process

Uncertainty

Reliability
 To discuss various methods for analyzing the uncertainty
 To compute reliability using various methods
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Water Resources Planning and Management: M6L2
D Nagesh Kumar, IISc
Uncertainty
 Most water resources decision problems face the risk of uncertainty
 Due to the randomness of the variables that influence the performance of the systems
 Uncertainty in water resources arises mainly due to the stochastic nature of
 Hydrological processes such as rainfall, evaporation, temperature
 Other variables like future population growth, per capita water usages, irrigation patterns
etc.
 Depending upon the severity of uncertainty of the quantities involved, they are
replaced by either their expected value or some critical value (eg., worst case value)
and proceed with a deterministic approach.
 The expected value or median value is used when the uncertainty is reasonably small
and the performance of the system is not much affected.
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Water Resources Planning and Management: M6L2
D Nagesh Kumar, IISc
Sensitivity Analysis
 Simplest method for assessing the effect of uncertainty
 System performance is analyzed by varying the magnitude of the more uncertain
parameters
 This will help to identify the most sensitive parameter or variable in a system
 For example let the cost required for a structure of capacity k is C(k) = akb
 The parameter b represents the elasticity of costs and
b
k dC
C dk
or
dC
dk
b
C
k
 For a value of b = 0.6, a change in capacity by 10% will result in a cost change of
only 6%.
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Water Resources Planning and Management: M6L2
D Nagesh Kumar, IISc
First-order Analysis or Delta Method
 Used to estimate the uncertainty in a deterministic model formulation in which the
parameters involved are uncertain
 For example, in the estimation of weir discharge Q = CLH1.5
 If the parameters C and H are not certain, then Q is also uncertain
 Through first order analysis, one would be able to estimate the mean and variance of
a r.v. which is related to other variables which may also be random
 Combined effect of uncertainty can thus be assessed
 Consider a r.v. X which is a function of n other r.v. s
 Mathematically,
X = f (Y)
where Y = (Y1, Y2, …, Yn) is a vector of n variables
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Water Resources Planning and Management: M6L2
D Nagesh Kumar, IISc
First-order Analysis or Delta Method…
 Through Taylor’s expansion about the means of n r.v., the first order approximation
of the r.v X can be expressed as (ignoring the second and higher order terms)
 f 
X  f y   
 Yi  y i 

Y
i 1 
i Y y
n
(1)
where y   y1 , y2 ,..., yn  and
 f 
  is the sensitivity coefficient which is the rate of change of function value
 Yi  y
f (y) at Y  y
 The mean and variance of the r.v. X are approximated as
 X  EX   f y 
 n  f 

VarX   Var f y   Var    Yi  yi 
 i 1  Yi  y

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(2)
(3)
D Nagesh Kumar, IISc
First-order Analysis or Delta Method…

Var f y   0 since is calculated using the mean values of and is constant.
 Therefore,
 f 
where ai   
 Yi  y
n

VarX   Var  ai Yi  yi 
 i 1

(4)
 Eqn. (4) can also be expressed as
n 1
VarX    a   2  ai a j CovYi ,Y j 
n
i 1
2
i
2
i
n
(5)
i 1 j i 1
n
where VarX    ai2 i2 is the variance of the r.v Yi.
i 1
For uncorrelated Yi s CovYi ,Y j   0
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Water Resources Planning and Management: M6L2
D Nagesh Kumar, IISc
First-order Analysis or Delta Method…
 Therefore,
n
VarX    ai2 i2
(6)
i 1
 The coefficient of variation can be expressed as
C 
2
v X
 yi
  a 
i 1
 Y
n
2
i

Cv Y2
i

(7)
 Equations (6) or (7) express the relative contribution of uncertainty of each
component to the total uncertainty
 These can be used to minimize the effects of uncertainties in model output X
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Water Resources Planning and Management: M6L2
D Nagesh Kumar, IISc
Example:
Analyse the uncertainty in the channel discharge Q 
1
AR 2 / 3 S 1 / 2 where the parameter
n
R is certain and the parameters n and S are uncertain.
Solution:
Since R and A are certain Q  C n 1 S 1 / 2 where the constant C  AR 2 /.3
First order approximation of Q is
 Q 
 Q 
Q  Q    n  n     S  S 
 n  n ,S 
 S  n ,S 

S 1/ 2 
1 

S  S 
 Q   C 2  n  n   C
1 / 2 
n 
 2n S
 n ,S 

where Q  C 1 S 1 / 2 ;
n
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Q
Q
and
are sensitivity coefficients.
S
n
Water Resources Planning and Management: M6L2
D Nagesh Kumar, IISc
Example…
The uncertainty of Q assuming n and S as independent variables, can be expressed by
the variance operator as
 Q 
 Q 
 Q2     n2     S2
 n  n ,S 
 S  n ,S 
2
2
Expressing uncertainty in the form on coefficient of variance gives (as per eqn. 7)
C 
2
v Q
 Q 
 
 n 
2
n
 Q 
  Cv 2n   
 S 
Q 
2
S

Q

 Cv 2S

 Cv n  0.25 Cv S
2
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D Nagesh Kumar, IISc
Reliability
 Hydrosystems are expected to exhibit resistance against any external stresses
 Resistance or strength of any system is its ability to function without failure against
the external stresses
 Reliability is the probability of the system functioning satisfactorily
 Consider the state of a system denoted by a random variable Xt at time t for t =
1,2,…,n.
 If the possible outcomes of Xt can be divided into two sets:
 (i) satisfactory outputs or successes, S and
 (ii) unsatisfactory outputs or failures, F.
 Then, reliability can be expressed as
  PX t  S 
 For example, the reliability of a water supply system depends on the conditions when
supply is greater than demand i.e., successes.
 Reliability is the ratio of non-failures to the total periods.
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Water Resources Planning and Management: M6L2
D Nagesh Kumar, IISc
Reliability Computation using load-resistance
method
 Reliability: Probability that the capacity of the system (or resistance) exceeds or
equals the loading
i.e., .
  PL  R
 Risk is just the opposite of reliability
 Risk is the probability of the loading exceeding the resistance
i.e., .
'  PL  R  1  
 In load-resistance method, reliability can be computed by
 By direct integration
 Using safety margin and safety factor
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Water Resources Planning and Management: M6L2
D Nagesh Kumar, IISc
Reliability Computation using load-resistance method…
Computation by direct integration:
 Let the joint PDF of load and resistance be
f L ,R l , r  .
 Then reliability is
 r
    f L ,R l , r  dl dr
0 0
 If loading and resistance are independent, then,

 
0
r

f R r   f L l  dl  dr
0


  f R r  FL r  dr
0
where FL r  is the CDF of the loading at L = r.
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Water Resources Planning and Management: M6L2
D Nagesh Kumar, IISc
Reliability Computation using load-resistance method…
Computation using safety margin and safety factor
 Safety margin (SM): Difference between the system’s resistance and the anticipated
loading
SM = R – L
i.e.,
 Reliability can be expressed as,   PR  L  0  PSM  0
 This requires the PDF of SM
 Assuming that the loading and resistance are independent normal r.v. s, the mean
and variance of SM are
 SM   R   L
 Then reliability is
and
 SM   SM   SM
  P


 SM
SM



 P Z  SM
 SM

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2
 SM
  R2   L2
Water Resources Planning and Management: M6L2


   SM

  SM






D Nagesh Kumar, IISc
Example:
The average surface runoff to a sewer is 3 m3/s with a standard deviation of 1.2 m3/s.
The mean capacity of the sewer is estimated to be 4.5 m3/s with a standard deviation of
0.8 m3/s. Compute the reliability using safety margin approach.
Solution:
μL = 3;
μR = 4.5;
σL = 1.2;
σR = 0.8
μSM = μR - μL = 1.5
σSM2 = σR2 + σL2 = 2.08
Therefore,

  Z 

1.5 
  0.851
2.08 
And risk is,
α’ = 1 – 0.851 = 0.149
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Water Resources Planning and Management: M6L2
D Nagesh Kumar, IISc
Reliability Computation using Time-to-Failure
Analysis
 Instead of considering the resistance and loading, only one r.v
i.e., time is considered
 Time-to-failure (T) of a system is the random variable with PDF fT(t) and is called
the failure density function.
 Then, the reliability within a time interval [0, t] can be expressed as

 t    fT t  dt
t
 Risk can be expressed as
t
' t    fT t  dt
0
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Water Resources Planning and Management: M6L2
D Nagesh Kumar, IISc
Example
The time to failure of a pump in a water distribution system is assumed to follow an
exponential distribution with the parameter λ = 0.0137/day (5 failures per year).
Compute the reliability for 10 days operation.
Solution
The failure density function is an exponential distribution function
fT(t) = λ e- λt = 0.0137 e- 0.0137 t , t ≥ 0
The reliability is

 t    e t dt
t
 e t  e 0.0137t ,
For 10 days,
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t0
α = e- 0.0137 *10 = 0.872
Water Resources Planning and Management: M6L2
D Nagesh Kumar, IISc
Thank You
Water Resources Planning and Management: M6L2
D Nagesh Kumar, IISc