Simulation and Forecasting of Hydrological Power
Generation: An Alternative Approach
J. Andrew Howe, PhD
TransAtlantic Petroleum
Presented at the
Palisade EMEA 2012 Risk Conference
London
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Presentation Outline
Introduction
What is Hydrological Power?
next: Our Model
Hydro Simulation Model
Exploratory Data Analysis
next: Model Development
Model Development
next: Model Validation
Model Validation
next: Forecasting
Forecasting
next: Conclusion
Concluding Remarks
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Introduction
What is Hydrological Power?
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Hydrological power (hydro) is electricity generated when a moving
source of water is dammed.
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Specified volumes of water are allowed to flow through a turbine,
generating electricity.
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In some areas of the world, hydro is a significant source of cheap
energy.
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A few examples: Norway - 99%, Brazil - 83%, New Zealand - 65%,
North Korea - 56%1 .
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Damming river systems also can provide other benefits - flood
control, irrigation, and recreation.
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The power that a damn can generate is limited by the amount of
water in the source.
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On the next slide, we see a simple diagram of how power is generated
from a dam.
1
http://www.nationmaster.com
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Introduction
What is Hydrological Power?
How power is generated from a dam.2
2
http://ga.water.usgs.gov/edu/wuhy.html
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Introduction
What is Hydrological Power?
How power is generated from a dam.2
2
http://ga.water.usgs.gov/edu/wuhy.html
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Introduction
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What is Hydrological Power?
Generally, power from dams is classified as baseload power.
Baseload power plants are used to supply much of a region’s
continuous energy demand.
Common examples include nuclear, coal-fired, hydro. These types of
plants generally take a significant amount of time to reach peak
production; they are expected to generate power at a constant rate.
Baseload power plants typically have high fixed costs, but low variable
costs, so it is most economical to run them continuously.
The difference between baseload supply and peak power demand is
generally met by combustion turbine / combined-cycle gas plants or
out-of-system purchases. These sources are said to be dispatchable,
because they can be dispatched to meet demand.
CC and CT plants are very dispatchable, as they spin up to peak
production quickly; they have lower fixed costs and higher variable
costs, however.
While nominally a baseload source, hydroelectric power can be
somewhat dispatched by controlling the number of sluice gates open,
and how much they are open.
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Introduction
What is Hydrological Power?
Example power demand curve for winter.
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The red curve here indicates an average power demand curve during
winter months, which could be met using
i) blue line - baseload hydroelectric, coal, nuclear
ii) black line - gas (CT / CC)
iii) remainder - purchases
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Because of its role in cheaply supplying baseload power, it is
important to be able to model and forecast hydro power.
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Introduction
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What is Hydrological Power?
Sources of variability in hydrological power generation include:
i) weather and precipitation characteristics
ii) scheduling and operating procedures
iii) economic drivers of supply and demand
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A typical approach to this kind of modeling would be to stochastically
model the meteorological and economic drivers, and deterministically
model the operating procedures.
Scheduling and operating procedures would generally be coded into a
seasonal model that balances constraining requirements such as:
i) reservoir levels must accommodate seasonal recreational and flood
control uses
ii) spill rates must not cause ecological damage
iii) planned maintenance
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Reservoir level variability is affected by weather and precipitation in
several ways, which we can see on the familiar hydrological cycle.
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Introduction
What is Hydrological Power?
The hydrological cycle.3
3
http://www.wpclipart.com/science/earth/water_cycle_USGS.png.html
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Introduction
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next: Our Model
In this case study, we present an alternate method to stochastic
modeling and forecasting of hydro power generation.
Our proposed approach infers the seasonal distributional and
variability characteristics of the system’s drivers directly from the
modeled data.
Our model breaks the calendar year into seven seasons of weeks based
on similarities in the distributional shapes, variabilities, and trends.
Each season is stochastically modeled independently, then stitched
together.
Descriptive statistics measured on 50 years of both modeled and
simulated data match very closely - average generation, variability,
and first-order correlation most notably.
There is no visible difference between simulated hydro generation
histories and the modeled data.
The model makes it easy to layer-in deterministic components, such
as trends and levels, for scenario testing.
We implement a novel “library” method to match actual power
generation at a specific season and forecast future results.
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Hydro Simulation Model
Exploratory Data Analysis
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The question underlying this case study is “How can we simulate a
history of power generation from dams in a way that gives us the
flexibility to model various scenarios, match reality, and provide
forecasts?”
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The source data is composed of 50 years (1954-2003) of modeled
weekly hydro generation from a large dammed river system in the
United States.
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This data, which we’ll call “actual”, was generated by a system
scheduling model based on actual weather / precipitation data and
dam operation procedures.
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Hydro Simulation Model
Exploratory Data Analysis
Actual data.
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Visual inspection of the data suggests a seasonal cyclical nature,
which is expected.
There is no other consistent pattern visible - note the irregularity of
the orange 200-week moving average line.
We initially focused on the cyclical behavior of the data; on the next
slide, we have the autocorrelation plot.
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Hydro Simulation Model
Exploratory Data Analysis
Weekly autocorrelations.
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In this context, it is much easier to directly model just the first lag
than it is the others, so we focused on the 82% 1-week correlation.
After modeling the first-order autoregressive relationship, the
time-ordered residuals show no sign of a trend, and variability seems
constant, as seen on the next slide
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Hydro Simulation Model
Exploratory Data Analysis
Residuals after first-order autoregressive model.
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Regression analysis of various rolling periods in this data show that
despite periodic higher/lower levels of power generation, it remains
constant on average.
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Thus, we justify building our @Risk simulation model to the mean
shape.
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Hydro Simulation Model
Exploratory Data Analysis
Seasonal patterns seen in five selected years of data.
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Here we have plotted five evenly-spaced years of weekly power so as
to visualize the seasonal patterns.
Each of these years shows a similar pattern:
i)
ii)
iii)
iv)
slightly positive trend with higher generation initially
downward trend as the season progresses
prolonged flat / whipsaw period of low generation
rapid up trend as summer ends
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Hydro Simulation Model
Exploratory Data Analysis
Average seasonal pattern suggesting seven distinct periods.
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With the previously mentioned autoregressive structure, it would
generally be a straightforward task to build a model around these
averages.
However, this problem is much more complicated than that.
Besides the seasonal trends and levels observed here, the shape and
scale of the week distributions varies dramatically from season to
season.
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Hydro Simulation Model
next: Model Development
Distributions for weeks 9, 26, 51.
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While only three examples are shown here, the general observations
apply to all 52 weeks, with gradual shifts in distribution shape:
i) February / March: Lower limit of 95% interval is highest, distribution is
skewed toward higher power generation
ii) June: 95% interval is the narrowest, with distribution more strongly
skewed toward lower generation
iii) December: Slight tendency for higher power generation, but
distribution is rather uniform
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Directly modeling an autoregressive process with this much
distributional variation can be a difficult proposition.
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Hydro Simulation Model
Model Development
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Rather than using a smooth autoregression, to model the seasonality,
we break the average into the seven disjoint seasons for modeling.
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In each season and for each week, we used the distribution fitting
functionality of @Risk to estimate and score various distributions of
weekly power generation.
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Each distribution fit was scored using the chi-squared,
Kolmogorov-Smirnov, and Anderson-Darling statistics; the
distribution that was selected by the majority of the tests was used.
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As a point of interest, in most seasons, the same distribution was
selected for each week.
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For each season, we estimated the slope from the average curve
already shown.
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The next slide shows details estimated for each season.
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Hydro Simulation Model
Model Development
Model details for each season.
Season
1
2
3
4
5
6
7
Weeks
1-9
10-19
20-23
24-29
30-31
32-43
44-52
Distribution Fit
BetaGeneral
Triang
Loglogistic
Invgauss
Pearson5
Logistic
Weibull
Slope
7, 390
−26, 542
19, 242
−10, 870
32, 629
−3, 203
16, 221
Standard Error
1, 651
1, 151
3, 702
1, 073
8, 013
714
1, 186
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In real life, seasons generally change gradually.
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If we simulate each season independently, then stitch them together,
we could end up with a very disjoint and unrealistic time series.
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Hydro Simulation Model
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Model Development
To solve this, we first computed the difference at each season
boundary, then used @Risk to select a best-fitting distribution for
each boundary.
These distributions are used in the simulation algorithm to adjust
seasons so the boundaries are all within the appropriate 98% intervals.
Seasonal boundary characteristics.
1-2
2-3
3-4
4-5
5-6
6-7
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Distribution
Loglogistic(-905946, 907310, 21.798)
Loglogistic(-334526, 358180, 6.4176)
Loglogistic(-4206.6, 48689, 1.8717)
Logistic(21061, 30445)
Logistic(-6908.6, 19485)
Loglogistic(-315443, 325251, 11.622)
98% Interval
−171, 090 214, 290
−159, 490 398, 410
−202, 080 164, 370
−118, 840 160, 960
−96, 444 167, 540
−96, 412 167, 540
For each year simulated, the algorithm is thus:
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Hydro Simulation Model
Model Development
i) For each season, draw a random value from the appropriate
distribution, this becomes the seed for the midpoint of the season.
ii) For each season, draw a Gaussian random value with µ = the slope
estimated for that season and with σ = t× the associated standard
error; this becomes the slope. t is a scaling factor drawn from a
triangular distribution =RiskTriang(0.5, 1, 5.5).
Because of the degree of right-skew, the scale factor will increase the
variability about 90% of the time. This scaling factor was tuned to
both match the variability and the first-order autocorrelation.
iii) Extrapolate each seed backward and forward in time, using the slope.
iv) If the boundary between any two seasons fall outside of the
appropriate 98% interval, generate a random value from the
appropriate distribution, and adjust the boundary week.
v) Add a translation factor generated by =RiskNormal(-15000, 35000) to
each observation; like t, this was tuned to match the source data.
vi) Finally, if any week in the raw simulated data falls outside of the
range of the actual data by more than 5%, it is truncated.
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Hydro Simulation Model
Howe
Model Development
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Hydro Simulation Model
next: Model Validation
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This @Risk model is run 1, 000 times for a burn-in period, then we
save the following 50 simulations to create a smooth record of
simulated hydro power generation.
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Computing 100 such 50-year simulations, after a single burn-in period,
requires slightly less than 2 minutes on a standard desktop computer.
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Model Validation
Example simulated year.
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Visual inspection of the green time series as compared to the earlier
chart showing actual years of data suggests this could very well be
from the actual data. It also models the average shape very well.
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Since we estimated much of the structure of our simulation from
averaged data, we had to iteratively tune certain parameters so as to
better match the actual data.
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Model Validation
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For validation, we simulated 100 50-year histories from our @Risk
model.
From each simulation, we computed the mean, median, 1st and 3rd
quartiles, standard deviation, interquartile range, and first-order
correlation, and averaged them all.
These averages were then compared to the same statistics estimated
from the actual data.
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Model Validation
Comparing average characteristics of simulated and actual data.
mean
median
1st quartile
3rd quartile
standard deviation
IQR
st
1 correlation
Howe
Actual
338, 143
329, 744
239, 619
428, 598
141, 837
188, 979
0.8229
Simulated
340, 278
323, 462
235, 296
436, 725
147, 238
201, 429
0.8317
Hydro Simulation
Deviation
0.63%
−1.91%
−1.80%
1.90%
3.81%
6.59%
1.08%
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Model Validation
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After simulating 50 years of data, we have the capability to add
deterministic components - trends, levels, and collars.
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At user-definable periods within the 50 years, we can add a trend (ex:
increase 5,000 MW per week) and / or a level (ex. decrease by
100,000 MW) adjustment.
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We also give the modeler the ability to enforce a minimum or
maximum on the entire simulated power generation history.
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All together, this allows us to simulate hydrological power generation
and make forecasts under a variety of scenarios.
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As an example, we modeled a slight uptrend from mid-1999, as well
as depressed generation between 1985 - 1988 and 1999 - 2002 in an
attempt to match the actual data.
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The next slide compares 30 years from one example simulation from
this scenario with the actual data.
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Model Validation
next: Forecasting
Which is actual and which is simulated?
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Model Validation
next: Forecasting
Which is actual and which is simulated? ↑ Actual, ↓ Simulation
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Forecasting
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A primary reason for modeling power generation is forecasting.
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Since meteorological data is not an input, how can we forecast future
power generation using this model?
For this purpose, we developed a novel “library” method to generate
forecasts and error bands. The algorithm is:
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i) Generate a large number of simulated 50-year hydro power generation
histories.
ii) Specify several weeks of real hydro generation, along with the week
number(s), and the number of forecast weeks desired, called T .
iii) Search all simulated histories, inspecting the specified weeks and
compute the difference between each simulated and real value.
iv) Simulation histories for which the differences are smaller than a
specified threshold are taken as starting points for the forecasts. The
next T weeks are taken as the forecast.
v) The average and error bands are computed from all selected sets of T
weeks.
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Forecasting
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next: Conclusion
We started with actual hydro generation in weeks 25 - 27 of 2009.
Using a cutoff of 5%, the system identified 34 portions of simulation
histories that matched.
In this plot, we see the first three black datapoints that were used to
match the histories. After that, the black line and red dashed lines
indicate the forecast and its error bands.
Example 17-week forecast with error bands.
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Forecasting
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next: Conclusion
We started with actual hydro generation in weeks 25 - 27 of 2009.
Using a cutoff of 5%, the system identified 34 portions of simulation
histories that matched.
In this plot, we see the first three black datapoints that were used to
match the histories. After that, the black line and red dashed lines
indicate the forecast and its error bands.
Almost all the out-of-sample observations fell inside the interval.
Example 17-week forecast with error bands.
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Concluding Remarks
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We have presented an alternate method to stochastic modeling and
forecasting of hydro power generation.
◮
Our proposed approach infers the seasonal distributional and
variability characteristics of the system’s drivers directly from the
modeled data.
◮
Descriptive statistics measured on 50 years of both actual and
simulated data match very closely.
◮
Our model makes it easy to layer-in deterministic components, such
as trends and levels, for scenario testing.
◮
Finally, we implemented a novel “library” method to match actual
power generation at a specific season and forecast future results.
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