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POWELDATA
~~~TECHNICAL REPORT
SUBJECTrrASK (title)
Review of the BC Hydro Marginal Cost Model (MCM)
SINTEF Energy Research
Address: 7034 Trondheim CONTRIBUTOR(S)
NORWAY
f;e Mo O(JjOSSOReception: Sem Scelands ve; 11Telephone: +4773597200Telefax: +4773597250 CLlENT(S)
http://ww .energy.sintef. no BC Hydro Power Supply Purchasing
E. No. : NO 939350 675
TRNO. DATE CLIENT'S REF.PROJECT NO.
F4754 1998- John W. Taylor lIX134.ELECTRONIC FILE CODE
CLASSIFICATION
I:\DOK\11 \BM\98004023.DOC J 0 Einar V remes , Powel Data AS RestrictedISBN NO. REPORT TYPE RES RCH DIRECTOR ( SIGN. ;:fl COPIES PAGES
82-594- 1318- f4V Nils Flatabfi at DIVISION LOCATION
LOCAL TELEFAX
Power Generation and Market Sem Srelandsvei 11 +4773 5972 50RESULT (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 supporttool and if the program development and maintenance are in accordance with the best industrypractices.
The MCM is a special purpose program used as decision support in the planning and operation of theWiliston Reservoir based on input representing: uncertain reservoir inflow, expected powerproduction from the rest of the system, uncertain load, uncertain export market, available importpossibilities and thermal power production. The program outputs are: incremental water values, powerproductions, imports , exports and several other results.
The program has a high level of detailed modellng of the BC Hydro system and produces results thatare useful for decision support in many application areas. The most important are: power trading,Willston Reservoir forecasting, financial forecasting, risk management and system operationplanning.
The MCM model has an input data representation and a mathematical formulation well in step withwhat is used in Norway and elsewhere. The processing appears to be correct
, and the main conclusionfrom the review is that the model is a useful tool and therefore a valuable asset for BC Hydro.
KEYWORDSSELECTED BY Hydro Operation Planning OptirnsationAUTHOR(S) Decision Support Simulation
(I~Executive Summary
The report contains a review of the Marginal Cost Model (MCM) developed by PowerSupply Operations of BC Hydro for resource optimization. The marginal cost is just one ofthe results from the model. The program has been used for 12 years and has beencontinuously developed to adapt to the changing planning environment.
The purpose of this work is to determne whether or not the MCM is an appropriate decisionsupport tool and if the program development and maintenance are in accordance with the bestindustry practices.
The MCM is a special purpose program used as decision support in the planning andoperation of the Wiliston Reservoir based on input representing: uncertain reservoir inflow
. expected power production from the rest of the system, uncertain load, uncertain exportmarket, available import possibilities and thermal power production. The program outputsare: 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 producesresults that are useful for decision support in many application areas. The most important are:power trading, Willston Reservoir forecasting, ' financial forecasting, risk management andsystem operation planning.
The main conclusions from this evaluation related to the Scope of Work are:
1. Appropriateness of input dataInflow forecasting and modellng are state of the arSame input data should be used consistently for all BC Hydro modelsUncertainty should be modelled-for the Columbia River and the Pacific North West(PNW) market
2. Test of model resultsInflow and cost parameters were changed and model response monitored. Prom thesetests , the processing within the MCM appears to be adequate. The models providing theinput data were not checked.
3. Computation methodology
The MCM optimisation par is based on a mathematical formulation that is well in stepwith 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 theoutput.
4. Dealing with uncertaintiesThe 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 alreadyimplemented in the MCM but the functionality must be activated.
5. Suitabilty, Significance and Usefulness
The MCM contains all relevant information concerning the physical and economicoperation of the system. It is therefore possible to build further on this model to get morecomprehensive tools for power system operation.The MCM is a useful tool for BC Hydro today and wil probably be even more useful inthe 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 modelsThe mathematical formulation is commonly used in Norway and elsewhere and is ideallysuited for the defined problem.
7. Proposed modificationsThe MCM would be strengthened by the changes currently being proposed.
The recommendations for future use of the MCM:
BC Hydro should take advantage of the probabilty that this type of tools wil be evenmore useful in the future with increased competition and trading.
The MCM results should be used more extensively than today. Especially theprobabilstic output should be util sed to get a picture of the uncertainty of importantquantities such as power production and income.
The user environment could be improved to provide better access to the model results forthe 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 abetter understanding of program output and interaction between different inputparameters. This would enhance the confidence in the MCM and increase theunderstanding of the operational complexity of hydro-thermal systems. Increased use wilalso uncover useful extensions and enhancements.
Models like the MCM need continuous development to adjust to changing marketconditions , to take advantage of better processing techniques , and to better simulate thehydrology. Input data must also be continuously refined. Sufficient resources should beallocated to do this work.
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TABLE OF CONTENT
INTRODUCTION..........................
...........................................
:..................................... 5
OVERVIEW OF SYSTEM AND PLANNING PROCEDURE ...................................... 7Power production system characteristics .............................................................. 7Planning procedure............................................................................................... 7
MARGINAL COST MODEL (MCM) ................. """""""'''''''' ........................... ........... 9Appropriateness of input data ............................................................................. 101.1 Inflow forecasting for Wiliston Reservoir.......................................... 101.2 Domestic load forecast.........................................................................
Generation forecast.................................................. ............................. 12Inport market.........................................
...... .......... ........ ......... .............
Export market................................................................... .................... 13Economic parameters...... ............ ...........
"""""'''''' .......... ........
""""'"'' 143 . Quality of input data............................................................................. 14Test of model results and output......................................................................... 14
Output from MCM ............... ........
....... .......
........................................... 15Case specification.......................... ....................................................... 15Case results........................................................................................... 17
Discussion of computation methodology..................... ............. "'" ........ ...... ....... 233.3. Optimisation algorithm......... ...........
.......... .............. .............
"""""""" 233.3. Simulation procedure............................................................................ 243.3 Discussion of assumptions ..................... ............................ ...... ............. 24
Adequacy of model in dealing with uncertainties............................................... 25Suitabilty, significance and usefulness of model............................................... 26
General comments................................................................................ 26Thermal operation planning ......................................................... ......... 26Hydro facilty maintenance scheduling ................................................ 275.4 Electricity trade purchases or sales ....................................................... 27Wiliston Reservoir elevation forecasting............................................. 28Financial forecasts................................................................................ 28Systems operation planning .................................................................. 29
Comparison with similar models ........................................................................ 29Discussion of proposed modifications.. ................... ........................................... 32
3.2
3.3
3.4
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 thePower Supply Operations deparment of BC Hydro. The program has a history of 12 yearsand 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 supporttool and if the program is developed and maintained in accordance with the best industrypractices.
The first par of this work was performed during a stay of 8 work days at BC Hydro wherediscussion of the input data, program performance and use of results , was the main focus. Thereport preparation has been done in Norway with valuable assistance from Donald Druce forthe 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 modeloutput and use.
Verify and test model results and output.
Examine and comment on the computation methodology and the capabilty andeffcacy of the optimisation algorithm, simulation procedure and assumptions.
Determne the adequacy of the model in dealing with input uncertainties andsensitivity related to: flow , load, gas supply, energy and capacity supplies from otherplants, and the electricity market.
Evaluate the suitabilty, significance and usefulness of the model for Power SupplyOperations and Corporate business decision makng.
Compare the MCM with other models that are used for similar purposes at hydro-electric utilties in other pars of the world. Comment on similarities and differenceswith respect to the physical system characteristics and modellng issues.
Review and comment on the appropriateness of any proposed modification to themodel , its inputs , outputs , computations and assumptions.
The complete Scope of Work is given in the section TR3 of the Invitation for Proposals withreference PSQ8-0 11. The approach and amount of work involved on each item is discussed ineach subsection.
Discussion of information technology standards is not par of the scope but a few commentsare 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 planningprocedures of BC Hydro.
The third chapter discusses the different items as specified in Scope of Work. Thesection numbering corresponds to the item number in the Scope. We have tried toseparate the different items as well as we can , but there might be some overlapping inthe report.
The fourth chapter discusses general observations regarding use and development ofthe 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 twocombustion turbine stations. The thermal plants account for about 9 percent of the capacitybut accounts for little energy.
The hydroelectric plants are grouped into three:
Small hydroelectric plantsColumbia River systemPeace River system
Small hydroelectric plants
The majority of these have limited storage capacity and their operation is related to the localhydrology rather than the system load. The year-to-year variabilty in the small hydrogeneration 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% ofproduction capacity). The energy produced is about 45% of the system s average yearlyenergy production. This river system has many operational constraints and despite its size , itdoes not contribute to the flexibilty of the system operation as much as expected from itsSIze.
Peace River system
The Peace River system consists of two plants with a total capacity of 3112 MW (32% of theproduction capacity). The energy produced is about 37% of the system s average yearlyenergy production. The total storage capacity is 40.000 millon m . The main flexibilty of theoperation 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 hydroresources. The hydraulic coupling, the uncertain resource availability, uncertain marketconditions and operational constraints make this to a very complicated optimisation problem.Simplifications are necessary to make such a problem computationally feasible. In theprocedure of simplification , it is important to take the system characteristics into account.
From the general characteristics listed above the following approach has been used: Schedulethe resources where the operating flexibilty is low first and then undertake the finalcoordination procedure.
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Using this approach, Wiliston Reservoir wil be the one that gets the role as slack reservoirand therefore defines the system marginal cost. There is no complex hydraulic coupling alongthe waterway and both plants on the Peace River have about the same flow capacity and canuse the discharged water. The average yearly inflow is about 75 percent of the reservoirstorage capacity.
The hierarchy applied in the planning has made it possible to prepare a tailor-made modelfor the long-term operation planing. It is assumed that a general pUrpose program would notbe appropriate to account for the special conditions in the system. The BC Hydro system is inthis planning structure very nice to model , but it is stil a challenge to come up with a modelformulation that takes maximum advantage of the system attributes.
The flexibilty in the operation of the Willston Reservoir makes it possible to use a timeresolution of one month, which again gives the opportunity to improve the modellng of otheritems such as importexport capabilities.
All these assumptions simpliy the whole planning problem and the crucial question is then ifthere is anything lost in the simplification process. We have evaluated the MCM modelassuming the assumptions to be correct. However, these basic assumptions are to some extentchallenged in section 3.3.3 where we conclude that some kind of interaction betweenColumbia 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 astochastic dynamic optimisation problem. The margInal cost is just one of the results fromthe model.
The goal function of the model is to maximise the expected net income for the company inthe 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 priceminus all imports multiplied by import priceminus thermal production multiplied by production costminus 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 theoptimisation.
The problem is dynamic since the reservoirs can be used to store water from one time periodto the next. It means that the decisions made in one time period have an impact on thedecisions made in the next time step.
Knowledge of the export price in one time step may also increase the possibilty to forecastthe price for the next time step, compared to a case where the price in the previous time is notknown.
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 dynamicoptimisation formulation would give the optimal and correct decisions. However, there is alimit to the size of the problems that can be solved with available computer technology. Thenumber of state variables that can modelled gives one of the major limitations. The statevariables 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 PacificNorthwest (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 occuronly 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 acurtailment. This functionality in the MCM serves to extend the feasible range of reservoirlevels , 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 onknowing the effect of weather on the domestic load and on the generation to be suppliedby 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 berandomised on a month to month basisLinking the historical small hydro generation to the weather year acknowledges that theseplants generally have small reservoirs with inflows primarily due to rainfall runoff, sotheir generation pattern is influenced more by weather conditions and local operatingconcerns than by system load or electricity trade opportunities - a realistic contributionfrom the small hydro plants is added to the MCM without have to spend any CPU timeleaving 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
It is always possible to marginally improve the modellng. Our evaluation of theappropriateness is based on what is practical and possible from our point of view.
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 modelfrom UBC for Wiliston Lake and the Columbia River system. A simpler regression basedmodel is applied for the small hydro plants. The snow pack correction changes the expectedspring/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
171495613
2179941
214637496
145573783
In order to be able to compare the inflows from year to year we have only used the inflowsfrom 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 14observations, 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 theexpected inflow to the system.
The actual numbers are somewhat smaller than indicated in the table since data from 1973 to1997 are used in the MCM model. This type of uncertainty wil be reduced as more inflowstatistics become available. This last comment is based on the assumption that there are noclimatic changes.
Columbia River systemPeace River s stemSmall 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 theregression model. Druce has published an article (1) where he shows that the UBC modelsdecrease 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 thequantity of the future obligations are not known through contracts. The load forecasting isdone once a year with a forecasting horizon of 20 years. Only the forecast for the first 6 yearsis important for the MCM model, since this is the planning horizon of the model. The loadforecast is referred to normal temperature , and the variations in load caused by uncertainty inthe temperature are included in the MCM model.
In order to be able to quantify the uncertainty in the load forecasting, we have comparedforecasts 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 forthe 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 forweather variations. The load forecast is referred to normal weather.
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Table 3. Different load forecasts (GWh) made for same periods and different momentsin time.
94/95 46736.95/96 48853. 48526.96/97 49255. 50064. 50036.97/98 50445. 50480. 51644. 51758.98/99 51888. 52472. 52590. 53648.99/00 53081.0 53680. 53767. 55651.000/01 53593. 54719. 55363. 57561.001/02 54708. 56056. 56614. 58481.002/03 55803. 57174. 57874. 59383.03/04 56885. 58058. 58833. 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 theuncertainty in the next years inflow.
Generation forecast
Generation forecasting is performed for all the generation units that are modelled as input tothe MCM model. This includes the small hydro plants, the Columbia River plants andBurrard generation.
Currently the Columbia River production is calculated using a spreadsheet model. Input to thecalculation 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 ColumbiaRiver system is generated.
The generation forecasts for the small hydro plants are given by the historical generation forthese plants (monthly generation for the period 1973- 1997).
Uncertainty in Columbia River generation forecast should be modelled and included into theMCM model. In principle , a simultaneous optimisation of these two reservoirs would havebeen preferable. However, due to the complexity of the planning problem this wil be verydifficult. Other procedures for coupling the model wil be required as for example iterationbetween 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 athigh load hours is linked to the swap prices for natural gas at AECO-C. Line capacities andthe number of light and heavy load hours give the maximum import quantity.
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Powerex forecasts , for the current year, the quantity and price of energy available fromPacific Northwest. For the subsequent years a typical market pattern is assumed that reflectsBP A's fish flush operation.
The import market is split from the export market in order to account for the price differencesbetween heavy and light load hours. The difference between import and export disappears
, ifmodellng of separate energy balances for heavy and light load hours are included in themodel as discussed in section 3.7. Wheeling costs and losses are included in the marketpnces.
Export market
The export markets are essentially the same markets as the import markets: the PacificNorthwest and Alberta. The export price in Alberta is connected to the gas price as describedin the previous section.
The export price in the Pacific Northwest is based on the COB index for the first 18 monthsand corrected for wheeling and losses , i.e. referred to the border. The export price after thefirst .18 months is described by two simple Markov models.
The first Markov model describes the water conditions in the Pacific Northwest (PNW). Themodel can take two values after the first calendar year, the value defines the market size usedin the model. After the first 18 months the value of the market size wil also influence on themarket price.
The second Markov model describes the prices in the PNW market. The model is currentlyused from the 18 to last month in the planning period. The model can take three values thatcombined 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 thephysical tie lines capacities wil be corrected for the quantity already used by commttedcontracts. This is a manual process.
The price in the Pacific Northwest is the most important market input parameter since the linecapacity (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 samemarket 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 priceforecasting methodology. The input to the MCM model should be the best available. TheMCM model does not constrain the price forecasting methodology.
Uncertainty in the PNW market should be modelled for the whole planning period asdiscussed further in section 3.
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Economic parameters
The economic parameters used in the MCM model are:
Foreign exchange rateInflation rate
Discount rate
Foreign exchange rateThe exchange rate shows significant moves in the period from January 1992 and up to April1998. 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 theforecasts from Nesbitt Burns , RBC- , Scotia McLeod and Wood Gundy. The rates for thesubsequent periods are taken from the 3 Year Plan - Rate assumptions.
Nominal discount and inflation ratesThe nominal discount rate is the sum of the inflation rate and the real discount rate. The realdiscount rate is the 8% corporate value and the inflation rate is taken from 3 Year Plan - RateAssumptions.
Important impacts of this economic parameter are:
The value of possible future income becomes lower. The value of storing water wil beless 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.
The users of the MCM results must be aware of the discount rate used and its impact on themarginal 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 inputdata. The input data to the MCM model should be based on the best that is available. Theinput 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 differentusers within BC-Hydro should use the same input data, same price forecast, same loadforecast etc.
Test of model results and output
The MCM model describes a complicated process and it is therefore difficult to make testswhere the result can be calculated in advance , i.e manual calculation ofthe results. We havetherefore tried to prepare different cases where we know how some of the results (watervalues) should change when the input is changed. It is not possible to quantify the change dueto complex interaction between model constraints , but we know in advance whether the water
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CJ~values should increase or decrease. This type of testing wil check the consistency of themodel
Output from MCM
Currently two different types of reports are distributed based on the results from the MCMmodel. Both reports are prepared by Donald Druce and distributed monthly.The first report contains least information and has the largest distribution list. The reportcontains median for marginal cost for the first 6 months and figures showing 10, 50 and 90percentiles for simulated reservoir level and marginal cost of water for the first 3 years. Thereport 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 LakeProbabilty distribution for Elevation in Wiliston LakeExpected discretionary sales and purchases GWh and $ for the next three fiscal yearsCommtted sales and purchases
In addition it is also possible to report:
Probability distributions for sales and purchasesProduction (GWh) and firm loadSpil 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 preparedand run by Donald Druce. The diffC:.tent cases are specified as follows:
1. Base CaseDescribes the import, export market and the physical production system by May 1998. Thedescription is identical to what was used in the May report by Donald Druce. Over the next18 months the export prices are relatively high, as a result of the water supply shortage in thePNW. The expected export prices beyond that time are lower, since the water supplyconditions are expected to be higher, on average . In the base case, the model already has thesignal 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 productionThe assumptions are identical to the base case except for that Burrard production is cut by600 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 situationinfluences on the future water conditions are given by the storage capacity and the dynamics of the PNWsystem.
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When Burrard production is reduced the water values should increase independently on thereservoir level.
3. Export priceThe export price to the PNW is increased by 20 % for the first 18 months compared to thebase case.
The water values should increase when the export price increases if the tie line capacities arenot already fully utilised. If the tie line capacities are already utilised, increased export pricewil not change the water values since extra water in the reservoir cannot be sold in the exportmarket. Remember that the water values describe the marginal value of the water, i.e thevalue of one unit extra of water in the reservoir.
4. InflowThe inflow forecast for May to September 1998 is increased by 3.7 %. All inflow scenariosare increased by the same amount Increased inflow should result in decreased water values since the probability of futureoverflow wil increase. The water value is zero (or even negative if flood damage costs areincluded) if there is 100 % probabilty of overflow. The water values before September 1998should therefore decrease. The water values after September 1998 should be independent onthe increase in inflow since the future inflow and thus also the probabilty of future overflowfor a given reservoir level from that time wil be unchanged.
5. LoadThe load is increased by 1 % for the first three years and 2 % for the rest of the planningperiod compared to the base case.
The water values should increase when the load is increased. Increased load should result inincreased water values since increased load wil increase the probability of curtailment in thefuture and reduce the probabilty of overflow. Both effects contribute to higher water values.
6. Unserved energyThe cost of unserved energy is increased by 20 % compared to the base case , this shouldresult in increased water values. Increased cost of unserved energy should increase the watervalues since the probabilty of curtailment in the future is higher than zero for most of thereservoirs. 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 themarginal value of the water. The water can either be used today or it can be stored and usedsome time in the future. When the discount rate increases , the future value of the waterdecreases. The value of current use wil be the same. The sensitivity to the discount rate wilamong 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 monthsSimulated reservoir level for the first three years of the simulation periodSimulated production for the first eight months
In addition , the water values for the year 2000 are observed for the base case as function ofreservoir volume.
The cases are such that they should all give a unique increase or decrease in the water valuesas described in the case description above. Our testing of the model is therefore based onchecking how the water values change compared with what we expected. We have checkedmainly 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 differentinput variables.
Discontinuous water values
The results from the cases show unexpected variation in the water values. The water valuesshould in theory be a continuously decreasing function of increasing reservoir volume ifnonlinearities are not included in the optimisation. Two types of discontinuities can beobserved:
- 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 wasquickly identified and new calculations were done. The original water value curves are shownin 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 reservoirand they wil therefore probably not have any significant effect on the model results. Thecause of the discontinuities should, however, be found. Use of debugger as discussed inchapter 4 wil make it easier to find the reason.
The normal operation range for the reservoir is shown is Appendix 3. The same appendix alsoshows 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 notbelieve this to be an indication of a fundamental problem, but an explanation should befound.
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120
100
3500 8500 13500 18500 23500 28500 33500 38500Reservoir volume (Mm3)
Figure 3. 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
Reservoir volume (Mm3)
27500 32500 37500 42500
Figure 3. Water values as a function reservoir volume for December 1998 for the basecase.
I :\DOK\ 11 \BM\98004023. DOC
(I~g160
140
120.
f 100
-; 80
2500 7500 12500 17500 22500 27500 32500 37500Reservoir volume (Mm3)
Figure 3. Water values as a function of reservoir volume for December 2000 for thebase case.
Stochastic modellng of export market
The stochastic market model for the PNW is currently used after the first 18 months of theplanning 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 pricerepresent the uncertainty in the future market price and are based on analyses of historicaldata. The values represent something similar to +/- one standard deviation. Thisdescription 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 tothe maximum export quantity for the given month and can take two values , the seconddescribing the smallest export quantity. The value is also coupled to the expected priceafter the 18 month.
Figure 3.4 shows the water values for different water conditions and price levels in the PNWfor December 2000. The values are in percent deviation from the mean of all price and waterstates. The results are as expected since price state equal 3 and water state equal 1 gives thehighest water value for the whole reservoir level. The lowest price combined with the highestwater state gives the lowest water values. The figure also shows that the uncertainty in exportprice 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 exportquantity effects the marginal value of the water.
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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 andwater states in the PNW.
Sensitivity studies
Figure 3.5 and 3.6 show changes in the water values from the base case for different casestudies. The water values should in theory increase for all cases except for the increasedinflow 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 basecase given by Figures 3. 1 and 3.2. It makes it easier to identify the different cases since theabsolute 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 unservedenergy case (20 % increase in curtailment cost) near empty reservoir. This is correct, since atthis time of the year, the probabilty of future curtailment is almost 100 % near emptyreservOIr.
The Burrard case gives lower water values than the base case for December 1998. This iscaused by the fact the inexpensive production (600 GWh) which is available in June and Julyin 1998 in the base case instead is available in year 2000. It is therefore correct that the watervalues 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 giveincreased water values for various pars of the reservoir. The increase for the export case isless than expected, but it is explained by the fact that the export capacity is almost fullyutil sed in the base case. The marginal value of the water wil therefore not increase when theexport 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 inflowcase.
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.15
-20
Figure 3.
go 5
Figure 3.
45'00
- .. ..
Burrard
-Increased load
. -...
-UnselVsd energy
Export price-Inflow
Discount rate
Reservoir volume (Mm3)
Changes in water values from base case for May 1998 for the different casestudies.
-.-.--
Burrard-Increased load
Unserved energy-Export price-Inflow
.. .. ..
Discount rate
45'00
.. .
Reservoir volume (Mm3)
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 reservoirlevels 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 springreservoir level as could be expected.
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~~~
672
670
, -'''. "' .. ;- '
! -Base case
\\ I . .. .. Burrard
' '
" , -h-"Exportprice
I-Inflowi -'--Load
Unserved energy
666
'C 664
662
: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 todifferent input variables. The results show for example that it might be more important thanexpected to update the load forecast more frequently since a change of 1-2 % is important forthe water values as shown in Figure 3.5 and Figure 3.6. The forecasting error seems to besystematic, which lead to that even a relatively small forecasting error accumulates to a largeamount of energy over the planning period.
It is impossible to give a general evaluation of the sensitivity to the different input variablessince the sensitivity wil depend on the current state. The sensitivity to the export price wouldfor 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 asdiscussed previously. We therefore conclude, based on the results from our case studies, thatthe model is correctly implemented. All the results seem reasonable.
The testing done is not a proof of that the model is correctly implemented. More testingwould always make us more certain that everything is correct. It is always possible to testmore.
We believe, however, that it is more efficient to train more people to run and understand themodel than it is to do more testing. If more people run the model , the probabilty ofdiscovering inconsistencies wil increase significantly.
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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)
v(k)
p(k)u(k)
Future net income as function of reservoir level in the first time periodReservoir level in period k Net income in period k as function of exportimport prices, production andreservoir level
Inflow in period kExpectation operator, stochastic variables are p(k) and v(k)Exportimport pricesProduction in period kTime period, month numberNumber of months in the planning horizonValue of water by end of month N
The described maximisation problep1 can be solved by use of stochastic dynamicprogramming, which is a standard optimisation technique. This method is perfect if theproblem can be solved by this approach. The number of state variables that can be included inthe model mainly gives the limitations. The number of state variables is in this case given bythe number of reservoirs (one) and the number of uncertain variables that are described byMarkov models (one for market size and one for market price). One of the advantages withthis method is that it wil always give a solution to the problem. For other methods thatdepend on iteration, convergence is not always guaranteed.
We have not checked how the solution algorithm is implemented and can therefore notcomment on the implementation and how effcient (meaning how fast it is compared to whatis possible) it is. Presently the optimisation takes approximately 1.5 cpu. minute for each yearin the planning period which is more than fast enough. Preparation of input data is withoutdoubt much more time consuming than running the model, days or hours compared tominutes.
The problem of efficiency may however be more important when the model is improved asdescribed 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 isimplemented today, be represented by the historical years (1973 to 1978), the second by 1974to 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 (priceand quantity limits) are made. There is only one draw for each historical scenario. Thesimulation 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 modelare the mean of the results from the 25 scenarios.
In our opinion the simulation methodology is good. In the future we believe that there wouldbe 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 credibleprobabilty distribution. More scenarios can easily be generated combining the historicalyears differently and makng new draws from the Markov models.
The simulation results are dependent on the draw from the Markov model. If more scenariosare generated based on new draws from the Markov model , the simulation results (expectedvalues etc.) wil be less dependent on the actual draw.
Uncertainty in Columbia River production should be included using the same historical yearsas for the other uncertain variables. It wil probably increase the uncertainty in the simulationresults.
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 theWiliston. It is therefore assumed that possible changes in Columbia River production donot 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. ThePNW system is a hydro-thermal production system with both hydrological and electricalcoupling to the BC Hydro system. The Markov type modellng of PNW is only correct ifthere 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 fromApril to May 1998 in the official results from the MCM model were to some extent explainedby changes in the production strategy for the Columbia River.
Table 3.4 shows some ' system statistics ' for different pars ofthe BC Hydro system. KelvinKetchum supplied the numbers. The values give a simplified description of the physicalsystem, but they indicate that the flexible storage capacity in Columbia River may not beinsignificant compared to Wiliston Lake.
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(t~Both the previous comments imply that some type of interaction between Columbia Riverand Peace River should be modelled. It should be noted that the operation of the MicaReservoir must, to some extent, be coordinated with downstream U.S. project operation underthe Columbia River Treaty and Non-Treaty Storage Agreement. This linkage reduces theflexibilty of Mica Reservoir storage for BC Hydro purposes.
Storage capacity\TWh)Maximumproduction(GWh/week)Inflow (TWh/ ear)
18.4 10.
570 640
17. 15. 11.5
It is not possible to quantify the error introduced by these assumptions. There are two slightlydifferent approaches to overcome them:
Model the PNW and BC system in one large model. Possible solution approaches arediscussed 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 PNWcould be estimated from a physical model of the PNW or historical variations and theforward price. If a physical model is used, possible hydrological connection orcorrelation could be included using the same historical years as reference in thesimulati ons.
BC-Hydro is currently developing a new model for the Columbia System. The interactionbetween 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 marketuncertainty, which are included both in optimisation and simulation. The uncertainty in thephysical input variables (temperature and inflow) are modelled as well as possible. TwoMarkov models model the market uncertainty. These models should also be used for the first18 months if necessary data is available.
It might also be possible to increase the number ofdiscrete price levels in the Markov model. In our own model we have found that 7 discretelevels 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 theminimum allowable reservoir levels. For the two multi-year reservoirs (Williston and Mica), it is practicallyimpossible to use all of this storage in one year. Furthermore , Columbia River Treaty and, to a smaller extentPeace River Ice constraints limit BC Hydro s flexibility at these two reservoirs.
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(J~The model results are also dependent on other uncertain input variables as ilustrated by thecase 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, mediumand 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 probabilitydistribution given by the simulation results are used in further analyses such as riskmanagement. This method can however disguise the result' s sensitivity to the different inputparameters.
We believe that the best practical method is to do manual sensitivity analyses for theuncertain input parameters , which are not already modelled. The MCM model is in principlewell adapted to such analyses since it is relatively fast. The user interface and resultpresentations are , however, based on old computer technology and this makes it in practiceimpossible for the users of the results to do such analysis themselves.
The model can relatively easily be expanded to include simulation of more uncertainty asdescribed above. However, it is probably more beneficial to star using the already availableuncertainty from the MCM model in the different applications than it is to simulate for moreuncertain variables.
Suitabilty, signifcance and usefulness of model
General comments
The users own explanation of how they currently use the results from the MCM model isdescribed in Appendix 1.
The model contains in principle all the relevant information concerning the physical andeconomical operation of the BC-Hydro system including production, transmission,consumption and importexport. The model can be applied to a wide rage of analysis wherethe economic consequence for BC Hydro of changes in some of the input variables to modelis wanted.
CUlTent use of the model as described in Appendix 1 and below, applies only the expectedresults 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 thedecision 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 themarginal 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 suchas reactive power support, air quality and water temperature.
There is nothing in the formulation of the Burrard operation that takes into account thehistory when the decision is made. If there is no contract on gas that is coupling the decisionon the different stages , this is probably an adequate approach. Starstop costs are not relevantwith the time resolution used. A state variable for the operation of Burrard would not havebeen computation ally feasible.
The Burrard is modelled in the MCM in a way that is common for thermal units in hydrodominated systems. A full physical model is never possible, but since the MCM takes intoaccount the main characteristics of the Burrard operation, the results from the MCM shouldbe useful for the refinement of the plan for Burrard.
Hydro facility maintenance scheduling
Maintenance scheduling is required for:
Small hydro unitsColumbia River systemPeace River system
The whole planning hierarchy is based on the fact that each of the small hydro units does notsignificantly affect the results of MCM. The planning of the small units are based on localconditions. These units wil see a market price where they wil principally be price takers. Formaintenance planning of these plants , the local conditions combined with the marginal cost ofwater wil define the timing of the maintenance.
The maintenance planning of the Columbia River system is more complicated. The powerproduction plans are input to the MCM, and the relative size of the systems cause significantinteraction. The marginal water value is important input when decisions about maintenance ofthe Columbia River are made. However, the use of Treaty and non-Treaty storage wil haveimpact on the maintenance decisions. It is important that major changes in the output from theColumbia River systems are checked with each new run of the MCM.
The consequences of maintenance schedules of the Peace River system must be evaluated bythe MCM itself. Major reductions in the power production capabilities may impact themarginal water values.
Reduction in transfer capacity due to maintenance or other reasons for derating may be theworst problem for the utilisation of the system. The MCM must then access the impact ofreduced market access capability.
Electricity trade purchases or sales
The MCM model gives the current marginal value of the water. If the current export price ishigher, the right decision is to sell as much as possible. The opposite decision is correct if theexportimport price is below the current water value. The model results are well suited to suchanalysis.
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The MCM model also outputs simulated reservoir levels , exportimport and marginal valuesof water for different future scenarios. These values are and can be used in decision supportfor today s trading future contracts. Currently only the expected future values are used byPowerex in their trading decisions as described in Appendix 1.
Today the simulated expected net importexport over a period of time (6- 12 months) ahead isused as a target for the net trading over the same period. The expected water values are usedto decide which months to buy and sell.
An alternative method would be to use the expected simulated water value as a signal fortoday s trading in the forward market with less consideration to the overall balance. Thismethod is in principle more correct, but it makes it more difficult to control the future energybalance (risk) as long as there is no risk management tools available to provide both price andvolume uncertainty.
The MCM model contains the connection between physical constraints (tie-line capacitiesmaximum production etc.), exportimport price uncertainty, load uncertainty, inflowuncertainty, i.e. the connection between price and quantity uncertainty. This is importantinformation that should be incorporated into risk management. Two different approaches canbe applied:
Enhancement of a MCM type model to also incorporate functionality for riskmanagement.Using the results (price and production scenarios etc.) from the MCM model as input toother available risk management tools.
Without going into detail the first method is the best, but the second method can probably beimplemented faster. Both are improvements from present practice.
Wiliston Reservoir elevation forecasting
The optimisation par of the MCM gives the incremental water values for different reservoirlevels and time stages. The simulation par, gives among other results , the reservoirtrajectories 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 marketscenarlOS
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 muchimpact on the results. For special cases , one may find that the extreme (maximum andminimum) trajectories of the reservoirs are underestimated. Despite this we believe that theMCM 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 aredescribed 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 theseare more related to how things are organised than to the MCM itself. The optimisation par ofthe MCM calculates the optimal strategy (incremental water values) and the simulation pargives an outcome for different scenarios. Much of the requested information can be foundfrom the basic model output. Total costs can be derived from the proposed production plansand 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. Webelieve a wider use of all the results would be more appropriate. The MCM gives thedistribution for different market and weather conditions and wil therefore provide results thatcan 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 thesimpliication in the planning procedures where the results in one evaluation is input to theMCM, use of this model wil be the last step to evaluate the impact of the decisions on themarginal cost values. Change of strategy for the operation of Columbia River plants wil haveto 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 planningproblem. There are several similar programs used around the world to solve hydro operationplanning problems. These programs would not necessarily be able to solve the same problemas the MCM due to special conditions within the BC Hydro s system. However, use ofsimilar mathematical formulation is a justification for that the MCM is in step with similarprograms. This section briefly describes some approaches used to solve the hydro operationplanning 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 andfrequently used solution method for similar problems also in other countries. Thecorresponding model used in Norway (2), (4), was originally based on a paper published asearly as 1962 (3), and is based on the same type of modellng and solution algorithm as theMCM model.
This model has gone through several improvement stages , one of the recent ones being themodellng of the market price using Markov models as a consequence of the marketliberalisation.
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 istherefore 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 onlyinflows and decisions that make you reach a pre-calculated value in the next time step ispossible. 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 levelswhich again could allow for improved market modelling (i.e. increase the number ofdiscrete price levels). Markov models for price are usually estimated as continuos autoregressive models whichare made discrete in the optimisation. The model allows simulation for continuous pricesenes.
A similar model is also developed by the Swedish company Kraftdata. The principle ofmodellng and optimisation of the single reservoir model is as far as we know identical to ourown model.
Both the SINTEF Energy Research model (EOPS) and the Swedish model contain simulationmodels that distribute the production to the different plants in the physical system using arather complicated set of.rules. We wil not discuss this aspect since the Peace River Systemis so simple that the simulation is no problem. The simulation aspects of these models arehowever interesting if more complicated systems are modelled.
Generally, methods based on SDP are superior to other solution methods if the basicstochastic dynamic optimisation problem can be described by a limited number of statevariables.
Multi-reservoir models
The problem can generally be formulated as a multi-state stochastic dynamic optimisationproblem. It has up to now been impossible to solve such problems in general. In order tosolve 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 , oftenused with shorter planning horizon.Use a combination of optimisatipn and simulation combined with user interference to findthe ' optimal' solution , as in our own multi-reservoir model described below.
The continuous development of computer technology has now made it possible to solvemulti-state stochastic dynamic optimisation problems with fewer simplifications thandescribed above. Some of the mathematical techniques used are the following:
Stochastic Dual Dynamic Programmng (SDDP)Scenario AggregationDetermnistic 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 byalmost all the major players in the Scandinavian market. The model contains a detailedphysical description of the hydraulic systems and includes tie line constraints betweendifferent areas in the market. The model is described in (4). Thermal production units aredescribed by their marginal cost, production capacity and probabilty of failure. One of theimportant 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 fromthe 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 StochasticDynamic Programng (SDP). The main advantage is that the method can solve problemsthat consist of many state variables. The method can therefore in theory be used to optimise aphysical system consisting of a number of different hydro plants and reservoirs connectedwith tie lines with limited capacity. This is the same problem as solved by the EMPS-modelthe difference being that the SDDP method in theory gives a more formal correct optimisationof the problem.
The method has been implemented and tested both by SINTEF Energy Research andCEPEL, Brazil (8) and (9).
Based on our own experience , it wil probably take a long time before the method can solveall long and medium-term planning problems. Currently we are implementing the methodinto 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 themethod is very time consuming compared to SDP approach.
Scenario Aggregation
The principle of this method is described in (5). Both scenario aggregation and the methodusing determnistic equivalents assume that a scenario tree can describe the stochasticprocesses. In practice this wil limit the number oftime periods in the model to 5- , since thenumber branches in the tree increas. exponentially. Some results from the University inTrondheim indicate that the problems can as well be solved as Deterministic Equivalentsusing commercial LP solvers as CPLEX.
Deterministic Equivalents
The stochastic problem is converted to a large determnistic problem that can be solved bystandard LP solvers. The number of periods and branches in the scenario tree gives thelimitations. If the scenario tree consist of three branches for each period, the number of nodesare 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 methodhas much in common with the SDDP method but it is required that the uncertainty can bedescribed along a scenario tree.
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(J~Discussion of proposed modifications
The following modifications of the MCM model are proposed:
1) Segregation of heavy load hours (HLH) and light load hours (LLH) and model thepossibility of load factoring the Peace River generation.
2) Use of physical tie-line capacities to define limits for export and import3) Modellng of uncertainty in Columbia River production4) Restore uncertainty to the export price forecast for the US market.
We believe that these proposed modifications of the model can and should be implementedwithin the current framework.
The first and the second modifications are tied together. Implementation of these extensionswil increase the computation time. However, the increase in computation time should not bea problem, due to rapidly increasing computer capacities.
The third modification can also easily be included if uncertain (generation planscorresponding to the historical years 1973- 1997) production plans for Columbia River areavailable.
The fourth extension is already implemented but not used. The uncertainty is not modelledfor the first eighteen months due to two different reasons:
Lack of market informationThe 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 availablehistorical data. However, more statistical data wil be available. An alternative could also beto apply physically based forecasting models for the PNW market. The stochastic pricemodel 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 futureprice for the first eighteen months. The MCM model maximises the long-term income of thewater resources. The correct value of the generation is the market price at the time each GWhis produced, i.e. the spot price. The forward price can be seen as the market' s best estimate offuture spot price. Appendix 2 shows the value of modellng uncertainty in the future marketprice , even though there is a known forward price. The example also to some extentilustrates 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 avery important tool for BC Hydro. The model should therefore be understood and used bymore 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 thanone person should be able to maintain the program code.
Productivity could be improved by changing the programing environment. It is for examplenot possible to run a debugger with the current tools. Modern programmng tools areavailable for Fortran on a PC.
The future user interface for the MCM type model should be run from a Pc. It makes it easierto use the model and the model results. It is also easier to implement a modern user interfaceon 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 theMCM model.
The input to and the output from the model should be stored in a database. More simplemodel reports on basic input and results can be available for non-specialists. In order to dosensitivity 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 andforecasts , and physical values given by line capacities and the production system. The usercan 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 thatrequires any knowledge of the algorithm to be understood.
If the different users themselves can do sensitivity analyses , the confidence in the modelresults wil increase. More people wil thereby gain insight into how the complicated hydro-thermal system functions.
It is impossible to prove that the MCM gives the optimal decision strategy by comparingwhat happened for instance the last year with what had happened if we had followed anotherdecision strategy (a kind of benchmarking). The other strategy could for example be based onthe forward price and expected inflow and temperature.
The reason for this is that the model gives a decision strategy that in the long run wiloptimise use of available resources. The simpler strategy could by chance be better for justthe 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 differentcombinations which are included in the statistics that the strategy is optimal for. The inflow isfor instance represented by 25 years. At least 25 new years must pass before the history hasbeen covered. If we combine this with the statistics for market price, we must conclude thatthe strategy must be followed for many years before it can be proved that one strategy isbetter than another with absolute certainty.
,- ~~~,;.
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Jr.
(I~An estimate of the profit gained by using the MCM model could be computed using asimulator that simulates the different decision strategies over many years.
The results from the MCM model are not better than the quality of the input. Qualitychecking of the forecasting methodology for different input (e.g. price forecasting), ifpossible , might be beneficiaL
The stochastic dynamic programng approach gives without doubt the optimal decision aslong as the input is correct. The problem may be both nonconvex and nonlinear which makeit difficult to solve with other methods.
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Ct CONCLUSIONS
Our conclusions are based on written documentation , discussions and the results from the testsimulations. It is easier to verify that the modellng is correct than to prove that the model isimplemented without errors.
If the basic assumptions for the model hold as discussed in section 3.3.3 , the modellng andsolution algorithm implemented in the MCM model is sound. The model gives severalimportant results , which should be used more extensively than today. The results from themodel include net income , reservoir levels , export, import, thermal production , and firm loaddelivery. Each output can be given for each month in the planning period (maximum 3 yearsahead), or as accumulated numbers for periods of time. The model also contains probabilitydistributions for the same outputs for each month or accumulated periods. These probabiltydistributions should be used further if it is interesting for the decision maker to for exampleknow that there is 10 % probabilty of the income for the next year to be below a certainnumber with the current contract portfolio and production strategy. The probabiltydistributions 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 maintainthe model.
The user interface should be improved and run on a Pc. It is probably possible to run also theoptimisation on the PC. Use of modern software tools as debuggers can simplify developmentand maintenance of the program code significantly.
Methods for integration of the planned Columbia River model with the MCM model must bedeveloped in parallel with the development of the new model.
The value of the MCM is much dependent on consistent use for different applications. If itcan be agreed upon that the input data is the best available and is adequately represented inthe 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, theMCM must be upgraded with a new user interface and should also get a database connectionto make preparation of tailor-made reports easier. A new user interface would make morepeople able to use the model and thereby more confident with what the results are and theimpact of different assumptions.
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~~~
REFERENCES
( 1) Druce , DJ., "Forecasting B.C. Hydros operation of Wilis tone Lake - How muchuncertainty is enough", Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering, Vol. 3 , pp. 63- , 1995. Kluwer Academic PublishersNetherlands.Wangensteen, I., Mo , B. , and Haugstad, A.
, "
Hydro Generation Planning in aDeregulated Electricity Market" , Proceedings from "Hydropower Into the NextCentury , Barcelona, Spain , June 1995. Sutton, Uk: Aqua-Media International.Lindquist, J. , 1962. "Operation of a hydrothermal electric system: A multistagedecision process , AlEE Journal , April 1962.Haugstad, A.
, "
System Modeling in a Hydrothermal Electrical System, PresentationSeptember 5 , 1997 , Melhus , Norway.Gjelsvik, A. , and Wallace, S.W: "Methods for stochastic medium-term scheduling inhydro-dominated power systems, EFI TR A4438 , 1996.Rfitting, T. and Gjelsvik, A.
, "
Stochastic dual dynamc programng for seasonalscheduling 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 energyplanning , Mathematical programmng 52, 1991.Pereira, M.
, "
Optimal stochastic operations scheduling of large hydroelectricsystems , Electrical Power & Energy Systems, vol. 43 , no.3, July 1989 , pp. 161- 169.Jacobs, J. , et al. "SOCRATES: A system for scheduling hydroelectric generationunder uncertainty , Annals of Operations Research, vol. 59 , 1995
, pp.
99. 113.
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CI~APPENDIX 1: USER DESCRIPTION OF USE OF MARGINAL COSTMODEL
Information provided by the Marginal Cost Model (MCM) has a wide range of users. Theseusers either utilize details from the MCM itself, or access other reports which containinformation which derives from the MCM. The diagram below highlights the range of usersof the MCM.
BC HydroFinancialReporting
PowerexPowerSupply
Operations
PowerFacilities
This appendix contains a description of how the results of the MCM currently are used withinin BC Hydro. The descriptions are provided by BC Hydro staff. Only minor editorialchanges have been made to the text. In the main report , there are some references to thisappendix 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 theoutput from the MCM. However, the end users of this information, either directly orindirectly, is quite broad. The diagram on the next page demonstrates the flow of informationwhich is impacted by MCM output.
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Power Supply
Business Services
BC Hydro FinancialReporting
ProvincialGovernment! BC
Utilties Commssion
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(J~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 linecomponent on BC Hydro s Statement of Operations.
Power Supply Business Services uses the MCM summary schedule which lists the threeyear expected generation at Burrard Thermal (in GWh and $) using valley gas, and spotgas. These are the variable components that go into determning the cost of gas forBUITard included in the Cost of Energy line item on the income statement.
This same schedule also provides expected energy volumes and dollar values forpurchases from the United States , Alberta and Alcan, as well as sales to the United Statesand Alberta. This information is not directly included in the Three Year Plan or CUITentYear Forecast because Powerex provides a forecast in its place. However, it is used as animportant check against the Powerex forecast as the net difference between sales andpurchases must be identical for both the MCM and Powerex forecasts otherwise thepredicted future Wiliston elevation levels wil not be valid.
Power Supply Business Services uses a monthly MCM schedule which presentscommtted purchases and BUITard Thermal generation using assumed swing gas (in GWhand $). This information is directly included in the Three Year Plan or Current YearForecast for Cost of Energy.
Ad-hoc Reports
Periodically, Power Supply is reque ted to prepare special ad-hoc reports evaluating thefinancial impact of potential events. Examples include the analysis of the opportunity cost ofthe sink-holes at Bennett Dam and the financial impact of measures taken to improve theoutlook for Wiliston Reservoir levels. The MCM is able to perform this work, but its outputcomes in the form of net economic cost. Power Supply Business Services takes thisinformation and translates it into financial reporting terms on a fiscal year basis.
An example includes where Power Supply would compare the "before and after" effecton the financial forecast of a significant event , such as the sink-holes found at BennettDam. Several assumptions may be run over a period of time as different assumptionschange.
1:\DOK\11\BM\98004023.DOC Prepared by Be Hydro. Power Supply
(I ~Opportunities for improvements in the current process for financial forecasting
The inability of the model to translate economic costs into a financial statementequivalency. For example, the total impact of a change in circumstances could be $20millon , but it is not easily known whether what the impact is in terms of sales andpurchases 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 forBurrard Thermal energy costs it is necessary to add to the costs provided by the MCMany other costs which are allocated, i.e. municipal gas tax , demand charge and prepaidtransportation costs which are considered to be sunk costs.
The Electricity trade sales and purchase numbers are not the final numbers adopted by BCHydro. These are prepared by Powerex which uses some MCM information and thendevelops their own forecast.
The MCM uses load forecasting volumes for domestic consumption prepared annually.Periodically, adjustments wil be made based on advice from Marketing & CustomerServices for major known changes , Le. industrial labour strikes , El Nino. The financialforecast (as prepared by Marketing & Customer Services) uses the same domestic loadforecast as its staring point, but makes refinements to total and monthly curving ofconsumption on an ongoing basis. These changes are not incorporated in the MCM socost 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 thiscan 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 totest for a large number of scenarios on a timely basis. Although the MCM is toocomplicated to be easily accessible to all potential users, an improved situation wouldhave additional specialists available to run different analysis and provide expertise for awide range of users.
Comparabilty of different scenarios run over a number of months was sometimesquestionable as other inputs would change during the period (Le. updated inflowprojections). Therefore , different results couldn t always be directly linked to themanipulated variable.
MCM is not useful for treasury s needs for cash flow information. For example , theMCM only provides marginal costs for running BUITard and does not include other cashpayments related to this generation such as transportation charges for valley gas which areassumed to be sunk costs by the MCM.
There is a limited information available as to what assumptions have been used in theMCM. 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, tie-line 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 usingupdated load forecasts.
Integration of electricity trade information (sales and purchases reported in the MCM andforecasted by Powerex.
Additional staffng resources for performng scenario and sensitivity analysis using theMCM.
Output to incorporate' all cost of energy components so that the different scenarioanalysis ' are directly translatable into financial statement form. Alternatively, anaccounting program should be developed which combines the output from the MCM withother costs to produce financial statements.
B) BC Hydro Corporate Finance
BC Hydro Corporate Finance takes the Three Year Plans, forecasts and ad-hoc reportsprepared by Power Supply and described in the last section and combines them withinformation provided by other business units of BC Hydro to develop consolidated Plansforecasts and reports for BC Hydro. The Cost of Energy expense , which is influenced by theMCM, 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 ManagementBC Utilties CommissionCrown Corporate SecretariatProvincial Government
It 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 forenergy payments. This work is done to assist with BC Hydro s cash flow management.
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Ci~~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 Wilistonreservoir levels. Powerex refines the monthly importexport projections while keeping thereservoir level the same at the end of the next water year with the goal of optimisingprofitabilty in imports and exports.
The water values provided by the MCM (Rbch) are used as price indicators in a given month.For instance , if Rbch is 12 and the market price is 10 Powerex wil not sell energy from the BCHydro system. Instead, Powerex would buy energy regardless of the monthly expectedvolume for the given month as determned by the MCM. Powerex would exceed this energyvolume value (if physically possible) and then sell the differenceat a later time in order tomaintain the end of water year reservoir elevation.
Another use that Powerex has for the MCM is where significant forward sales or purchaseshave been made or are being considered. Powerex would request that a model run beperformed to determne the sensitivities of the MCM results as a consequence of theseactions.
Contribution from Murray Margolis
Powerex uses both the forecast expected purchase and sales volumes by month and thecalculated Rbch value when makng sales and purchase decisions. Commtted sales andpurchase contracts are included in both Powerex and MCM forecasts and since Powerexforecasts always leave the ending Wiliston Level unchanged the only difference is in theexpected sales and purchases by month. This might include simply transferring sales to timeperiods that we expect wil have the most value and also increasing both purchases and salesto levels that we feel we can obtain, nd are not captured because of the current limitations ofthe model. Volumes are used primarily for planning purposes and we do not make purchaseand sales decisions within the month solely for meeting volume targets. We do howeveradjust our marketing decision on a continual bases takng into account aggregate sales andpurchase volumes to date and forecasts to the end of the fiscal year including any updates weget 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 hasongoing requirements for financial forecast information. Powerex provides Electricity TradeSales and a significant part of the energy purchase component of Cost of Energy to theconsolidated forecast. As discussed in this section , the MCM impacts the development of thePowerex forecast and , as a consequence, has an influence on significant and volatile aspectsof the consolidated forecast.
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Ct~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 whererequired for capacity or system support these plants wil only be operated in the marginal costto 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 powerproducers within B.C , energy associated with Non-Treaty Storage activity, etc.). The valueof 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 pricefloor 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, thisresults in different price signals for energy from different reservoirs.
for performance measurement, keeping track of the margin between the system marginalcost and the actual exportimport transaction price. This is done to evaluate howexportimport decisions have added value to the BC Hydro system.
to forecast the Wiliston elevation levels for multiple years. This is a very importantfunction due to the sensitivity of business operations in the town of Mackenzie toreservoir levels. There is an ongoing need for elevation forecasts for both the businessesand BC Hydro to plan their operations accordingly. In addition, this information is reliedupon by BC Hydro to communicate the expected levels and to explain the anticipatedresponse, if any, which may be required.
to determne the prudent over-winter Peace discharge over the ice regime. This isimportant for maximizing the hydraulic capacity of the Peace River during the wintermonths and mitigating against the chance of unwanted flooding in the spring.
to perform numerious scenario analysis ' to estimate and compare the cost of possibleoutcomes. Analysis work of nature occurs on a regular basis and can have a significantinfluence on the ultimate plan of action under-taken by BC Hydro. The analysis caninclude a range of activities which have included the following:
the options for managing the Bennett Dam sink-hole issuethe cost of maintaining minimum elevation levels at Wilistonthe options for running or not running the non-SCR units at Burrardtesting the options of various longer term export contract proposals
to provide a number which indicates the percentage probability of a spil occurring atWiliston due to high reservoir levels. This is a powerful number for easily describing thestate of the reservoir. This is just one of the determnants that go into making Rbch.
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(J~~the MCM identifies the uncertainties of future operations by providing a range ofprojected reservoir levels , not just the expected level. This is an important systemplanning 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 toPower Facilities is a major operational planning document. The System Operating Planprovides information on planned generation at all BC Hydro facilities and additional detailsfor major plants which may include projected reservoir elevation levels, discharge rates andinflows.
Information from the MCM is used in developing the System Operating Plan. Assumptionsused 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 ofpurposes, but do not use any of the output for specific decision makng purposes. These usersinclude the following:
Energy Plans & Investments. Power Supply
In the past, Energy Plans & Investment (EPI) used the long range marginal cost model toprovide energy values for the first two years when calculating CONES (Cost of New EnergySources). However, the MCM is no longer required as more developed market informationhas become available in recent years; and a forward price curve has been developed.
1:\DOK\11\BM\98004023.DOC Prepared by Be Hydro , Power Supply
CI ~Marketing & Customer Services
Marketing & Customer Services (M&CS) reviews theMCM study report to ensureconsistency 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) ratestructure. The MCM value was used as a reasonableness check in evaluating various RTPrate 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 responsibiltyfallng under the Risk Operating Committee which is expected to evolve from the BC HydroEnergy Risk Management Project. While it is difficult to project these uses until theCommttee is formed and its agenda determned there should be some similarities in theirrequirements as with those already presented in Section A of this Appendix , Power SupplyBusiness Services.
Performance Measurement (Transfer Pricing)
Although currently not used as a transfer pricing measure, Rbch has been used in the past as ameans of allocating a purchase and sales price for energy transactions between Power Supplyand Powerex. There remain many strong arguments in favour of using the MCM fordetermning the transfer price. This is true for both Powerex electricity trade transactions andfor market based transactions operated by M&CS such as RTP.
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(J APPENDIX 2: DETERMINISTIC VERSUS STOCHASTICMODELLING
Determnistic and stochastic approach to evaluation of a load factor contract
In Norway, a type of contract called 'brukstidskontrakt' (load factor contract) is quitepopular. 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 periodThe contract price (mils/kWh)
The flexibility of the contract is given by the number of time periods needed to use the totalenergy 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 10GWh/period as maximum withdraw. The contract is assumed to be valid for three periods. Itmeans that all the energy can be withdraw in either the first, second or the third period. Thiscontract is equal to a simple hydro production system with 10 GWh storage capacity, 10GWh/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 be10 mills/kWh in the first period and the figure shows the probabilties for prices in thefollowing two time periods. The expected price is 10 mills/kWh for each time period sincethe price model is symetric. It is reasonable to believe the forward price to be equal toexpected 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 10mills/kWh. The value of the energy is therefore independent of which time period the energyis withdrawn and the value must be:
10 mills /kWh* 10 GWh = 100 000 $
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(J
Market price
Figure A.I
14 milslk
10 mills/kWh
6 mills/k
Time period
Markov model for future spot market price. The figure shows for example thatprice is 14 millslkWh with probabilty 0.2 in the third time period if the priceis 6 millslkWh in second time period
Correct value using SDP
By using stochastic dynamic progratng similar to what is used in the MCM model , thecorrect value of the load factor contract is calculated to 108 400 $ (corresponding to amarginal 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 starsat the last (third) time period.
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Ct ~EJR
Table A.I
14 millslkWh10 millslkWh6 millslkWh
10.
The reason for the difference is that the flexibility of the reservoir combined with the spotprice uncertainty is incorporated into the calculation. If the price is 6 millslkWh in the secondperiod and the energy in the contract is not used, the optimal thing to do is to wait to the lasttime 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 theprobabilty 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 additionalvalue compared to a forward contract.
The example shows that modellng of uncertainty in spot price may give an optimal valuewhich is different from what can be calculated from the forward price. The difference is givenby the flexibility of the contract combined with the price uncertainty. More flexibility and/orincreased uncertainty add to the error caused by the determnistic (forward) modelling ofmarket price.
The load factor contract is just a simple hydro production system, and the same conclusionsare therefore also valid for the MCM model. The example shown above is a simple examplethat ilustrates the principle, and cannot be used to quantify how much modellng ofuncertainty 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
6402500 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 Lake
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CI~~APPENDIX 4: WATER VALUES FROM THE FIRST CASE STUDIES
The appendix shows the water values as function of rtservoir volume for the first casestudies. The cause of the irregularities shown in these figures was quickly ideritified and theprevailing figures are shown in the report.
100
g: 50
:; 40
3500 6000 8500 11 000 13500 16000 18500 21000 23500 ' 26000
Reservoir volume (Mm3)
28500 31000 33500 36000 38500
Figure A4. Water as function of reservoir volume for May 1998 for the base case in thefirst 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 42000Reservoir volume (Mm3)
Figure A4. Water as function of reservoir volume for December 2000 for the base case inthe first case studies (corresponding to Figure 3.3 in the report).
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