Stochastic Optimization ESI 6912

1
Stochastic OptimizationESI 6912
NOTES 5 CAPACITY EXPANSION MODEL

• Instructor Prof. S. Uryasev

2
Load Duration Curve
Demand
time
3
Piecewise Constant Approximation of Load Duration
Curve
Demand
Demand during time
time
4
Static Deterministic Analysis
Optimal power plant i(j) for segment j can be
found from equation

• type of power plant
• investment cost per 1 MW
• operating cost per 1 MW per unit time
• availability factor

5
Motivation for Multistage Model

• Long-term evolution of equipment costs
• Long-term evolution of load curve
• Appearance of new technologies
• Obsolescence of currently available equipment

6
Multistage Model Notations

• index of stages (periods)
• index of available
  technologies
• index of operating modes in
  load duration curve

• availability factor of technology
• lifetime of technology
• existing capacity of at time
  , decided before
• unit investment cost for at time
  (assuming a fixed kkkkk plant
  life cycle for each type of plant )

7
Multistage Model Notations (contd)

• unit production cost for i at time t
• maximal power demanded in mode j at
  time t
• duration of mode j at time t

Decision variables

• new capacity made available for
  technology at time
• total additional capacity of
  available at time
• capacity effectively used at time
  in mode

8
H-Stage Problem Deterministic Statement
s.t.
9
H-Stage Problem Stochastic Statement
s.t.

• new capacity decided at time for
  equipment ,
• total additional capacity of
  available and in order at time
• vector of random parameters at time
  (demands )

10
H-Stage Problem Stochastic Statement (contd)
If decisions variables do not
depend upon then problem is
deterministic
11
Example Problem Variables

• technologies
• construction delay
• existing equipment
• full availability
• takes values 3, 5, 7 with probabilities 0.3,
  0.4, 0.3

12
Example Problem Model
investment costs
production costs
s.t.
investment constraint
random demand, first duration mode
deterministic demand
13
Optimal Solution of Stochastic Problem
Value of performance function 381.85
Expected value solution
Value of performance function 365
is not feasible in stochastic problem
14
Alternative Approaches
Chance constraints
Quantile constraints
where is the distribution function of
In example