1. 1 Systems Analysis Advisory Committee (SAAC) Thursday, October 24, 2002 Michael Schilmoeller John Fazio

2. Northwest Power Planning Council 2 Original Agenda Metrics –Stakeholders –Risk measures –Timing Representations in the portfolio model –thermal generation –hydro generation –conservation and renewables –loads –contracts –reliability –** Plan Issues **

3. Northwest Power Planning Council 3 Plan Issues incentives for generation capacity price responsiveness of demand sustained investment in efficiency information for markets fish operations and power transmission and reliability resource diversity role of BPA global change

4. Northwest Power Planning Council 4 Revised Agenda Approval of the Oct 4 meeting minutes Price Processes Representations in the portfolio model –thermal generation Metrics –Stakeholders –Risk measures –Timing Representations in the portfolio model –** Plan Issues ** : price responsive demand

5. Northwest Power Planning Council 5 Revised Agenda Approval of the Oct 4 meeting minutes Price Processes Representations in the portfolio model –thermal generation Metrics –Stakeholders –Risk measures –Timing Representations in the portfolio model –** Plan Issues ** : price responsive demand

6. Northwest Power Planning Council 6 Price Processes Problem: There are mathematical difficulties with describing prices and price processes statistically. To show: The natural logarithm of prices (or price ratios) provides a solution to the problem Price Processes

7. Northwest Power Planning Council 7 Price Processes Problem: Naïve attempts to describe prices statistically lead to nonsense. For example, a symmetric distribution, unbounded on the high side, must be unbounded on the low side Price Processes Oops

8. Northwest Power Planning Council 8 Price Processes Problem: More seriously, if we try to describe variation in a price process from historical data, we run into seasonality problems. Suppose we wanted to estimate the daily variation in prices from a price series over 90 days: Price Processes Standard deviation of the price curve (black) would be quite large and would not describe the daily variation (red) Clearly, we want something that more closely resembles daily price returns

9. Northwest Power Planning Council 9 Price Processes Problem: Returns themselves, however, have bad statistical properties. For example, what is the meaning of the average of a 50% increase in prices and a 50% decrease in prices? Price Processes

10. Northwest Power Planning Council 10 Price Processes Solution: What does work is the logarithm of returns (and the inverse transformation, exponentiation) Price Processes If price ends where it started out, the ratio of the prices is one, and the logarithm of returns is zero.

11. Northwest Power Planning Council 11 Price Processes Other nice properties of the logarithm of returns Price Processes

12. Northwest Power Planning Council 12 GBM Geometric Brownian Motion (GBM) –independent draws of ln(p) are made from a normal distribution –grows as Time T--> Makes sense, because the standard deviation of the sum of T draws from a distribution is Price Processes

13. Northwest Power Planning Council 13 GBM And the normal distribution of ln(p) gives us a reasonable distribution of prices Price Processes

14. Northwest Power Planning Council 14 Conclusion The natural logarithm of prices (or price ratios) overcome the mathematical difficulties with describing prices and price processes statistically Normal log returns produce lognormal price distributions, which have desirable distribution GBM is perhaps the simplest description of a stochastic process, where draws are independent, random, normal. May describe processes like stock prices well, where daily returns should be about normal Price Processes

15. Northwest Power Planning Council 15 Revised Agenda Approval of the Oct 4 meeting minutes Price Processes Representations in the portfolio model –thermal generation Metrics –Stakeholders –Risk measures –Timing Representations in the portfolio model –** Plan Issues ** : price responsive demand

16. Northwest Power Planning Council 16 Thermal Generation Objectives –We need a way to quickly estimate the dispatch factor for thermal generation, so that we can calculate variable cost. –Should have certain basic properties If average monthly prices ($/MWh) for gas are about the same as average monthly prices for electricity, the dispatch factor should be about 50 percent. If average monthly prices ($/MWh) for gas are well above the average monthly prices for electricity, but there is a good deal of uncertainty in the prices, the plant should dispatch, albeit a small amount If average monthly prices ($/MWh) for gas are well below the average monthly prices for electricity, but there is a good deal of uncertainty in the prices, the plant should run close to, but not quite 100% capacity factor (disregarding maintenance and forced outage)

17. Northwest Power Planning Council 17 Thermal Generation Objectives –To show: Thermal dispatch can be reasonably well using a spread call option on electricity and gas –To show: The monthly capacity factor of the thermal unit is provided by the “delta” of the option, that is, the change in the option’s price with respect to the underlying spread –To show: The standard Black-Scholes model for option pricing gives a good estimate of the capacity factor, with these adjustments: Discount rate r = 0 Volatility incorporates terms for the uncertainty in and the expected variation of the spread over the specified time frame

18. Northwest Power Planning Council 18 Typical Dispatchable Example of 1 MW Single Cycle Combustion Turbine (no dispatch constraints) Natural Gas price, $3.33/MBTU Heat rate, 9000 BTU/kWh Price of electricity generated from gas, $30/MWh Representations - thermal Value: note: we assuming the entire capacity is switched on when the turbine runs

19. Northwest Power Planning Council 19 Price Duration Curve If we assume each hour’s dispatch is independent, we can ignore the chronological structure. Sorting by price yields the market price duration curve (MCD) Value V is this area Representations - thermal

20. Northwest Power Planning Council 20 Variability viewed as CDF Turning the MCD curve on its side, we get something that looks like a cumulative probability density function (CDF) Value V is this area Representations - thermal

21. Northwest Power Planning Council 21 Uncertainty To this point, we have assumed we know what the hourly electricity price will be in each hour. However, we could similarly calculate the expected capacity factor for fixed hour using a CDF that described our uncertainty about prices within that hour. The preceding results still apply. Representations - thermal ??

22. Northwest Power Planning Council 22 Transformation of variables Everything we have done to this point still holds if we used transformed prices Representations - thermal

23. Northwest Power Planning Council 23 Simplification #1 The expected capacity factor over the time period will be a function of expected variation over the period and the uncertainty associated with each hour. It will be determined by the CDF of Representations - thermal

24. Northwest Power Planning Council 24 Simplification #1 Assume distributions of uncertainty in, say, are identical across all hours of the time period (e.g., month). Note is still a vector and has covariance structure Representations - thermal

25. Northwest Power Planning Council 25 Simplification #1 Note that constant uncertainty for z implies greater price uncertainty during times of high prices than during times of low prices (cool) Representations - thermal

26. Northwest Power Planning Council 26 Digression to Options European call option: Confers the right, but not the obligation to purchase a specified quantity of an instrument or commodity at a fixed price at a specified time in the future Representations - thermal Example of a call option on a stock with a strike price of $30 Below $30, the option is worthless For each dollar over $30 that the price of the stock reaches, the value of the option increases a dollar Intrinsic value

27. Northwest Power Planning Council 27 Digression to Options Because the option will expire at some specified time in the future and because we are uncertain what the value of the stock will be when the option expires, the value of the option is greater than the intrinsic value Representations - thermal max(0,p-X)

28. Northwest Power Planning Council 28 Digression to Options The value of an option is the expected value of the discounted payoff (See Hull, 3 rd ed., p. 295) Representations - thermal

29. Northwest Power Planning Council 29 Digression to Options The challenge in solving for the value explicitly (in closed form) is determining the discount rate r. The problem was essentially solved otherwise much earlier by A.J. Boness in his Ph.D. thesis. Fischer Black and Myron Scholes in 1973 showed that if certain assumptions held, the discount rate r should be the risk-free discount rate: –the stock pays no dividends, markets are efficient, interest rates are known –returns are normally distributed (Geometric Brownian Motion), so prices are lognormally distributed –prices change continuously so the option can be hedged Representations - thermal

30. Northwest Power Planning Council 30 Digression to Options The Black Scholes pricing formula Representations - thermal

31. Northwest Power Planning Council 31 Simplification #2 We will discount the payoff externally, so let r = 0 Representations - thermal This now closely resembles our calculation for the value of a thermal plant, assuming the CDF is the CDF of

32. Northwest Power Planning Council 32 Simplification #3 We assume that the distributions of Representations - thermal can be adequately approximated by normal distributions. Seems reasonable that could be described by a multivariate normal, because our uncertainty is largely symmetric, continuous, and unbounded above and below

33. Northwest Power Planning Council 33 Simplification #3 Case of Representations - thermal

34. Northwest Power Planning Council 34 Almost there... Then the B-S formula for the value the plant is Representations - thermal with the variance of playing the role of

35. Northwest Power Planning Council 35 Option “Delta” The change in value of the option with respect to the price of the underlying is the option “Delta.” It is just the slope of the price curve at a specified price Representations - thermal

36. Northwest Power Planning Council 36 The payoff The B.S. formula for the capacity factor the plant is Representations - thermal

37. Northwest Power Planning Council 37 Two issues remain –Gas prices are not constant (X is not fixed) –Most of what we may think we know about future price uncertainty might be expressed in terms of average monthly prices Solution –Use a European “spread” option instead of a standard European call option –Try to estimate the volatility of the hourly spread from the monthly volatilities and correlations Ah, Darn It Representations - thermal

38. Northwest Power Planning Council 38 Use a European spread call option instead of a standard European call option European spread option Representations - thermal

39. Northwest Power Planning Council 39 European spread option The Margrabe pricing formula for the value of a spread option, assuming no yields Representations - thermal

40. Northwest Power Planning Council 40 European spread option The delta for the Margrabe spread option, assuming no yields Representations - thermal

41. Northwest Power Planning Council 41 Hourly Volatilities from Monthly We are dealing with expected variation of electricity and gas price over the specific time period and with uncertainties in these, as well. Using our assumption that the hourly uncertainties are constant and independent of the temporal variations in the respective commodities, Representations - thermal

42. Northwest Power Planning Council 42 Hourly Volatilities from Monthly The first term is determined by the expected temporal covariance in commodity prices over the period, due to normal “seasonality” Representations - thermal

43. Northwest Power Planning Council 43 Hourly Volatilities from Monthly Question: Is unlikely new information will become available that would influence our view of temporal structure? If not, we do not expect uncertainties in monthly averages to be affected too much by assumptions about expected temporal variations in price. Representations - thermal

44. Northwest Power Planning Council 44 Hourly Volatilities from Monthly The second term is determined by the hourly uncertainties surrounding the hourly values for gas and electricity and their expected covariance. Clearly, uncertainty factors can swamp temporal variations Representations - thermal

45. Northwest Power Planning Council 45 Hourly Volatilities from Monthly Assumptions about hourly covariance in uncertainties is important here. They will affect the uncertainty in the monthly average price. If the hourly uncertainties for electricity (gas) are perfectly correlated, the relationship between monthly variance and hourly variance is If the hourly uncertainties for electricity (gas) are uncorrelated, the relationship between monthly variance and hourly variance is Representations - thermal

46. Northwest Power Planning Council 46 Hourly Volatilities from Monthly Assignment for the next class: –If we know the correlations between the monthly average returns for electricity and gas, what can we conclude about the correlation between the hourly returns? Representations - thermal

47. Northwest Power Planning Council 47 Example If we use the example we started out with, and the BS equation for delta, we use a normal distribution on ln(prices) that looks like But exact CF is 50% and B-S calculated CF is 49.7% (not bad) 100% uncertainty in the log returns (prices could be higher or lower by a factor of 2.7) gives us a 69% CF Representations - thermal

48. Northwest Power Planning Council 48 Conclusion –Thermal dispatch can be reasonably well using a spread call option on electricity and gas –The monthly capacity factor of the thermal unit is provided by the “delta” of the option, that is, the change in the option’s price with respect to the underlying spread –The standard Black-Scholes model for option pricing gives a good estimate of the capacity factor, with these adjustments: Discount rate r = 0 Volatility incorporates terms for the uncertainty in and the expected variation of the spread over the specified time frame –We need to better understand not only the expected correlation of uncertainties in electricity and gas prices, but how hourly prices are self-correlated over the time period of interest. Representations - thermal

49. Northwest Power Planning Council 49 Revised Agenda Approval of the Oct 4 meeting minutes Price Processes Representations in the portfolio model –thermal generation Metrics –Stakeholders –Risk measures –Timing Representations in the portfolio model –** Plan Issues ** : price responsive demand

50. Northwest Power Planning Council 50 Objective Objectives of this section: –Develop a risk metric for the region –To show: Risk metric should be minimum total power cost, subject to an annual CVaR constraint Metrics

51. Northwest Power Planning Council 51 Stakeholders Candidate Stakeholders –Load Serving Entity (Investor-Owned Utility or Public Utility District) –Customer –Regulatory Agency (Public Utility Commission) –BPA Metrics

52. Northwest Power Planning Council 52 Stakeholders Proposed Stakeholder Perspective –Total societal costs, to include –Capital costs, including those of transmission –Variable costs, –Internalized emission costs Metrics

53. Northwest Power Planning Council 53 Metric Candidates Candidates –Value at Risk (VaR) –Standard deviation –Expected shortfall –Conditional VaR (CVaR) –Van Neumann utility functions –Block maxima Metrics

54. Northwest Power Planning Council 54 V@R Metrics 95% one-day V@R V@R=4 cost

55. Northwest Power Planning Council 55 Example of Power Plants Metrics Consider an ensemble of 1MW power plants, each with a forced outage rate of 0.10, equal to that of the 10MW plant. A Paradox, because we know a system of smaller plants are better V@R is not “subadditive” Var85=2 Var85=0

56. Northwest Power Planning Council 56 Desirable Properties Metrics Subadditivity – For all random losses X and Y,  (X+Y)   (X)+  (Y) Monotonicity – If X  Y for each scenario, then  (X)   (Y) Positive Homogeneity – For all  0 and random loss X  ( X) = (Y) Translation Invariance – For all random losses X and constants   (X+  ) =  (X) + 

57. Northwest Power Planning Council 57 A List of Loss Scenarios Define a measure of risk  (X) = Maximum{X i } Metrics

58. Northwest Power Planning Council 58 Subadditivity  (X+Y)   (X)+  (Y) Metrics

59. Northwest Power Planning Council 59 Monotonicity If X  Y for each scenario, then  (X)   (Y) Metrics

60. Northwest Power Planning Council 60 Positive Homogeneity For all  0 and random loss X,  ( X) = (Y) Metrics

61. Northwest Power Planning Council 61 Translation Invariance For all random losses X and constants   (X+  ) =  (X) +  Metrics

62. Northwest Power Planning Council 62 Axioms for Coherent Measures Subadditivity – For all random losses X and Y,  (X+Y)   (X)+  (Y) Monotonicity – If X  Y for each scenario, then  (X)   (Y) Positive Homogeneity – For all  0 and random loss X  ( X) = (Y) Translation Invariance – For all random losses X and constants   (X+  ) =  (X) +  Metrics

63. Northwest Power Planning Council 63 Value at Risk/Probability of Ruin violates subadditivity Metrics

64. Northwest Power Planning Council 64 Standard Deviation violates monotonicity Metrics

65. Northwest Power Planning Council 65 Conditional Value at Risk - (CVaR) 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 Subject Loss Cumulative Probability Value At Risk CVaR is the average of all losses above the Value at Risk Metrics

66. Northwest Power Planning Council 66 What about the upside potential? CVaR is coherent But ! variation from year to year can be large! What about minimizing variation from year to year? We expect that upside will be sold to minimize cost, and variation will be automatically reduced Metrics

67. Northwest Power Planning Council 67 Timing Study? No Annual? Yes: Rates are often recalculated annually Metrics

68. Northwest Power Planning Council 68 Conclusion Risk metric is CVAR<X Objective function is Min costs More on coherent measures: See Artzerner, Delbaen, Eber, Heath, “Coherent Measures of Risk,” July 22, 1998, preprint http://www.math.ethz.ch/~delbaen/ftp/preprints/CoherentMF.pdf Metrics