Modelling inflows
for SDDP
Dr. Geoffrey Pritchard
University of Auckland / EPOC
Inflows – where it all starts
CATCHMENTS
thermal
generation
reservoirs
hydro
generation
transmission
consumption
In hydro-dominated power systems, all modelling and evaluation
depends ultimately on stochastic models of natural inflow.
Why models?
• Raw historical inflow sequences get us only so far.
- they can’t deal with situations that have never happened before.
• Autumn 2014 :
- Mar ~ 1620 MW
- Apr ~ 2280 MW
- May ~ 4010 MW
Past years (if any) with
this exact sequence are
not a reliable forecast
for June 2014.
What does a model need?
1. Seasonal dependence.
- Everything depends what time of year it is.
Waitaki catchment (above Benmore dam) 1948-2010
What does a model need?
2. Serial dependence.
- Weather patterns persist, increasing probability of shortage/spill.
- Typical correlation length ~ several weeks (but varying seasonally).
Iterated function systems
Let
 xt   NorthIsland inflow
   
 in week t
 yt   South Island inflow
Make this a Markov process by applying randomly-chosen linear
transformations, as in:
  0.2  0.4  xt   0.5 
     , 50% chance
 
 xt 1    0.7 0.6  yt   0.6 

  
 yt 1    0.8 0.5  xt   0.4 

   
  0.1  0.9  y    0.7  , 50% chance
 t   

(numerical values are only to illustrate the form of the model).
IFS inflow models
Differences from IFS applications in computer graphics:
• Seasonal dependence
- the “image” varies periodically, a repeating loop.
• Serial dependence
- the order in which points are generated matters.
Single-catchment version
Model for inflow Xt in week t :
X t  Rt  St X t 1
- where (Rt, St) is chosen at random from a small collection of
(seasonally-varying) scenarios.
The possible (Rt, St) pairs can be devised by quantile regression:
- each scenario corresponds to a different inflow quantile.
Scenario functions for the Waitaki
High-flow scenarios differ in intercept (current rainfall).
Low-flow scenarios differ mainly in slope.
Extreme scenarios have their own dependence structure.
Exact mean model inflows
• We can specify the exact mean of the IFS inflow model.
Inflow Xt in week t :
X t  Rt  St X t 1
Take averages to obtain mean inflow mt in week t :
mt  rt  st mt 1
where (rt, st) are the averages of (Rt, St) across scenarios.
• Usually we know what we want mt (and mt-1) to be; the resulting
constraint on (rt, st) can be incorporated into the model fitting process,
guaranteeing an unbiased model.
• Similarly variances.
• Control variates in simulation.
Inflow distribution over 4-month timescale.
(Model simulated for 100 x 62 years, dependent weekly inflows, general linear form.)
Hydro-thermal scheduling by SDDP
• The problem: Operate a combination of hydro and thermal power stations
- meeting demand, etc.
- at least cost (fuel, shortage).
• Assume a mechanism (wholesale market, or single system operator)
capable of solving this problem.
• What does the answer look like?
Structure of SDDP
Week 6
Week 7
Week 8
Structure of SDDP
Week 6
Week 7
min (present cost) + E[ future cost ]
s.t. (satisfy demand, etc.)
Week 8
Structure of SDDP
Week 6
S ps
Week 7
Week 8
min (present cost) + E[ future cost ]
s.t. (satisfy demand, etc.)
s
- Stage subproblem is (essentially) a linear program with discrete scenarios.
Why IFS for SDDP inflows?
• The SDDP stage subproblem is (essentially) a linear program with discrete
scenarios.
• Most stochastic inflow models must be modified/approximated to make them fit
this form, but ...
•
… the IFS inflow model already has the final form required to be usable in SDDP.