Transcript lecture4

Stochastic Optimization
Chance Constrained LP - II
Water Resources Planning and Management: M6L4
D Nagesh Kumar, IISc
Objectives
 To formulate chance constrained LP (CCLP) for reservoir
operation
 To discuss linear decision rules
 To discuss deterministic equivalent of a chance constraint
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Water Resources Planning and Management: M6L4
D Nagesh Kumar, IISc
Introduction
 In reservoir planning and operation problems, future inflow, Qt, is a random variable
and is not known with certainty
 Its probability distribution may be estimated from the historical sequence of inflows
 Storage St and release Rt, are functions of inflow, Qt
 St and Rt are also random variables
 In a constraint containing two random variables, if the probability distribution of
one is known, the probabilistic behavior of the second can be expressed as a measure
of chance in terms of the probability of the first variable
 If a constraint contains more than two random variables, we get into computational
complications
 We need to understand the specific problem clearly to reformulate the problem, if
necessary, and avoid those complications.
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Water Resources Planning and Management: M6L4
D Nagesh Kumar, IISc
Chance Constraint
 The constraint, relating the release, Rt, (random) and demand, Dt, (deterministic), is
expressed as a chance constraint
P [Rt ≥ Dt] ≥ 1
 Probability of release equaling or exceeding the known demand is at least equal to
1; 1 is the reliability level
 Interpretation of this chance constraint is simply that the reliability of meeting the
demand in period t is at least 1
 Similarly, chance constraints for the maximum release and the maximum and
minimum storage is
P [Rt ≤ Rtmax] ≥ 2
P [St ≤ K] ≥ 3
P [St ≥ Smin] ≥ 4
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Water Resources Planning and Management: M6L4
D Nagesh Kumar, IISc
Chance Constraint…
 To use the above chance constraints in an optimization algorithm, first determine the
probability distribution of Rt and St
 Probability distribution of Qt is known
 Probability distribution of Rt and St can be derived from that of Qt
 Since St , Qt and Rt are all interdependent through the continuity equation, it is not
possible to derive the probability distributions of both St and Rt simultaneously.
 To overcome this difficulty and to enable the use of linear programming in the
solution, a linear decision rule is appropriately defined
 Linear decision rule (LDR) relates the release, Rt, from the reservoir as a linear
function of the water available in period t.
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Water Resources Planning and Management: M6L4
D Nagesh Kumar, IISc
Linear Decision Rule
 Simplest form of such an LDR is
Rt = St + Qt - bt
where bt is a deterministic parameter called the decision parameter.
 In this LDR, the entire amount, Qt, is taken into account while making the release
decision.
 Depending on the proportion of inflow, Qt, used in the linear decision rule, a number
of such LDRs may be formulated.
 General form of this LDR may be written as
Rt = St + t Qt - bt
0  t  1
t = 0 yields a relatively conservative release policy with release decisions related
only to the storage, St;
t = 1 yields an optimistic policy where the entire amount of water available
(St + Qt), is used in the LDR.
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Water Resources Planning and Management: M6L4
D Nagesh Kumar, IISc
Linear Decision Rule…
 Consider the LDR
Rt = St + Qt - bt
 Storage continuity equation is
St+1 = St + Qt - Rt
 Using above two equations
St+1 = bt
 Random variable, St+1, is set equal to a deterministic parameter bt
 Thus, the role of the LDR in this case is to treat St, as deterministic in formulation
 Advantage: To do away with one of the random variables, St, and the distribution of
the other random variable, Rt, may be expressed in terms of the known distribution of
Qt.
 This implies that the variance of Qt is entirely transferred to the variance of Rt.
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Water Resources Planning and Management: M6L4
D Nagesh Kumar, IISc
Linear Decision Rule…
 For eg. assume that the evaporation loss Et is a linear function of the average storage
volume,
e0t + et (St + St+1)/2
where e0t is the fixed evaporation volume loss and
et is the evaporation volume loss per unit average storage volume in period t
 Then the linear decision rule can be written as,
 e 
 e 
Rt  Qt  e0t  1  t bt 1  1  t bt
2
2


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Water Resources Planning and Management: M6L4
D Nagesh Kumar, IISc
Deterministic Equivalent of Chance
Constraint
Knowing the probability distribution of inflow, Qt , it is possible to obtain the
deterministic equivalents of the chance constraints using the LDR, as follows:
P [Rt ≥ Dt] ≥ 1
P [St + Qt - bt ≥ Dt] ≥ 1
P [bt-1 + Qt - bt ≥ Dt] ≥ 1
P [Qt ≥ Dt + bt - bt-1] ≥ 1
P [Qt ≤ Dt + bt - bt-1] ≤ 1- 1
The term Dt + bt – bt-1, is deterministic with bt and bt-1 being decision variables and Dt
being a known quantity for the period t.
FQt (Dt + bt – bt-1)  1 - 1
(Dt + bt – bt-1)  FQt-1 (1 - 1)
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Water Resources Planning and Management: M6L4
D Nagesh Kumar, IISc
Deterministic Equivalent of Chance Constraint…
(Dt + bt – bt-1)  FQt-1 (1 - 1)
FQt-1 (1 - 1) is the flow, qt, at which the CDF value is (1 - 1).
Similarly,
Deterministic equivalent of chance constraint P(Rt  Rtmax)  2 is
Rtmax+ bt – bt-1  FQt-1 (2)
Since the storage, St+1, is set equal to the deterministic parameter, bt, the chance
constraints containing only the storage random variable are written as deterministic
constraints (without using the probability distribution of inflows).
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Water Resources Planning and Management: M6L4
D Nagesh Kumar, IISc
Chance Constrained LP
Complete deterministic equivalent of the CCLP can be written as
Minimize K
Subject to
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(Dt + bt – bt-1)  FQt-1 (1 - 1)
for all t
Rtmax+ bt – bt-1  FQt-1 (2)
for all t
bt-1  K
for all t
bt-1  Smin
for all t
bt  0
for all t
K 0
for all t
Water Resources Planning and Management: M6L4
D Nagesh Kumar, IISc
Chance Constrained LP…
 While solving this model, for a problem with 12 periods (months) in a year, we also
set b0 = b12 for a steady state solution.
 Further, depending on the nature of LDR used, the decision parameters, bt, may be
unrestricted in sign.
 For example, if we use the LDR,
Rt = St - bt,
the decision parameter, bt, may be allowed to take negative values.
 CCLP can also be applied for reliability based reservoir sizing
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Water Resources Planning and Management: M6L4
D Nagesh Kumar, IISc
Example:
For the following chance constrained optimization problem, formulate the equivalent
deterministic optimization problem using the LDR, Rt = St – bt. Storage continuity
should be maintained. Neglect losses. Following table gives the F-1( ) values for the
inflows and Rmax and Rmin values for different periods.
Minimize K
P [Smin ≤ St ≤ K] ≥ 0.8  t
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P [Rt ≤ Rtmax] ≥ 0.85
 t
P [Rt ≥ Dt ] ≥ 0.75
 t
t
F-1(0.0)
F-1(0.2)
F-1(0.25) F-1(0.75) F-1(0.8)
F-1(0.85) Rmax
Rmin
Smin
1
0
12
33
60
90
93
90
24
2
2
0
3
20
48
60
80
84
20
2
3
0
6
21
36
72
85
84
20
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Water Resources Planning and Management: M6L4
D Nagesh Kumar, IISc
Example…
Solution
Deterministic equivalent of P [Smin ≤ St ≤ K] ≥ 0.8  t.
This can be divided into two constraints as
P [Smin ≤ St ] ≥ 0.8 and P [ St ≤ K] ≥ 0.8.
Linear decision rule is given as Rt = St – bt.
Then, continuity equation can be written as
St+1 = St + Qt –R
= St + Qt – St + bt
= Qt + bt
Hence,
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St = Qt-1 + bt-1
Water Resources Planning and Management: M6L4
D Nagesh Kumar, IISc
Example…
Deterministic equivalent of P [Smin ≤ St ] ≥ 0.8
P [Smin ≤ Qt-1 + bt-1] ≥ 0.8
P [Qt-1 ≥ Smin - bt-1] ≥ 0.8
P [Qt-1 ≤ Smin - bt-1] ≤ (1- 0.8)
P [Qt-1 ≤ 2 – bt-1] ≤ 0.2
FQt-1 (2 – bt-1)  0.2
(2– bt-1)  FQt-1-1 (0.2)
Therefore,
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(2– b3)  6
t =1
(2– b1)  12
t=2
(2– b2)  3
t=3
Water Resources Planning and Management: M6L4
D Nagesh Kumar, IISc
Example…
Deterministic equivalent of P [ St ≤ K] ≥ 0.8
P [St ≤ K] ≥ 0.8
P [Qt-1 + bt-1≤ K] ≥ 0.8
P [Qt-1 ≤ K - bt-1] ≥ 0.8
K - bt-1 ≥ FQt-1-1 (0.8)
Therefore,
K – b3 ≥ 72
t =1
K – b1 ≥ 90
t=2
K – b2 ≥ 60
t=3
Deterministic equivalent of P [Rt ≤ Rtmax] ≥ 0.85
P [St - bt ≤ Rtmax] ≥ 0.85
P [Qt-1 + bt-1 - bt ≤ Rtmax] ≥ 0.85
P [Qt-1 ≤ Rtmax - bt-1 +bt] ≥ 0.85
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Water Resources Planning and Management: M6L4
D Nagesh Kumar, IISc
Example…
Rtmax - bt-1 +bt ≥ FQt-1-1 (0.85)
90 – b3 +b1 ≥ 85
t =1
84 – b1 +b2 ≥ 93
t=2
84 – b2 +b3 ≥ 80
t=3
Deterministic equivalent of P [Rt ≥ Dt ] ≥ 0.75
P [St - bt ≤ Dt] ≥ 0.75
P [Qt-1 + bt-1 - bt ≤ Dt] ≥ 0.75
P [Qt-1 ≥ Dt - bt-1 +bt] ≥ 0.75
Dt - bt-1 +bt ≤ FQt-1-1 (0.25)
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24 – b3 +b1 ≤ 21
t =1
20 – b1 +b2 ≤ 33
t=2
20 – b2 +b3 ≤ 20
t=3
Water Resources Planning and Management: M6L4
D Nagesh Kumar, IISc
Thank You
Water Resources Planning and Management: M6L4
D Nagesh Kumar, IISc