Modelling inflows for SDDP

Dr. Geoffrey PritchardUniversity of Auckland / EPOC

Inflows – where it all starts

In hydro-dominated power systems, all modelling and evaluation depends ultimately on stochastic models of natural inflow.

CATCHMENTSCATCHMENTS

hydro generation

thermal generation

transmission consumption

reservoirs

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

2. Serial dependence.

- Weather patterns persist, increasing probability of shortage/spill.

- Typical correlation length ~ several weeks (but varying seasonally).

What does a model need?

Iterated function systems

(numerical values are only to illustrate the form of the model).

Make this a Markov process by applying randomly-chosen linear transformations, as in:

ty

x

t

t week in inflow Island South

inflow Island North

Let

chance 50% , 7.0

4.0

9.01.0

5.08.0

chance 50% , 6.0

5.0

6.07.0

4.02.0

1

1

t

t

t

t

t

t

y

x

y

x

y

x

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 :

- 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.

1 tttt XSRX

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

1 tttt XSRX

• We can specify the exact mean of the IFS inflow model.

Inflow Xt in week t :

Take averages to obtain mean inflow mt in week t :

1 tttt msrm 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.

(Model simulated for 100 x 62 years, dependent weekly inflows, general linear form.)

Inflow distribution over 4-month timescale.

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?

Week 6 Week 7 Week 8

Structure of SDDP

Week 6 Week 7 Week 8

min (present cost) + E[ future cost ]

s.t. (satisfy demand, etc.)

Structure of SDDP

- Stage subproblem is (essentially) a linear program with discrete scenarios.

Week 6 Week 7 Week 8

min (present cost) + E[ future cost ]

s.t. (satisfy demand, etc.) ps

s

Structure of SDDP

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.