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TECHNICAL REPORT
SUBJECTrrASK (title)
Review of the BC Hydro Marginal Cost Model (MCM)
SINTEF Energy Research
Address:
Reception:
Telephone:
Telefax:
7034 Trondheim
NORWAY
Sem Scelands ve; 11
+4773597200
+4773597250
http://ww . energy. sintef. no
E. No. :
CONTRIBUTOR(S)
f;e Mo
O(JjOSSO
CLlENT(S)
BC Hydro Power Supply Purchasing
NO 939350 675
TRNO.
DATE
CLIENT'S REF.
F4754
1998-
John W. Taylor
PROJECT NO.
lIX134.
ELECTRONIC FILE CODE
at;:fl
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J 0 Einar V remes , Powel Data AS
ISBN NO.
RES
REPORT TYPE
RCH DIRECTOR (
82- 594- 1318-
Nils Flatabfi
DIVISION
LOCATION
Power Generation and Market
Sem Srelandsvei 11
f4V
SIGN.
CLASSIFICATION
Restricted
COPIES
PAGES
LOCAL TELEFAX
+4773 5972 50
RESULT (summary)
The report contains a review of the Marginal Cost Model (MCM) developed by Power Supply
Operations of BC Hydro for resource optirnsation.
The purpose of this work is to determne whether or not the MCM is an appropriate decision support
tool and if the program development and maintenance are in accordance with the best industry
practices.
The MCM is a special purpose program used as decision support in the planning and operation of the
Wiliston Reservoir based on input representing: uncertain reservoir inflow
, expected power
production from the rest of the system , uncertain load , uncertain export market
, available import
possibilities and thermal power production. The program outputs are: incremental water values, power
productions, imports , exports and several other results.
The program has a high level of detailed modellng of the BC Hydro system and produces results that
are useful for decision support in many application areas. The most important are: power trading,
Willston Reservoir forecasting, financial forecasting, risk management and system operation
planning.
The MCM model has an input data representation and a mathematical formulation well in step with
what is used in Norway and elsewhere. The processing appears to be correct
, and the main conclusion
from the review is that the model is a useful tool and therefore a valuable asset for BC Hydro.
KEYWORDS
SELECTED BY
AUTHOR(S)
Hydro Operation Planning
Decision Support
Optirnsation
Simulation
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Executive Summary
The report contains a review of the Marginal Cost Model (MCM) developed by Power
Supply Operations of BC Hydro for resource optimization. The marginal cost is just one of
the results from the model. The program has been used for 12 years and has been
continuously developed to adapt to the changing planning environment.
The purpose of this work is to determne whether or not the MCM is an appropriate decision
support tool and if the program development and maintenance are in accordance with the best
industry practices.
The MCM is a special purpose program used as decision support in the planning and
operation of the Wiliston Reservoir based on input representing: uncertain reservoir inflow
. expected power production from the rest of the system, uncertain load , uncertain export
market , available import possibilities and thermal power production. The program outputs
are: incremental water values , power productions , imports , exports and several other results.
The program has a high level of detailed modellng of the BC Hydro system and produces
results that are useful for decision support in many application areas. The most important are:
power trading, Willston Reservoir forecasting, ' financial forecasting, risk management and
system operation planning.
The main conclusions from this evaluation related to the Scope of Work are:
1. Appropriateness of
input data
Inflow forecasting and modellng are state of the ar
Same input data should be used consistently for all BC Hydro models
Uncertainty should be modelled- for the Columbia River and the Pacific North West
(PNW) market
2. Test of
model results
Inflow and cost parameters were changed and model response monitored. Prom these
tests , the processing within the MCM appears to be adequate. The models providing the
input data were not checked.
3. Computation methodology
The MCM optimisation par is based on a mathematical formulation that is
well in step
with what is used in similar programs in Norway and elsewhere.
The simulation methodology is good.
The number of input scenarios should be increased to improve the usefulness of the
output.
4. Dealing with
uncertainties
The uncertainties of the physical input variables are modelled as well as possible.
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Uncertainty in the spot price for the first 18 months should be modelled. This is already
implemented in the MCM but the functionality must be activated.
5. Suitabilty, Significance and Usefulness
The MCM contains all relevant information concerning the physical and economic
operation of the system. It is therefore possible to build further on this model to get more
comprehensive tools for power system operation.
The MCM is a useful tool for BC Hydro today and wil probably be even more useful in
the future with increased competition and trading.
'. The MCM gives a number of results that should be used more extensively than today.
Currently only the expected (mean) values are used.
Probabilstic output of the model should be used for risk management.
6. Compare with
other models
The mathematical formulation is commonly used in Norway and elsewhere and is ideally
suited for the defined problem.
modifications
The MCM would be strengthened by the changes currently being proposed.
7. Proposed
The recommendations for future use of the MCM:
BC Hydro should take advantage of the probabilty that this type of tools wil be even
more useful in the future with increased competition and trading.
The MCM results should be used more extensively than today. Especially the
probabilstic output should be util sed to get a picture of the uncertainty of important
quantities such as power production and income.
The user environment could be improved to provide better access to the model results for
the user. A new user interface based on current technology should be implemented.
More people should be trained to use the program for sensitivity studies in order to get a
better understanding of program output and interaction between different input
parameters. This would enhance the confidence in the MCM and increase the
understanding of the operational complexity of hydro- thermal systems. Increased use wil
also uncover useful extensions and enhancements.
Models like the MCM need continuous development to adjust to changing market
conditions , to take advantage of better processing techniques , and to better simulate the
hydrology. Input data must also be continuously refined. Sufficient resources should be
allocated to do this work.
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TABLE OF CONTENT
INTRODUCTION..........................
:..................................... 5
OVERVIEW OF SYSTEM AND PLANNING PROCEDURE ...................................... 7
Power production system characteristics .............................................................. 7
Planning procedure............................................................................................... 7
MARGINAL COST MODEL (MCM) ................. """""""'''''''' ........................... ........... 9
Appropriateness of input data ............................................................................. 10
Inflow forecasting for Wiliston Reservoir.......................................... 10
1.1
1.2 Domestic load forecast.........................................................................
Generation forecast.................................................. ............................. 12
Inport market.........................................
Export market................................................................... .................... 13
Economic parameters...... ............ ...........
""""'"'' 14
Quality of input data............................................................................. 14
Test of model results and output......................................................................... 14
Output from MCM ............... ........
........................................... 15
Case specification.......................... ....................................................... 15
3 .
3.2
Case results........................................................................................... 17
3.3
Discussion of computation methodology..................... ............. "'" ........ ...... ....... 23
3.3.
Optimisation algorithm......... ...........
"""""""" 23
3.3.
Simulation procedure............................................................................ 24
Discussion of assumptions ..................... ............................ ...... ............. 24
Adequacy of model in dealing with uncertainties............................................... 25
Suitabilty, significance and usefulness of model............................................... 26
General comments................................................................................ 26
Thermal operation planning ......................................................... ......... 26
Hydro facilty maintenance scheduling ................................................ 27
Electricity trade purchases or sales ....................................................... 27
Wiliston Reservoir elevation forecasting............................................. 28
Financial forecasts................................................................................ 28
Systems operation planning .................................................................. 29
Comparison with similar models ........................................................................ 29
Discussion of proposed modifications.. ................... ........................................... 32
3.3
3.4
5.4
GENERAL COMMENTS ........ ........................... .............. ........ ........ ................ ......... .... 33
CONCLUSIONS............................................................................................................ 35
REFERENCES .......... ........ ................
.... 36
APPENDIX I: USER DESCRIPTION OF USE OF MARGINAL COST MODEL.............. 37
APPENDIX 2: DETERMINISTIC VERSUS STOCHASTIC MODELLING........................ 46
APPENDIX 3: NORMAL RESERVOIR ELEVATION ......................................................... 49
APPENDIX 4: WATER VALUES FROM THE FIRST CASES STUDIES ..........................
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INTRODUCTION
The report contains an evaluation of the Marginal Cost Model (MCM) developed by the
Power Supply Operations deparment of BC Hydro. The program has a history of 12 years
and has been continuously developed to adapt to the changing planning environment.
The purpose of this work is to determne whether or not the MCM is a useful decision support
tool and if the program is developed and maintained in accordance with the best industry
practices.
The first par of this work was performed during a stay of 8 work days at BC Hydro where
discussion of the input data , program performance and use of results , was the main focus. The
report preparation has been done in Norway with valuable assistance from Donald Druce for
the discussion and interpretation of the program results.
The scope can be summarised as:
Validate the appropriateness and accuracy of the input data relative to the model
output and use.
Verify and test model results and output.
Examine and comment on the computation methodology and the capabilty and
effcacy of the optimisation algorithm , simulation procedure and assumptions.
Determne the adequacy of the model in dealing with input uncertainties and
sensitivity related to: flow , load , gas supply, energy and capacity supplies from other
plants, and the electricity market.
Evaluate the suitabilty, significance and usefulness of the model for Power Supply
Operations and Corporate business decision makng.
Compare the MCM with other models that are used for similar purposes at hydroelectric utilties in other pars of the world. Comment on similarities and differences
with respect to the physical system characteristics and modellng issues.
Review and comment on the appropriateness of any proposed modification to the
model , its inputs , outputs , computations and assumptions.
The complete Scope of Work is given in the section TR3 of the Invitation for Proposals with
reference PSQ8- 0 11. The approach and amount of work involved on each item is discussed in
each subsection.
Discussion of information technology standards is not par of the scope but a few comments
are given.
We have chosen to organise the report as follows:
The first chapter gives a brief introduction.
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The second chapter briefly summarises the adopted assumptions and planning
procedures of BC Hydro.
The third chapter discusses the different items as specified in Scope of Work. The
section numbering corresponds to the item number in the Scope. We have tried to
separate the different items as well as we can , but there might be some overlapping in
the report.
The fourth chapter discusses general observations regarding use and development of
the MCM model.
The fifth chapter concludes the findings.
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OVERVIEW OF SYSTEM AND PLANNING PROCEDURE
Power production system characteristics
The production system consists of 29 hydroelectric plants, one conventional thermal and two
combustion turbine stations. The thermal plants account for about 9 percent of the capacity
but accounts for little energy.
The hydroelectric plants are grouped into three:
Small hydroelectric plants
Columbia River system
Peace River system
Small hydroelectric plants
The majority of these have limited storage capacity and their operation is related to the local
hydrology rather than the system load. The year-to- year variabilty in the small hydro
generation for a given month is usually less that five percent.
Columbia River svstem
The Columbia River system consists of four plants with a total capacity of 4722 MW (44% of
production capacity). The energy produced is about 45% of the system s average yearly
energy production. This river system has many operational constraints and despite its size , it
does not contribute to the flexibilty of the system operation as much as expected from its
SIze.
Peace River system
The Peace River system consists of two plants with a total capacity of 3112 MW (32% of the
production capacity). The energy produced is about 37% of the system s average yearly
energy production. The total storage capacity is 40. 000 millon m . The main flexibilty of the
operation of the BC Hydro system is connected to this river system.
Planning procedure
The objective of operation planning is to maximise the net income from the available hydro
resources. The hydraulic coupling, the uncertain resource availability, uncertain market
conditions and operational constraints make this to a very complicated optimisation problem.
Simplifications are necessary to make such a problem computationally feasible. In the
procedure of simplification , it is important to take the system characteristics into account.
From the general characteristics listed above the following approach has been used: Schedule
the resources where the operating flexibilty is low first and then undertake the final
coordination procedure.
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Using this approach , Wiliston Reservoir wil be the one that gets the role as slack reservoir
and therefore defines the system marginal cost. There is no complex hydraulic coupling along
the waterway and both plants on the Peace River have about the same flow capacity and can
use the discharged water. The average yearly inflow is about 75 percent of the reservoir
storage capacity.
The hierarchy applied in the planning has made it possible to prepare a tailor-made model
for the long- term operation planing. It is assumed that a general pUrpose program would not
be appropriate to account for the special conditions in the system. The BC Hydro system is in
this planning structure very nice to model , but it is stil a challenge to come up with a model
formulation that takes maximum advantage of the system attributes.
The flexibilty in the operation of the Willston Reservoir makes it possible to use a time
resolution of one month , which again gives the opportunity to improve the modellng of other
items such as importexport capabilities.
All these assumptions simpliy the whole planning problem and the crucial question is then if
there is anything lost in the simplification process. We have evaluated the MCM model
assuming the assumptions to be correct. However, these basic assumptions are to some extent
challenged in section 3.3. 3 where we conclude that some kind of interaction between
Columbia River and the MCM model should be modelled.
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MARGINAL COST MODEL (MCM)
The marginal cost model (MCM), which is a resource optimisation model , is formulated as a
stochastic dynamic optimisation problem. The margInal cost is just one of the results from
the model.
The goal function of the model is to maximise the expected net income for the company in
the long run using the generation resources and the possible trading options.
The net income for a period is given by:
The sum of all sales in the export markets multiplied by sales price
minus all imports multiplied by import price
minus thermal production multiplied by production cost
minus unserved energy multiplied by cost of unserved energy
The cost of not fulfiling the firm obligations is included in the goal function as shown above.
The physical restrictions of the hydraulic system are modelled and included in the
optimisation.
The problem is dynamic since the reservoirs can be used to store water from one time period
to the next. It means that the decisions made in one time period have an impact on the
decisions made in the next time step.
Knowledge of the export price in one time step may also increase the possibilty to forecast
the price for the next time step, compared to a case where the price in the previous time is not
known.
The problem is stochastic since the future inflow , load and importexport prices are uncertain.
If all the input factors were includeej and modelled correctly, the stochastic dynamic
optimisation formulation would give the optimal and correct decisions. However , there is a
limit to the size of the problems that can be solved with available computer technology. The
number of state variables that can modelled gives one of the major limitations. The state
variables are given by the variables that couple the time periods (e. g. reservoir level).
The number of different uncertain variables that can be modelled gives another major limit.
In the MCM model , the Wiliston Lake storage volume , the water supply in the Pacific
Northwest (PNW) and the market price the PNW are all modelled as state variables.
The MCM uses the weather as a random variable with the following main implications:
I Whenever firm load exceeds system resources plus identified import opportunities
, the
MCM wil cut load in order to balance load and resources. This condition is likely to occur
only when the level of Wiliston Reservoir is extremely low and head losses are substantial.
The unserved load is priced at 120 mills/kWh, to reflect the expected loss to industry of a
curtailment. This functionality in the MCM serves to extend the feasible range of reservoir
levels , but at the same time heavily discourages the release of water at such low levels.
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It makes it possible to choose a release from Wiliston Reservoir that is conditional on
knowing the effect of weather on the domestic load and on the generation to be supplied
by the small hydro plants for the current month - since
GMSIPCN
usually load follow
. this adds a sense of realism and variabilty to the operation - weather is assumed to be
randomised on a month to month basis
Linking the historical small hydro generation to the weather year acknowledges that these
plants generally have small reservoirs with inflows primarily due to rainfall runoff, so
their generation pattern is influenced more by weather conditions and local operating
concerns than by system load or electricity trade opportunities - a realistic contribution
from the small hydro plants is added to the MCM without have to spend any CPU time
leaving more time for analysing market conditions
The MCM model uses mDntWy time resolution and has a planning horizon of six years.
Appropriateness of input data
is always possible to marginally improve the modellng. Our evaluation of the
appropriateness is based on what is practical and possible from our point of view.
It
Inflow forecasting for Wiliston Reservoir
The inflow forecast to all the reservoirs is given by historical observations from 1973 to 1997.
The historical values are corrected for snow pack information using a hydrological model
from UBC for Wiliston Lake and the Columbia River system. A simpler regression based
model is applied for the small hydro plants. The snow pack correction changes the expected
spring/summer inflow for the first year and also reduces the uncertainty in the forecast.
The historical expected annual inflows and the standard deviation are shown in Table 3.
Table 3.
Inflow statistics from 1984- 1997
Columbia River system
(MCA+REV +KCL+SEV)
Peace River system
(GMS+PCN)
Small h dro lants
21613
4320
30841
17570
17149
5613
2179
21463
7496
14557
3783
941
In order to be able to compare the inflows from year to year we have only used the inflows
from 1984 to 1997 since the system has been expanded in the years previous to 1984.
If we use the numbers from Table 3. 1 and the fact that these number are based on 14
observations, the expected inflow is with 68 % certainty between the numbers shown in Table
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2. The numbers in the table show that there is considerable uncertainty also regarding the
expected inflow to the system.
The actual numbers are somewhat smaller than indicated in the table since data from 1973 to
1997 are used in the MCM model. This type of uncertainty wil be reduced as more inflow
statistics become available. This last comment is based on the assumption that there are no
climatic changes.
Columbia River system
Peace River s stem
Small h dro
lants
(20458 22767)
(16566 17731)
(5361 5864)
+/- 5.3
+/- 3.4
+/- 4.
In our experience the inflow forecasting and modellng are in accordance with state of the
industry practice. However , we have not studied the hydrological snow models and the
regression model. Druce has published an article (1) where he shows that the UBC models
decrease the error in the inflow forecasting.
Domestic load forecast
In principle the domestic load forecast is a forecast for all the end users sales where the
quantity of the future obligations are not known through contracts. The load forecasting is
done once a year with a forecasting horizon of 20 years. Only the forecast for the first 6 years
is important for the MCM model , since this is the planning horizon of the model. The load
forecast is referred to normal temperature , and the variations in load caused by uncertainty in
the temperature are included in the MCM model.
In order to be able to quantify the uncertainty in the load forecasting, we have compared
forecasts for the same year made at different points in time. The forecasts are shown in Table
3. If we use the difference in the load forecast from one year to the next as an indication for
the forecasting uncertainty, the maximum change in the forecast for 96/97 is 1.6 % and 2. 6 %
for 97/98. We have not compared with measured load since these data are not corrected for
weather variations. The load forecast is referred to normal weather.
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Table 3.
94/95
95/96
96/97
97/98
98/99
99/00
00/01
01/02
02/03
03/04
Different load forecasts (GWh) made for same periods and different moments
in time.
50445.
51888.
53081.0
53593.
54708.
55803.
56885.
49255.
50480.
52472.
53680.
54719.
56056.
57174.
58058.
48853.
50064.
51644.
52590.
53767.
55363.
56614.
57874.
58833.
46736.
48526.
50036.
51758.
53648.
55651.0
57561.0
58481.0
59383.
60496.
If we use this limited number of old forecasts as indication of the load forecast uncertainty,
we see that uncertainty in the next years forecast is relatively small compared to the
uncertainty in the next years inflow.
Generation forecast
Generation forecasting is performed for all the generation units that are modelled as input to
the MCM model. This includes the small hydro plants, the Columbia River plants and
Burrard generation.
Currently the Columbia River production is calculated using a spreadsheet model. Input to the
calculation are Treaty rules and inflow forecast from the UBC model as described in section
1.1. Due to the complexity of the ''freaty rules , only the expected forecast for the Columbia
River system is generated.
The generation forecasts for the small hydro plants are given by the historical generation for
these plants (monthly generation for the period 1973- 1997).
Uncertainty in Columbia River generation forecast should be modelled and included into the
MCM model. In principle , a simultaneous optimisation of these two reservoirs would have
been preferable. However , due to the complexity of the planning problem this wil be very
difficult. Other procedures for coupling the model wil be required as for example iteration
between individual marginal cost models.
Import market
The import market consists of two separate markets, the Alberta and the Pacific Northwest
(PNW) markets. The prices in these two markets are specified independently.
The import price from Alberta at light load hours is given by the coal price and the price at
high load hours is linked to the swap prices for natural gas at AECO- C. Line capacities and
the number of light and heavy load hours give the maximum import quantity.
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The future coal price is much more certain than the future gas price.
Powerex forecasts , for the current year , the quantity and price of energy available from
Pacific Northwest. For the subsequent years a typical market pattern is assumed that reflects
BP A's fish flush operation.
The import market is split from the export market in order to account for the price differences
between heavy and light load hours. The difference between import and export disappears
, if
modellng of separate energy balances for heavy and light load hours are included in the
model as discussed in section 3. 7. Wheeling costs and losses are included in the market
pnces.
Export market
The export markets are essentially the same markets as the import markets: the Pacific
Northwest and Alberta. The export price in Alberta is connected to the gas price as described
in the previous section.
The export price in the Pacific Northwest is based on the COB index for the first 18 months
and corrected for wheeling and losses , i. e. referred to the border. The export price after the
first . 18 months is described by two simple Markov models.
The first Markov model describes the water conditions in the Pacific Northwest (PNW). The
model can take two values after the first calendar year, the value defines the market size used
in the model. After the first 18 months the value of the market size wil also influence on the
market price.
The second Markov model describes the prices in the PNW market. The model is currently
used from the 18 to last month in the planning period. The model can take three values that
combined with the first Markov mQ,qel defines six possible prices.
The market size for the first calendar year is given based on what Powerex believes the
physical tie lines capacities wil be corrected for the quantity already used by commtted
contracts. This is a manual process.
The price in the Pacific Northwest is the most important market input parameter since the line
capacity (3150 MW) is much larger compared to Alberta (600 MW).
If separate energy balances for the high load and light load hours are modelled
, the same
market model can be used for both import and export.
We have checked how the prices are modelled in the MCM model, not the quality of the price
forecasting methodology. The input to the MCM model should be the best available. The
MCM model does not constrain the price forecasting methodology.
Uncertainty in the PNW market should be modelled for the whole planning period as
discussed further in section 3.
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Economic parameters
The economic parameters used in the MCM model are:
Foreign exchange rate
Inflation rate
Discount rate
Foreign exchange rate
The exchange rate shows significant moves in the period from January 1992 and up to April
1998. The lowest value is 1.1748 (January 1992) and highest value is 1.4302 (April 1998).
There are large fluctuations within this period despite the main increasing trend.
For the first four quarters of the study period , the exchange rate is based on the average of the
forecasts from Nesbitt Burns , RBC- , Scotia McLeod and Wood Gundy. The rates for the
subsequent periods are taken from the 3 Year Plan - Rate assumptions.
Nominal discount and inflation rates
The nominal discount rate is the sum of the inflation rate and the real discount rate. The real
discount rate is the 8% corporate value and the inflation rate is taken from 3 Year Plan - Rate
Assumptions.
Important impacts of this economic parameter are:
The value of possible future income becomes lower. The value of storing water wil
less and the MCM wil therefore tend to use more water in the first year.
A high discount rate wil reduce the impact of an uncertain future.
be
The users of the MCM results must be aware of the discount rate used and its impact on the
marginal value of the water.
1.7
Quality of input data
The quality of the output from the MCM model results are not better than quality of the input
data. The input data to the MCM model should be based on the best that is available. The
input data could be improved by for example increasing the length of the inflow statistics,
applying more resources and models to exportimport price forecasting etc. The different
users within BC- Hydro should use the same input data , same price forecast , same load
forecast etc.
Test of model results and output
The MCM model describes a complicated process and it is therefore difficult to make tests
where the result can be calculated in advance , i.e manual calculation ofthe results. We have
therefore tried to prepare different cases where we know how some of the results (water
values) should change when the input is changed. It is not possible to quantify the change due
to complex interaction between model constraints , but we know in advance whether the water
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values should increase or decrease. This type of testing wil
model
Output
check the consistency
of the
from MCM
Currently two different types of reports are distributed based on the results from the MCM
model. Both reports are prepared by Donald Druce and distributed monthly.
The first report contains least information and has the largest distribution list. The report
contains median for marginal cost for the first 6 months and figures showing 10 , 50 and 90
percentiles for simulated reservoir level and marginal cost of water for the first 3 years. The
report also summarises some of the input to the study.
The second report is sent to Powerex. The following values are reported in tables:
Probability distribution for outflow from Wiliston Lake
Probabilty distribution for Elevation in Wiliston Lake
Expected discretionary sales and purchases GWh and $ for the next three fiscal years
Commtted sales and purchases
In addition it is also possible to report:
Probability distributions for sales and purchases
Production (GWh) and firm load
Spil conditions
The results have a time resolution of one month and can be relied upon for the first 3 years.
Case specification
In order to test the model we have defined six different cases. Each case has been prepared
and run by Donald Druce. The diffC:.tent
cases are specified as follows:
1. Base Case
Describes the import, export market and the physical production system by May 1998. The
description is identical to what was used in the May report by Donald Druce. Over the next
18 months the export prices are relatively high , as a result of the water supply shortage in the
PNW. The expected export prices beyond that time are lower, since the water supply
conditions are expected to be higher , on average . In the base case , the model already has the
signal to export heavily when the prices are relatively high and is generally doing that.
However , it is running up against constraints on market size (reflecting transmission limits)
and system generation limitations in the winter.
2. Burrard production
The assumptions are identical to the base case except for that Burrard production is cut by
600 GWh in June and July 1998.
2 The expected water conditions after the first 18 months are given by statistics and are assumed to be
independent of the current situation. The current water situation is below average. How far ahead this situation
influences on the future water conditions are given by the storage capacity and the dynamics of the PNW
system.
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When Burrard production is reduced the water values should increase independently on the
reservoir level.
3. Export price
The export price to the PNW is increased by 20 % for the first 18 months compared to the
base case.
The water values should increase when the export price increases if the tie line capacities are
not already fully utilised. If the tie line capacities are already utilised , increased export price
wil not change the water values since extra water in the reservoir cannot be sold in the export
market. Remember that the water values describe the marginal value of the water, i. e the
value of one unit extra of water in the reservoir.
4. Inflow
The inflow forecast for May to September 1998 is increased by 3. 7 %. All inflow scenarios
are increased by the same amount
Increased inflow should result in decreased water values since the probability of future
overflow wil increase. The water value is zero (or even negative if flood damage costs are
included) if there is 100 % probabilty of overflow. The water values before September 1998
should therefore decrease. The water values after September 1998 should be independent on
the increase in inflow since the future inflow and thus also the probabilty of future overflow
for a given reservoir level from that time wil be unchanged.
5. Load
The load is increased by 1 % for the first three years and 2 % for the rest of the planning
period compared to the base case.
The water values should increase when the load is increased. Increased load should result in
increased water values since increased load wil increase the probability of curtailment in the
future and reduce the probabilty of overflow. Both effects contribute to higher water values.
6. Unserved energy
The cost of unserved energy is increased by 20 % compared to the base case , this should
result in increased water values. Increased cost of unserved energy should increase the water
values since the probabilty of curtailment in the future is higher than zero for most of the
reservoirs. For the par of the reservoir where the probabilty is almost 100 % in the base case
(e. g the lower par of the reservoir in figure 3. ) should the water values increase by 20 %.
7. Discount rate
The discount rate is increased from 8 % to 12 %.
Increased discount rate should result in decreased water values. The water values describe the
marginal value of the water. The water can either be used today or it can be stored and used
some time in the future. When the discount rate increases , the future value of the water
decreases. The value of current use wil be the same. The sensitivity to the discount rate wil
among other factors depend on the storage capacity.
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2.3
Case results
General
For each case the following values were observed:
Water values as a function of reservoir volume for the first eight months
Simulated reservoir level for the first three years of the simulation period
Simulated production for the first eight months
In addition , the water values for the year 2000 are observed for the base case as function of
reservoir volume.
The cases are such that they should all give a unique increase or decrease in the water values
as described in the case description above. Our testing of the model is therefore based on
checking how the water values change compared with what we expected. We have checked
mainly the water values since these values are the basis for all other model results.
The case results also give an indication of the results sensitivity to uncertainty in different
input variables.
Discontinuous water values
The results from the cases show unexpected variation in the water values. The water values
should in theory be a continuously decreasing function of increasing reservoir volume if
nonlinearities are not included in the optimisation. Two types of discontinuities can be
observed:
- Near full reservoir as shown in Figure 3.
Minor discontinuities in the middle of the reservoir as shown in Figure 3. 2
and 3.3
There were more of these irregularitfes in the first case studies. However , the reason was
quickly identified and new calculations were done. The original water value curves are shown
in Appendix 4. The irregularities in these figures were connected to some special constraints.
The discontinuities near full reservoir are outside the normal operation range of the reservoir
and they wil therefore probably not have any significant effect on the model results. The
cause of the discontinuities should , however, be found. Use of debugger as discussed in
chapter 4 wil make it easier to find the reason.
The normal operation range for the reservoir is shown is Appendix 3. The same appendix also
shows the connection between reservoir volume and reservoir elevation in Wiliston Lake.
The minor discontinuities in the middle of the reservoir have not been explained. We do not
believe this to be an indication of a fundamental problem, but an explanation should be
found.
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120
100
3500
Figure 3.
8500
13500
18500
23500
Reservoir volume (Mm3)
28500
33500
38500
Water values as a function of reservoir volume for May 1998 for the base case.
160
140
120
:2 100
= 80
2500
7500
12500
17500
22500
27500
32500
37500
42500
Reservoir volume (Mm3)
Figure 3.
Water values as a function reservoir volume for December 1998 for the base
case.
I :\DOK\ 11 \BM\98004023. DOC
(I~g
160
140
120.
f 100
-; 80
2500
7500
12500
17500
22500
27500
32500
37500
Reservoir volume (Mm3)
Figure 3.
Water values as a function of reservoir volume for December 2000 for the
base case.
Stochastic modellng of export market
The stochastic market model for the PNW is currently used after the first 18 months of the
planning period. The stochastic market model is described by the following values:
A variable for market price that can take three values (below (1), on (2) and above (3)
average price for the given month). These values below and above average market price
represent the uncertainty in the future market price and are based on analyses of historical
data. The values represent something similar to +/- one standard deviation. This
description is not completely correct , but ilustrates the principle of the modellng.
A state variable for water conditions in the PNW area. This variable is directly coupled to
the maximum export quantity for the given month and can take two values , the second
describing the smallest export quantity. The value is also coupled to the expected price
after the 18 month.
Figure 3.4 shows the water values for different water conditions and price levels in the PNW
for December 2000. The values are in percent deviation from the mean of all price and water
states. The results are as expected since price state equal 3 and water state equal 1 gives the
highest water value for the whole reservoir level. The lowest price combined with the highest
water state gives the lowest water values. The figure also shows that the uncertainty in export
price that are modelled results in maximum +/- 5 % change in the water values.
The figure also gives an indication of how the different price levels and maximum export
quantity effects the marginal value of the water.
I :\DOK\ 11 \BM\98004023. DOC
.._-.
-.
CI~
Price 1, water 1
-Price 2; water 1
-Price 3; water 1
-Price 1; water 2
Price 2; water 2
--Price 3;
water 2
Reservoir volume (Mm3)
Figure 3.4
Deviation in water values as function of reservoir level for different price and
water states in the PNW.
Sensitivity studies
Figure 3. 5 and 3. 6 show changes in the water values from the base case for different case
studies. The water values should in theory increase for all cases except for the increased
inflow case and increased discount rate case for all reservoir levels as described previously.
The changes are presented as percentage changes from the absolute water value in the base
case given by Figures 3. 1 and 3. 2. It makes it easier to identify the different cases since the
absolute change is relatively small for different pars of the reservoir.
All the results are as expected , i.e the water values increase and decrease for the correct
cases. Figure 3. 6 also shows almost 20 % increase in the water values for the unserved
energy case (20 % increase in curtailment cost) near empty reservoir. This is correct
, since at
this time of the year , the probabilty of future curtailment is almost 100 % near empty
reservOIr.
The Burrard case gives lower water values than the base case for December 1998. This is
caused by the fact the inexpensive production (600 GWh) which is available in June and July
in 1998 in the base case instead is available in year 2000. It is therefore correct that the water
values for December 1998 is lower than in the base case.
Figure 3. 5 shows that both the export case , the load case and the unserved energy case give
increased water values for various pars of the reservoir. The increase for the export case is
less than expected , but it is explained by the fact that the export capacity is almost fully
util sed in the base case. The marginal value of the water wil therefore not increase when the
export price is increased. The value of the export wil of course increase.
Modellng of flood losses probably causes the variations near full reservoir for the inflow
case.
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Burrard
-Increased load
-UnselVsd energy
Export price
45'00
-Inflow
Discount rate
.15
-20
Reservoir volume (Mm3)
Figure 3.
Changes in water values from base case for May 1998 for the different case
studies.
Burrard
-Increased load
Unserved energy
-Export price
-Inflow
go 5
Discount rate
45'00
Reservoir volume (Mm3)
Figure 3.
Changes in water values for December 1998 for the different case studies.
Figure 3. 7 shows the simulated median reservoir level for the different cases. The reservoir
levels are very similar for the different cases and it is not easy to separate them in the figure.
The cases with increased cost of unserved energy and increased load give the highest spring
reservoir level as could be expected.
1:\DOK\11\BM\98004023. DOC
~~~
;- '
.
, "'
'''
..
672
670
666
! -Base case
'C 664
\\ I . .. ..Burrard
" , -h-"Exportprice
662
I-Inflow
i -'--Load
Unserved energy
:g 660
658
656
654
652
Time period
Figure 3.
Simulated median reservoir level for the different case studies.
The case studies have been used to verify the model results and to indicate the sensitivity to
different input variables. The results show for example that it might be more important than
expected to update the load forecast more frequently since a change of 1- 2 % is important for
the water values as shown in Figure 3. 5 and Figure 3. 6. The forecasting error seems to be
systematic, which lead to that even a relatively small forecasting error accumulates to a large
amount of energy over the planning period.
It is impossible to give a general evaluation of the sensitivity to the different input variables
since the sensitivity wil depend on the current state. The sensitivity to the export price would
for instance be larger if the tie line capacity were not already fully utili sed in the base case.
Conclusions
The results from the case studies are as expected except for the minor discontinuities as
discussed previously. We therefore conclude , based on the results from our case studies, that
the model is correctly implemented. All the results seem reasonable.
The testing done is not a proof of that the model is correctly implemented. More testing
would always make us more certain that everything is correct. It is always possible to test
more.
We believe, however , that it is more efficient to train more people to run and understand the
model than it is to do more testing. If more people run the model , the probabilty of
discovering inconsistencies wil increase significantly.
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3.3
Discussion of computation methodology
Optimisation algorithm
The MCM model can simplified be described by the following maximisation problem:
J(x(1))
= max
(I.:=1
L(x(k), p(k), u(k), k)
S(x(N), N)
Subject to
x(k+l)=x(k) -u(k) + v (k)
and constraints to possibl x(k) and u(k)
where:
J(x (1))x(k)
Future net income as function of reservoir level in the first time period
Reservoir level in period k
Net income in period k as function of exportimport prices, production and
reservoir level
v(k)
p(k)
u(k)
Inflow in period k
Expectation operator , stochastic variables are p(k) and v(k)
Exportimport prices
Production in period k
Time period , month number
Number of months in the planning horizon
Value of water by end of month N
The described maximisation problep1 can be solved by use of stochastic dynamic
programming, which is a standard optimisation technique. This method is perfect if the
problem can be solved by this approach. The number of state variables that can be included in
the model mainly gives the limitations. The number of state variables is in this case given by
the number of reservoirs (one) and the number of uncertain variables that are described by
Markov models (one for market size and one for market price). One of the advantages with
this method is that it wil always give a solution to the problem. For other methods that
depend on iteration , convergence is not always guaranteed.
We have not checked how the solution algorithm is implemented and can therefore not
comment on the implementation and how effcient (meaning how fast it is compared to what
is possible) it is. Presently the optimisation takes approximately 1.5 cpu. minute for each year
in the planning period which is more than fast enough. Preparation of input data is without
doubt much more time consuming than running the model , days or hours compared to
minutes.
The problem of efficiency may however be more important when the model is improved as
described in section 3.
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3.3.
Simulation procedure
A number of historical years put together specify a scenario. The first scenario would , as it is
implemented today, be represented by the historical years (1973 to 1978), the second by 1974
to 1979 etc. The historical year gives the inflow and load for each month. To each scenaro
corresponding draw from the two Markov models describing the PNW export market (price
and quantity limits) are made. There is only one draw for each historical scenario. The
simulation results wil be dependent on the draw from the Markov model.
The future is simulated for 25 scenaros. The expected values that are output from the model
are the mean of the results from the 25 scenarios.
In our opinion the simulation methodology is good. In the future we believe that there would
be a need for more scenarios, especially connected to possible future use in risk management.
The 25 different outcomes currently used , are a too small number to give a credible
probabilty distribution. More scenarios can easily be generated combining the historical
years differently and makng new draws from the Markov models.
The simulation results are dependent on the draw from the Markov model. If more scenarios
are generated based on new draws from the Markov model , the simulation results (expected
values etc. ) wil be less dependent on the actual draw.
Uncertainty in Columbia River production should be included using the same historical years
as for the other uncertain variables. It wil probably increase the uncertainty in the simulation
results.
The simulation takes 20 seconds cpu time for each year in the planning period.
Discussion of assumptions
The MCM model relies on two impprtant assumptions:
The flexible storage capacity in the Columbia River system is small compared to the
Wiliston. It is therefore assumed that possible changes in Columbia River production do
not influence significantly on the water values in Wiliston Lake.
The PNW market can be modelled with sufficient accuracy using a Markov type model.
Both assumptions neglect possible interaction between the different physical systems. The
PNW system is a hydro- thermal production system with both hydrological and electrical
coupling to the BC Hydro system. The Markov type modellng of PNW is only correct if
there is no hydrological coupling and if BC Hydro is a price taker in the PNW market.
It is not possible to quantify the effect of these assumptions , but significant changes from
April to May 1998 in the official results from the MCM model were to some extent explained
by changes in the production strategy for the Columbia River.
Table 3.4 shows some ' system statistics ' for different pars ofthe BC Hydro system. Kelvin
Ketchum supplied the numbers. The values give a simplified description of the physical
system , but they indicate that the flexible storage capacity in Columbia River may not be
insignificant compared to Wiliston Lake.
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Both the previous comments imply that some type of interaction between Columbia River
and Peace River should be modelled. It should be noted that the operation of the Mica
Reservoir must , to some extent , be coordinated with downstream U. S. project operation under
the Columbia River Treaty and Non- Treaty Storage Agreement. This linkage reduces the
flexibilty of Mica Reservoir storage for BC Hydro purposes.
Storage capacity
18.4
10.
570
640
17.
15.
\TWh)
Maximum
production
(GWh/week)
Inflow (TWh/ ear)
11.5
It is not possible to quantify the error introduced by these assumptions. There are two slightly
different approaches to overcome them:
Model the PNW and BC system in one large model. Possible solution approaches are
discussed briefly in section 3.
Expand the MCM model to include both Peace River and Columbia River production.
Include PNW market into the MCM as it is done today. The price model for the PNW
could be estimated from a physical model of the PNW or historical variations and the
forward price. If a physical model is used , possible hydrological connection or
correlation could be included using the same historical years as reference in the
simulati ons.
BC- Hydro is currently developing a new model for the Columbia System. The interaction
between the MCM model and this n w model is not decided upon.
Adequacy of model in dealing with uncertainties
The MCM model explicitly includes modellng of inflow , temperature and market
uncertainty, which are included both in optimisation and simulation. The uncertainty in the
physical input variables (temperature and inflow) are modelled as well as possible. Two
Markov models model the market uncertainty. These models should also be used for the first
18 months if necessary data is available.
It might also be possible to increase the number of
discrete price levels in the Markov model. In our own model we have found that 7 discrete
levels seems to be appropriate.
3 Small hydro in Table 3.4 includes the Kootenay Canal and Seven Mile.
4 Storage capacity is defined here as live storage
, equal to the difference between the maximum and the
minimum allowable reservoir levels. For the two multi- year reservoirs (Williston and Mica), it is practically
impossible to use all of this storage in one year. Furthermore
, Columbia River Treaty and, to a smaller extent
Peace River Ice constraints limit BC Hydro s flexibility at these two reservoirs.
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The model results are also dependent on other uncertain input variables as ilustrated by the
case studies. The uncertainty in these variables could be included in three different ways:
Include stochastic models of these parameters similar to the uncertainty already modelled.
Automatic simulation for different values of the uncertain variables (e. g. high , medium
and low load forecast).
The user specifies and run the sensitivity cases manually.
The first method is only possible in theory. The second method is useful if the probability
distribution given by the simulation results are used in further analyses such as risk
management. This method can however disguise the result' s sensitivity to the different input
parameters.
We believe that the best practical method is to do manual sensitivity analyses for the
uncertain input parameters , which are not already modelled. The MCM model is in principle
well adapted to such analyses since it is relatively fast. The user interface and result
presentations are , however , based on old computer technology and this makes it in practice
impossible for the users of the results to do such analysis themselves.
The model can relatively easily be expanded to include simulation of more uncertainty as
described above. However , it is probably more beneficial to star using the already available
uncertainty from the MCM model in the different applications than it is to simulate for more
uncertain variables.
Suitabilty, signifcance and usefulness of model
General comments
The users own explanation of how they currently use the results from the MCM model is
described in Appendix 1.
The model contains in principle all the relevant information concerning the physical and
economical operation of the BC- Hydro system including production , transmission,
consumption and importexport. The model can be applied to a wide rage of analysis where
the economic consequence for BC Hydro of changes in some of the input variables to model
is wanted.
CUlTent use of the model as described in Appendix 1 and below , applies only the expected
results from the model. The model contains also probability distributions for all the results.
We believe this probabilty distribution should be distributed to the users and applied in the
decision process.
Thermal operation planning
The operation of the Burrard thermal plant is handled in the same way as possible imports.
There is a time dependent availability of power production where the price is defined by the
marginal cost. The expected price and availability of gas give the marginal cost.
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The available capacity of the Burrard plant is also defined by other system requirements such
as reactive power support , air quality and water temperature.
There is nothing in the formulation of the Burrard operation that takes into account the
history when the decision is made. If there is no contract on gas that is coupling the decision
on the different stages , this is probably an adequate approach. Starstop costs are not relevant
with the time resolution used. A state variable for the operation of Burrard would not have
been computation ally feasible.
The Burrard is modelled in the MCM in a way that is common for thermal units in hydro
dominated systems. A full physical model is never possible , but since the MCM takes into
account the main characteristics of the Burrard operation , the results from the MCM should
be useful for the refinement of the plan for Burrard.
Hydro facility maintenance scheduling
Maintenance scheduling is required for:
Small hydro units
Columbia River system
Peace River system
The whole planning hierarchy is based on the fact that each of the small hydro units does not
significantly affect the results of MCM. The planning of the small units are based on local
conditions. These units wil see a market price where they wil principally be price takers. For
maintenance planning of these plants , the local conditions combined with the marginal cost of
water wil define the timing of the maintenance.
The maintenance planning of the Columbia River system is more complicated. The power
production plans are input to the MCM, and the relative size of the systems cause significant
interaction. The marginal water value is important input when decisions about maintenance of
the Columbia River are made. However , the use of Treaty and non-Treaty storage wil have
impact on the maintenance decisions. It is important that major changes in the output from the
Columbia River systems are checked with each new run of the MCM.
The consequences of maintenance schedules of the Peace River system must be evaluated by
the MCM itself. Major reductions in the power production capabilities may impact the
marginal water values.
Reduction in transfer capacity due to maintenance or other reasons for derating may be the
worst problem for the utilisation of the system. The MCM must then access the impact of
reduced market access capability.
Electricity trade purchases or sales
The MCM model gives the current marginal value of the water. If the current export price is
higher , the right decision is to sell as much as possible. The opposite decision is correct if the
exportimport price is below the current water value. The model results are well suited to such
analysis.
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The MCM model also outputs simulated reservoir levels , exportimport and marginal values
of water for different future scenarios. These values are and can be used in decision support
for today s trading future contracts. Currently only the expected future values are used by
Powerex in their trading decisions as described in Appendix 1.
Today the simulated expected net importexport over a period of time (6- 12 months) ahead is
used as a target for the net trading over the same period. The expected water values are used
to decide which months to buy and sell.
An alternative method would be to use the expected simulated water value as a signal for
today s trading in the forward market with less consideration to the overall balance. This
method is in principle more correct , but it makes it more difficult to control the future energy
balance (risk) as long as there is no risk management tools available to provide both price and
volume uncertainty.
The MCM model contains the connection between physical constraints (tie- line capacities
maximum production etc. ), exportimport price uncertainty, load uncertainty, inflow
uncertainty, i. e. the connection between price and quantity uncertainty. This is important
information that should be incorporated into risk management. Two different approaches can
be applied:
Enhancement of a MCM type model to also incorporate functionality for risk
management.
Using the results (price and production scenarios etc. ) from the MCM model as input to
other available risk management tools.
Without going into detail the first method is the best , but the second method can probably be
implemented faster. Both are improvements from present practice.
Wiliston Reservoir elevation forecasting
The optimisation par of the MCM gives the incremental water values for different reservoir
levels and time stages. The simulation par , gives among other results
, the reservoir
trajectories for different weather and market price scenarios.
There wil always be some uncertainty involved these forecasts due to:
Adequate selection of weather sequences and appropriate pairing of these with market
scenarlOS
Non modelled uncertainty of important input variables as for example Columbia River.
For normal cases , the uncertainty introduced by such conditions does not have too much
impact on the results. For special cases , one may find that the extreme (maximum and
minimum) trajectories of the reservoirs are underestimated. Despite this we believe that the
MCM is a useful tool for the reservoir elevation forecasting.
Financial forecasts
A detailed explanation of current use of the model together with wishes for the future are
described in Appendix 1. This description contains current use , deficiencies of the current
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process and desired situation. For the deficiencies of the current process quite many of these
are more related to how things are organised than to the MCM itself. The optimisation par of
the MCM calculates the optimal strategy (incremental water values) and the simulation par
gives an outcome for different scenarios. Much of the requested information can be found
from the basic model output. Total costs can be derived from the proposed production plans
and some additional information (fixed costs , sales income from firm load etc.
As for the other applications , only the expected values from the MCM model are used. We
believe a wider use of all the results would be more appropriate. The MCM gives the
distribution for different market and weather conditions and wil therefore provide results that
can be used to check the range of possible future income.
Systems operation planning
The model is definitely a useful tool in the system operation planning. Despite the
simpliication in the planning procedures where the results in one evaluation is input to the
MCM, use of this model wil be the last step to evaluate the impact of the decisions on the
marginal cost values. Change of strategy for the operation of Columbia River plants wil have
to be checked with the MCM to estimate the impact on the system operation.
Comparison with similar models
The MCM is a speciai purpose model developed to solve a specific operation planning
problem. There are several similar programs used around the world to solve hydro operation
planning problems. These programs would not necessarily be able to solve the same problem
as the MCM due to special conditions within the BC Hydro s system. However , use of
similar mathematical formulation is a justification for that the MCM is in step with similar
programs. This section briefly describes some approaches used to solve the hydro operation
planning problem for the purpose of ilustrating techniques in use.
Single-reservoir models
The MCM model is based on stochastic dynamic programming (SDP). This is a common and
frequently used solution method for similar problems also in other countries. The
corresponding model used in Norway (2), (4), was originally based on a paper published as
early as 1962 (3), and is based on the same type of modellng and solution algorithm as the
MCM model.
This model has gone through several improvement stages , one of the recent ones being the
modellng of the market price using Markov models as a consequence of the market
liberalisation.
The Norwegian model is principally the same but has also some differences:
The model stores and computes the marginal value of water directly (the method is
therefore referred to as the water value method). Future income is not computed.
Linearization is used for state values between the discrete values. In the MCM model only
inflows and decisions that make you reach a pre-calculated value in the next time step is
possible. The inflow is pre- processed in order to make it possible to reach these values.
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Linearization could make it possible to reduce the number of discrete reservoir levels
which again could allow for improved market modelling (i. e. increase the number of
discrete price levels).
Markov models for price are usually estimated as continuos autoregressive models which
are made discrete in the optimisation. The model allows simulation for continuous price
senes.
A similar model is also developed by the Swedish company Kraftdata. The principle of
modellng and optimisation of the single reservoir model is as far as we know identical to our
own model.
Both the SINTEF Energy Research model (EOPS) and the Swedish model contain simulation
models that distribute the production to the different plants in the physical system using a
rather complicated set of.rules. We wil not discuss this aspect since the Peace River System
is so simple that the simulation is no problem. The simulation aspects of these models are
however interesting if more complicated systems are modelled.
Generally, methods based on SDP are superior to other solution methods if the basic
stochastic dynamic optimisation problem can be described by a limited number of state
variables.
Multi-reservoir models
The problem can generally be formulated as a multi-state stochastic dynamic optimisation
problem. It has up to now been impossible to solve such problems in general. In order to
solve the problems, different types of simplifications have been made:
Simplify the physical description to one or two reservoirs , as in the MCM model.
Assume that all the uncertain input parameters (price , inflow and load) are known , often
used with shorter planning horizon.
Use a combination of optimisatipn and simulation combined with user interference to find
the ' optimal' solution , as in our own multi-reservoir model described below.
The continuous development of computer technology has now made it possible to solve
multi-state stochastic dynamic optimisation problems with fewer simplifications than
described above. Some of the mathematical techniques used are the following:
Stochastic Dual Dynamic Programmng (SDDP)
Scenario Aggregation
Determnistic equivalents of the stochastic problem
SINTEF Energy Research' s multi-reservoir model (EMPS-model)
This model is used for price forecasting, operation planning and investment planning by
almost all the major players in the Scandinavian market. The model contains a detailed
physical description of the hydraulic systems and includes tie line constraints between
different areas in the market. The model is described in (4). Thermal production units are
described by their marginal cost, production capacity and probabilty of failure. One of the
important results from the model is a price forecast for each defined area.
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The optimisation algorithm is based on a combination of stochastic dynamc programng,
linear programmng and heuristic. The user can to some extent influence on the results from
the model. This can be an advantage since it is not possible to formally describe all the
. players behaviour in the market.
Stochastic Dual Dynamic Programming (SDDP)
The method is described in (6), (7), (8) and (9) and has much in common with Stochastic
Dynamic Programng (SDP). The main advantage is that the method can solve problems
that consist of many state variables. The method can therefore in theory be used to optimise a
physical system consisting of a number of different hydro plants and reservoirs connected
with tie lines with limited capacity. This is the same problem as solved by the EMPS-model
the difference being that the SDDP method in theory gives a more formal correct optimisation
of the problem.
The method has been implemented and tested both by SINTEF Energy Research and
CEPEL , Brazil (8) and (9).
Based on our own experience , it wil probably take a long time before the method can solve
all long and medium-term planning problems. Currently we are implementing the method
into our own commercial medium term planning model , which is used for local planning.
The SDDP method does in contrast to the SDP method rely on an iteration algorithm and the
method is very time consuming compared to SDP approach.
Scenario Aggregation
The principle of this method is described in (5). Both scenario aggregation and the method
using determnistic equivalents assume that a scenario tree can describe the stochastic
processes. In practice this wil limit the number oftime periods in the model to 5- , since the
number branches in the tree increas. exponentially. Some results from the University in
Trondheim indicate that the problems can as well be solved as Deterministic Equivalents
using commercial LP solvers as CPLEX.
Deterministic Equivalents
The stochastic problem is converted to a large determnistic problem that can be solved by
standard LP solvers. The number of periods and branches in the scenario tree gives the
limitations. If the scenario tree consist of three branches for each period , the number of nodes
are given by (3) 8 - 1= 6560. Each branch needs a separate set of decision variables.
Reference (10) describes an implementation based in Benders decomposition. The method
has much in common with the SDDP method but it is required that the uncertainty can be
described along a scenario tree.
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Discussion of proposed modifications
The following modifications of the MCM model are proposed:
heavy load hours (HLH) and light load hours (LLH) and model the
possibility of load factoring the Peace River generation.
2) Use of physical tie- line capacities to define limits for export and import
3) Modellng of uncertainty in Columbia River production
4) Restore uncertainty to the export price forecast for the US market.
1) Segregation of
We believe that these proposed modifications of the model can and should be implemented
within the current framework.
The first and the second modifications are tied together. Implementation of these extensions
wil increase the computation time. However , the increase in computation time should not be
a problem, due to rapidly increasing computer capacities.
The third modification can also easily be included if uncertain (generation plans
corresponding to the historical years 1973- 1997) production plans for Columbia River are
available.
The fourth extension is already implemented but not used. The uncertainty is not modelled
for the first eighteen months due to two different reasons:
Lack of market information
The misunderstanding that the forward price includes all the useful information
It has been diffcult to estimate a credible stochastic model for the price based on available
historical data. However , more statistical data wil be available. An alternative could also be
to apply physically based forecasting models for the PNW market. The stochastic price
model can then be estimated from tQe results of the physical model.
Some people assume that the forward price gives all the information there is about the future
price for the first eighteen months. The MCM model maximises the long- term income of the
water resources. The correct value of the generation is the market price at the time each GWh
is produced , i.e. the spot price. The forward price can be seen as the market'
s best estimate of
future spot price. Appendix 2 shows the value of modellng uncertainty in the future market
price , even though there is a known forward price. The example also to some extent
ilustrates the principle of stochastic dynamic programming.
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GENERAL COMMENTS
The MCM model is implemented and run by only one. person. The model is and wil be a
very important tool for BC Hydro. The model should therefore be understood and used by
more people than what is the case today.
The basic principle of the current model should be possible to grasp also for non-specialists.
More people must therefore be trained in order to understand the methodology. More than
one person should be able to maintain the program code.
Productivity could be improved by changing the programing environment. It is for example
not possible to run a debugger with the current tools. Modern programmng tools are
available for Fortran on a PC.
The future user interface for the MCM type model should be run from a Pc. It makes it easier
to use the model and the model results. It is also easier to implement a modern user interface
on a Pc. The computation par of the MCM model can run on a Unix server or at the local
, both solutions are possible. The present user interface consists of a number of Ascii fies.
This makes it relatively easy to implement a modern Windows based user interface to the
MCM model.
The input to and the output from the model should be stored in a database. More simple
model reports on basic input and results can be available for non-specialists. In order to do
sensitivity studies , it is important to keep track of the connection between input and output.
More people should run the model. The inputs to the model are market statistics and
forecasts , and physical values given by line capacities and the production system. The user
can have a good understanding of these values wIthout understanding the MCM algorithm.
When physical tie line constraints are modelled , the model does not contain any input that
requires any knowledge of the algorithm to be understood.
If the different users themselves can do sensitivity analyses , the confidence in the model
results wil increase. More people wil thereby gain insight into how the complicated hydrothermal system functions.
It is impossible to prove that the MCM gives the optimal decision strategy by comparing
what happened for instance the last year with what had happened if we had followed another
decision strategy (a kind of benchmarking). The other strategy could for example be based on
the forward price and expected inflow and temperature.
The reason for this is that the model gives a decision strategy that in the long run wil
optimise use of available resources. The simpler strategy could by chance be better for just
the combination of temperature , inflow and market price that occurred last year.
The long run is in this context specified by the time needed to run into the different
combinations which are included in the statistics that the strategy is optimal for. The inflow is
for instance represented by 25 years. At least 25 new years must pass before the history has
been covered. If we combine this with the statistics for market price
, we must conclude that
the strategy must be followed for many years before it can be proved that one strategy is
better than another with absolute certainty.
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An estimate of the profit gained by using the MCM model could be computed using a
simulator that simulates the different decision strategies over many years.
The results from the MCM model are not better than the quality of the input. Quality
checking of the forecasting methodology for different input (e. g. price forecasting), if
possible , might be beneficiaL
The stochastic dynamic programng approach gives without doubt the optimal decision as
long as the input is correct. The problem may be both nonconvex and nonlinear which make
it difficult to solve with other methods.
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CONCLUSIONS
Our conclusions are based on written documentation , discussions and the results from the test
simulations. It is easier to verify that the modellng is correct than to prove that the model is
implemented without errors.
If the basic assumptions for the model hold as discussed in section 3.3.3 , the modellng and
solution algorithm implemented in the MCM model is sound. The model gives several
important results , which should be used more extensively than today. The results from the
model include net income , reservoir levels , export , import , thermal production , and firm load
delivery. Each output can be given for each month in the planning period (maximum 3 years
ahead), or as accumulated numbers for periods of time. The model also contains probability
distributions for the same outputs for each month or accumulated periods. These probabilty
distributions should be used further if it is interesting for the decision maker to for example
know that there is 10 % probabilty of the income for the next year to be below a certain
number with the current contract portfolio and production strategy. The probabilty
distributions are especially interesting in connection to future use in risk management.
The modellng should be improved as suggested by Donald Druce and described in section
Because the model is a very important tool , more people should be able to run and maintain
the model.
The user interface should be improved and run on a Pc. It is probably possible to run also the
optimisation on the PC. Use of modern software tools as debuggers can simplify development
and maintenance of the program code significantly.
Methods for integration of the planned Columbia River model with the MCM model must be
developed in parallel with the development of the new model.
The value of the MCM is much dependent on consistent use for different applications. If it
can be agreed upon that the input data is the best available and is adequately represented in
the model , tbe strategies and forecasts developed should be used in the different applications.
If new information becomes available, the model should be rerun. To make this possible , the
MCM must be upgraded with a new user interface and should also get a database connection
to make preparation of tailor-made reports easier. A new user interface would make more
people able to use the model and thereby more confident with what the results are and the
impact of different assumptions.
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,"
, pp.
REFERENCES
( 1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
Druce , DJ., "Forecasting B. C. Hydros operation of Wilis tone Lake - How much
uncertainty is enough" , Stochastic and Statistical Methods in Hydrology and
Environmental Engineering, Vol. 3 , pp. 63- , 1995. Kluwer Academic Publishers
Netherlands.
Wangensteen , I., Mo , B. , and Haugstad , A. Hydro Generation Planning in a
Deregulated Electricity Market" , Proceedings from "Hydropower Into the Next
Century , Barcelona , Spain , June 1995. Sutton , Uk: Aqua-Media International.
Lindquist , J. , 1962. " Operation of a hydrothermal electric system: A multistage
decision process , AlEE Journal , April 1962.
Haugstad , A. System Modeling in a Hydrothermal Electrical System , Presentation
September 5 , 1997 , Melhus , Norway.
Gjelsvik , A. , and Wallace , S. W: "Methods for stochastic medium- term scheduling in
hydro- dominated power systems, EFI TR A4438 , 1996.
Rfitting, T. and Gjelsvik , A. Stochastic dual dynamc programng for seasonal
scheduling in the Norwegian power system. vol. 16 , 1991 , pp. 199- 147.
Gjelsvik , A. et. aI A case of hydro scheduling with stochastic price model"
Proceedings of the 3 international conference on hydropower , Trondheim 1997. A.
A. Balkema , Rotterdam , Brookfield , 1997.
Pereira , M. , and Pinto , L. Multi-stage stochastic optimization applied to energy
planning , Mathematical programmng 52 , 1991.
Pereira, M. Optimal stochastic operations scheduling of large hydroelectric
systems , Electrical Power & Energy Systems, vol. 43 , no.3, July 1989 , pp. 161- 169.
Jacobs, J. , et al. " SOCRATES: A system for scheduling hydroelectric generation
under uncertainty , Annals of Operations Research , vol. 59 , 1995
99. 113.
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APPENDIX 1: USER DESCRIPTION OF USE OF MARGINAL COST
MODEL
Information provided by the Marginal Cost Model (MCM) has a wide range of users.
These
users either utilize details from the MCM itself, or access other reports which contain
information which derives from the MCM. The diagram below highlights the range of users
of the MCM.
BC Hydro
Financial
Reporting
Powerex
Power
Supply
Operations
Power
Facilities
This appendix contains a description of how the results of the MCM currently are used within
in BC Hydro. The descriptions are provided by BC Hydro staff. Only minor editorial
changes have been made to the text. In the main report , there are some references to this
appendix where the current use and future requirements are discussed.
1 Use of the Marldnal Cost Model (MCM) BC Hvdro Financial Reporting
Power Supply Business Services is', the main interface between the BC Hydro Finance and the
output from the MCM. However , the end users of this information , either directly or
indirectly, is quite broad. The diagram on the next page demonstrates the flow of information
which is impacted by MCM output.
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Power Supply
Business Services
BC Hydro Financial
Reporting
Provincial
Government! BC
Utilties Commssion
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What information is currently used and for what purpose
A) Power Supplv Business Services
Three Year Financial Plan / Current Year Forecast
The MCM is a key source for projections involving the Cost of Energy expense line
component on BC Hydro s Statement of Operations.
Power Supply Business Services uses the MCM summary schedule which lists the three
year expected generation at Burrard Thermal (in GWh and $) using valley gas , and spot
gas. These are the variable components that go into determning the cost of gas for
BUITard included in the Cost of Energy line item on the income statement.
This same schedule also provides expected energy volumes and dollar values for
purchases from the United States , Alberta and Alcan , as well as sales to the United States
and Alberta. This information is not directly included in the Three Year Plan or CUITent
Year Forecast because Powerex provides a forecast in its place. However , it is used as an
important check against the Powerex forecast as the net difference between sales and
purchases must be identical for both the MCM and Powerex forecasts otherwise the
predicted future Wiliston elevation levels wil not be valid.
Power Supply Business Services uses a monthly MCM schedule which presents
commtted purchases and BUITard Thermal generation using assumed swing gas (in GWh
and $). This information is directly included in the Three Year Plan or Current Year
Forecast for Cost of Energy.
Ad- hoc Reports
Periodically, Power Supply is reque ted to prepare special ad- hoc reports evaluating the
financial impact of potential events. Examples include the analysis of the opportunity cost of
the sink- holes at Bennett Dam and the financial impact of measures taken to improve the
outlook for Wiliston Reservoir levels. The MCM is able to perform this work , but its output
comes in the form of net economic cost. Power Supply Business Services takes this
information and translates it into financial reporting terms on a fiscal year basis.
An example includes where Power Supply would compare the "before and after
" effect
on the financial forecast of a significant event , such as the sink- holes found at Bennett
Dam. Several assumptions may be run over a period of time as different assumptions
change.
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Opportunities for improvements in the current process for financial forecasting
The inability of the model to translate economic costs into a financial statement
equivalency. For example , the total impact of a change in circumstances could be $20
millon , but it is not easily known whether what the impact is in terms of sales and
purchases and in which fiscal years the components of this total would be realized.
The model only looks at marginal cost and not total costs. For example , in planning for
Burrard Thermal energy costs it is necessary to add to the costs provided by the MCM
any other costs which are allocated , i. e. municipal gas tax , demand charge and prepaid
transportation costs which are considered to be sunk costs.
The Electricity trade sales and purchase numbers are not the final numbers adopted by BC
Hydro. These are prepared by Powerex which uses some MCM information and then
develops their own forecast.
The MCM uses load forecasting volumes for domestic consumption prepared annually.
Periodically, adjustments wil be made based on advice from Marketing & Customer
Services for major known changes , Le. industrial labour strikes , El Nino. The financial
forecast (as prepared by Marketing & Customer Services) uses the same domestic load
forecast as its staring point , but makes refinements to total and monthly curving of
consumption on an ongoing basis. These changes are not incorporated in the MCM so
cost of energy forecastscan
t simply use the numbers provided from the MCM
chedules.
While the MCM is capable of performng scenario analysis and sensitivity analysis this
can only be done by Don Druce. Other users of the output are unable to do any " what if'
analysis. In addition , due to Don Druce s availability constraints it is often difficult to
test for a large number of scenarios on a timely basis. Although the MCM is too
complicated to be easily accessible to all potential users, an improved situation would
have additional specialists available to run different analysis and provide expertise for a
wide range of users.
Comparabilty of different scenarios run over a number of months was sometimes
questionable as other inputs would change during the period (Le. updated inflow
projections). Therefore , different results couldn t always be directly linked to the
manipulated variable.
MCM is not useful for treasury s needs for cash flow information. For example , the
MCM only provides marginal costs for running BUITard and does not include other cash
payments related to this generation such as transportation charges for valley gas which are
assumed to be sunk costs by the MCM.
There is a limited information available as to what assumptions have been used in the
MCM. The monthly Marginal Cost Study states assumptions made for inflation , U.
exchange rates , nominal discount rates and the forward prices for gas , Alberta and U.
electricity. However , many other assumptions are not stated such as unit availabilty, tieline constraints and operations for Peace River ice.
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Desired Situation
Predicted load inputs used by MCM should be identical to those provided by Marketing &
Customer Service s finance deparment (this is an improvement needed in Marketing &
Customer Service , not the MCM). The model should be updated on a regular basis using
updated load forecasts.
Integration of electricity trade information (sales and purchases reported in the MCM and
forecasted by Powerex.
Additional staffng resources for performng scenario and sensitivity analysis using the
MCM.
Output to incorporate' all cost of energy components so that the different scenario
analysis ' are directly translatable into financial statement form. Alternatively, an
accounting program should be developed which combines the output from the MCM with
other costs to produce financial statements.
B) BC Hydro Corporate Finance
BC Hydro Corporate Finance takes the Three Year Plans, forecasts and ad- hoc reports
prepared by Power Supply and described in the last section and combines them with
information provided by other business units of BC Hydro to develop consolidated Plans
forecasts and reports for BC Hydro. The Cost of Energy expense , which is influenced by the
MCM, has a significant impact on these consolidated reports.
The end users of these Corporate reports may include some or all of the following:
BC Hydro Senior Management
BC Utilties Commission
Crown Corporate Secretariat
Provincial Government
should also be noted that the Treasury deparment is a user of the Power Supply reports
(and , therefore , an indirect user of MCM output) for projecting cash flow expenditures for
energy payments. This work is done to assist with BC Hydro s cash flow management.
It
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Use of the Mandnal Cost Model (MCM) by Powerex
Contribution from Mikael Buchko
Powerex reviews the imports and exports determned by the MCM and the resulting Wiliston
reservoir levels. Powerex refines the monthly importexport projections while keeping the
reservoir level the same at the end of the next water year with the goal of optimising
profitabilty in imports and exports.
The water values provided by the MCM (Rbch) are used as price indicators in a given month.
if Rbch is 12 and the market price is 10 Powerex wil not sell energy from the BC
Hydro system. Instead , Powerex would buy energy regardless of the monthly expected
volume for the given month as determned by the MCM. Powerex would exceed this energy
volume value (if physically possible) and then sell the differenceat a later time in order to
maintain the end of water year reservoir elevation.
For instance ,
Another use that Powerex has for the MCM is where significant forward sales or purchases
Powerex would request that a model run be
performed to determne the sensitivities of the MCM results as a consequence of these
actions.
have been made or are being considered.
Contribution from Murray Margolis
Powerex uses both the forecast expected purchase and sales volumes by month and the
calculated Rbch value when makng sales and purchase decisions. Commtted sales and
purchase contracts are included in both Powerex and MCM forecasts and since Powerex
forecasts always leave the ending Wiliston Level unchanged the only difference is in the
expected sales and purchases by month. This might include simply transferring sales to time
periods that we expect wil have the most value and also increasing both purchases and sales
to levels that we feel we can obtain, nd are not captured because of the current limitations of
the model. Volumes are used primarily for planning purposes and we do not make purchase
and sales decisions within the month solely for meeting volume targets. We do however
adjust our marketing decision on a continual bases takng into account aggregate sales and
purchase volumes to date and forecasts to the end of the fiscal year including any updates we
get on domestic load and system energy.
BC Hydro Corporate Finance Requirements
As described in Section 1. , par B of this Appendix , BC Hydro Corporate Finance has
ongoing requirements for financial forecast information. Powerex provides Electricity Trade
Sales and a significant part of the energy purchase component of Cost of Energy to the
consolidated forecast. As discussed in this section , the MCM impacts the development of the
Powerex forecast and , as a consequence , has an influence on significant and volatile aspects
of the consolidated forecast.
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3 Use of the Marginal Cost Model (MCM) bv Power Supply Operations (PSO)
The MCM has a major impact on how Power Supply Operations (PSO) operates BC Hydro
. generating facilities. PSO uses the results of the model primarily for:
decisions on running BC Hydro s thermal plants (Burrard , Rupert , etc); except in cases where
required for capacity or system support these plants wil only be operated in the marginal cost
to do so exceeds the marginal cost of energy in Wiliston , Rbch, as determned by MCM.
for decisions on imports/exports that are under PSO' s control (e. g. independent power
producers within B.C , energy associated with Non- Treaty Storage activity, etc. ). The value
of undertakng these transactions are estimated and compared with Rbch .
to determne the costlenefit of compressing maintenance outages in the B. C. Hydro system.
If the benefit of a shorter maintenance period , as calculated using Rbch exceeds the
incremental cost than the maintenance schedule wil be revised.
to provide guidance to PQwerex as to the price ceiling for energy purchases and the price
floor for sales. This price is Rbch , as determned by the MCM.
as the basis for determning the marginal cost of energy in reservoirs other than Wiliston.
There is currently no other means for estimating the value of this energy. On occasion, this
results in different price signals for energy from different reservoirs.
for performance measurement, keeping track of the margin between the system marginal
cost and the actual exportimport transaction price. This is done to evaluate how
exportimport decisions have added value to the BC Hydro system.
to forecast the Wiliston elevation levels for multiple years. This is a very important
function due to the sensitivity of business operations in the town of Mackenzie to
reservoir levels. There is an ongoing need for elevation forecasts for both the businesses
and BC Hydro to plan their operations accordingly. In addition , this information is relied
upon by BC Hydro to communicate the expected levels and to explain the anticipated
response, if any, which may be required.
to determne the prudent over-winter Peace discharge over the ice regime. This is
important for maximizing the hydraulic capacity of the Peace River during the winter
months and mitigating against the chance of unwanted flooding in the spring.
to perform numerious scenario analysis ' to estimate and compare the cost of possible
outcomes. Analysis work of nature occurs on a regular basis and can have a significant
influence on the ultimate plan of action under- taken by BC Hydro. The analysis can
include a range of activities which have included the following:
the options for managing the Bennett Dam sink- hole
issue
the cost of maintaining minimum elevation levels at Wiliston
the options for running or not running the non- SCR units at Burrard
testing the options of various longer term export contract proposals
to provide a number which indicates the percentage probability of a spil occurring at
Wiliston due to high reservoir levels. This is a powerful number for easily describing
state of the reservoir. This is just one of the determnants that go into making Rbch.
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the MCM identifies the uncertainties of future operations by providing a range of
projected reservoir levels , not just the expected level. This is an important system
planning input.
1.4 Use of the Mandnal Cost Model (MCM) by Power Facilties
Power Facilities does not use to Marginal Cost Study or the relevant Rbch value as determned
by the MCM. However , the System Operating Plan which is prepared by PSO and issued to
Power Facilities is a major operational planning document. The System Operating Plan
provides information on planned generation at all BC Hydro facilities and additional details
for major plants which may include projected reservoir elevation levels, discharge rates and
inflows.
Information from the MCM is used in developing the System Operating Plan. Assumptions
used in the MCM and output from its report is incorporated into PSO' s system planning.
Therefore , Power Facilities is an indirect user of the MCM.
5 Other Recipients of Marl!inal Cost Model (MCM) Information
There are recipients of the MCM Study report which review the information for a variety of
purposes, but do not use any of the output for specific decision makng purposes. These users
include the following:
Energy Plans & Investments. Power Supply
In the past , Energy Plans & Investment (EPI) used the long range marginal cost model to
provide energy values for the first two years when calculating CONES (Cost of New Energy
Sources). However , the MCM is no longer required as more developed market information
has become available in recent years; and a forward price curve has been developed.
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Marketing & Customer Services
Marketing & Customer Services (M&CS) reviews theMCM study report to ensure
consistency in assumption and inputs with those used for M&CS load forecast.
The MCM Rbch value was used by M&CS when developing a Real- Time Pricing (RTP) rate
structure. The MCM value was used as a reasonableness check in evaluating various RTP
rate structures.
6 Other Potential Users of the Mandnal Cost Model (MCM
Risk Management
It is probable that there would be many uses for the MCM in the areas of responsibilty
Committee which is expected to evolve from the BC Hydro
Energy Risk Management Project. While it is difficult to project these uses until the
Commttee is formed and its agenda determned there should be some similarities in their
requirements as with those already presented in Section A of this Appendix , Power Supply
fallng under the Risk Operating
Business Services.
Performance Measurement (Transfer Pricing)
Although currently not used as a transfer pricing measure , Rbch has been used in the past as a
means of allocating a purchase and sales price for energy transactions between Power Supply
and Powerex. There remain many strong arguments in favour of using the MCM for
determning the transfer price. This is true for both Powerex electricity trade transactions and
for market based transactions operated by M&CS such as RTP.
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APPENDIX 2: DETERMINISTIC VERSUS STOCHASTIC
MODELLING
Determnistic and stochastic approach to evaluation of a load factor contract
In Norway, a type of contract called ' brukstidskontrakt' (load factor contract) is quite
popular. Each contract is specified with the following values:
The energy (GWh) in the contract.
The maximum withdraw from the contract for each time period. (GWh/period).
The contract period
The contract price
(mils/kWh)
The flexibility of the contract is given by the number of time periods needed to use the total
energy compared to the length of contract period , i. e. the load factor.
Assume that we want to sell a load factor contract with energy equal to 10 GWh and 10
GWh/period as maximum withdraw. The contract is assumed to be valid for three periods. It
means that all the energy can be withdraw in either the first , second or the third period. This
contract is equal to a simple hydro production system with 10 GWh storage capacity, 10
GWh/period as maximum production capacity, zero inflow and initial reservoir of 10 GWh.
The rest value of the water in the reservoir is zero at the end of the contract period.
Assume that the model shown in Figure A.l describes the spot price for the contract period.
The model is similar to the model implemented in the MCM model. The price is known to be
10 mills/kWh in the first period and the figure shows the probabilties for prices in the
following two time periods. The expected price is 10 mills/kWh for each time period since
the price model is symetric. It is reasonable to believe the forward price to be equal to
expected spot price, i.e. 10 mills/kWh for the second and the third time period.
If we are risk neutral , what is the value of this contract?
First we can use the forward price to evaluate the load factor contract.
Forward price evaluation
The forward price is 10 mils for the second and the third period. Toady s price is also 10
mills/kWh. The value of the energy is therefore independent of which time period the energy
is withdrawn and the value must be:
10 mills /kWh* 10 GWh = 100 000 $
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Market price
14
10
milslk
mills/kWh
6 mills/k
Time period
Figure A.I
Markov model for future spot market price. The figure shows for example that
price is 14 millslkWh with probabilty 0. 2 in the third time period if the price
is 6 millslkWh in second time period
Correct value using SDP
By using stochastic dynamic progratng similar to what is used in the MCM model , the
correct value of the load factor contract is calculated to 108 400 $ (corresponding to a
marginal water value of 10. 84 milsIkWh). The correct value of the contract is therefore 8.4
% higher than the value given by the simple forward price evaluation.
The optimal strategy specified by marginal values is shown in Table A.l. The algorithm stars
at the last (third) time period.
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Table A.I
14 millslkWh
10 millslkWh
6 millslkWh
10.
The reason for the difference is that the flexibility of the reservoir combined with the spot
price uncertainty is incorporated into the calculation. If the price is 6 millslkWh in the second
period and the energy in the contract is not used , the optimal thing to do is to wait to the last
time period (i.e. use the reservoir , see Table A.l in combination with the Markov model).
The value of the flexibilty in the load factor contract is reduced in our example if the
probabilty of transition from 6 millslkWh in the second period to 14 millslkWh is reduced.
If only the forward price is used in the evaluation , the load factor contract has no additional
value compared to a forward contract.
The example shows that modellng of uncertainty in spot price may give an optimal value
which is different from what can be calculated from the forward price. The difference is given
by the flexibility of the contract combined with the price uncertainty. More flexibility and/or
increased uncertainty add to the error caused by the determnistic (forward) modelling of
market price.
The load factor contract is just a simple hydro production system , and the same conclusions
are therefore also valid for the MCM model. The example shown above is a simple example
that ilustrates the principle , and cannot be used to quantify how much modellng of
uncertainty wil improve the MCM model.
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,,
(J
674
672
670
668
666
664
- 662
660
J 658
.J 656
654
0: 652
650
648
646
644
642
640
2500
5000
7500 10000 12500 15000
17500 20000 22500 25000 27500 30000 32500 35000
37500 40000
Reservoir volume (Mm3)
Figure A 3.
Reservoir elevation as function of reservoir volume for Willston
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Lake
CI~~
APPENDIX 4: WATER VALUES FROM THE FIRST CASE STUDIES
The appendix shows the water values as function of rtservoir volume for the first case
studies. The cause of the irregularities shown in these figures was quickly ideritified and the
prevailing figures are shown in the report.
100
g: 50
:; 40
3500 6000 8500 11 000 13500
16000
18500 21000 23500 ' 26000
28500 31000 33500 36000
38500
Reservoir volume (Mm3)
Figure A4.
Water as function of reservoir volume for May 1998 for the base case in the
first case studies (corresponding to Figure 3. 1 in the report)
160
140
120
:c 100
m 80
2000
6000
10000
14000
18000
22000
26000
30000
34000
38000
42000
Reservoir volume (Mm3)
Figure A4.
Water as function of reservoir volume for December 2000 for the base case in
the first case studies (corresponding to Figure 3. 3 in the report).
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