Customers. State Regulated. FERC Regulated. Competitive PSERC ISO LSE

Transcription

1 PSERC Shmuel Oren IEOR Det., University of California at Berkeley and Power Systems Engineering Research Center (PSerc) (Based on joint work with Yumi Oum and Shijie Deng) Centre for Energy and Environmental Markets University of New South Wales October 9, 8 Oren - October 9, 8 1 Cometitive FERC Regulated State Regulated Customers Demand Management Generation Transmission Distribution Micro-Grids Retail Providers Power Generation SC LSE ISO PX TO SC - Scheduling Coordinator PX - Power Exchange ISO - Inde. System Oerator TO - Transmission Owner LSE - Load Serving Entity Oren - October 9, 8 1

2 CAISO (1998) Poulation: 3 Million Peak Load: 45,5MW Annual Total Energy: 36,45GWh Generation Caacity: 5,MW Average Net Imorts: 6,5MW (Peak 8,3MW) Wholesale Average rice: $56.7/MWh Annual Wholesale Market: $14 Billion PJM (1999) Poulation: 51 million Peak load: 144,644 MW Annual Total Energy: 79,GWh Generating caacity: 164,95 MW Transmission lines - 56,5 miles Members/customers Annual Wholesale Market: $4 Billion ERCOT (1) (not under FERC) Poulation: 18 Million (85% of Texas) Peak Load: 63,MW Annual Total Energy: 3,GWh Generation Caacity: 8,MW Average Net Imorts: Non Wholesale Average rice: ~$7/MWh Annual Wholesale Market: $ Billion Oren - October 9, 8 3 $1 $9 Marginal Cost ($/MWh) Marginal Cost $8 $7 $6 Resource stack includes: Hydro units; Nuclear lants; Coal units; Natural gas units; Misc. $5 $4 $3 $ $1 $ Demand (MW) Oren - October 9, 8 4

3 $4 ERCOT Energy Price On eak Balancing Market vs. Seller s Choice January thru July 7 $35 $3 UBES /5/3 $53.45 Balancing Energy (UBES) Price Sot Price Energy- ERCOT SC Energy Price ($/MWh) $5 $ $15 $1 $5 Jan- Mar- May- Jul- Se- Nov- Jan-3 Mar-3 May-3 Jul-3 Se-3 Nov-3 Jan-4 Mar-4 May-4 Jul-4 Se-4 Nov-4 Jan-5 Mar-5 May-5 Jul-5 Se-5 Nov-5 Jan-6 Mar-6 May-6 Jul-6 Se-6 Nov-6 Jan-7 Mar-7 May-7 Jul-7 $ Shmuel Oren, - UC October Berkeley 9, Oren - October 9, 8 6 3

4 Oren - October 9, 8 7 Generators Customers (end users served at fixed regulated rate) Wholesale electricity market (sot market) LSE (load serving entity Similar exosure is faced by a trader with a fixed rice load following obligation (such contracts were auctioned off in New Jersey and Montana to cover default service needs) Oren - October 9, 8 8 4

5 Real Time Price ($/MWh) 1 1 Period: 4/1998 1/ Actual load (MW) Oren - October 9, 8 9 Electricity Demand and Price in California Load 16 Price Electricity Price ($/MWh) Demand (MW) 16 9, July ~ 16, July 1998 Correlation coefficients:.539 for hourly rice and load from 4/1998 to 3/ at Cal PX.7,.58,.53 for normalized average weekday rice and load in Sain, Britain, and Scandinavia, resectively Oren - October 9, 8 1 5

6 Proerties of electricity demand (load) Uncertain and unredictable Weather-driven volatile Sources of exosure Highly volatile wholesale sot rice Flat (regulated or contracted) retail rates & limited demand resonse Electricity is non-storable (no inventory) Electricity demand has to be served (no busy signal ) Adversely correlated wholesale rice and load Covering exected load with forward contracts will result in a contract deficit when rices are high and contract excess when rices are low resulting in a net revenue exosure due to load fluctuations Oren - October 9, 8 11 Electricity derivatives Forward or futures Plain-Vanilla otions (uts and calls) Swing otions (otions with flexible exercise rate) Temerature-based weather derivatives Heating Degree Days (HDD), Cooling Degree Days (CDD) Power-weather Cross Commodity derivatives Payouts when two conditions are met (e.g. both high temerature & high sot rice) Demand resonse Programs Interrutible Service Contracts Real Time Pricing Oren - October 9, 8 1 6

7 One-eriod model At time : construct a ortfolio with ayoff x() At time 1: hedged rofit Y(,q,x()) = (r-)q+x() Sot market r LSE Load (q) x() Portfolio for a delivery at time 1 Objective Find a zero cost ortfolio with exotic ayoff which maximizes exected utility of hedged rofit under no credit restrictions. Oren - October 9, 8 13 Objective function max E U x( ) [ [( r ) q + x( )] ] max x( ) Utility function over rofit U[( r ) q + x( )] f (, q) dqd Joint distribution of and q Constraint: zero-cost constraint 1 Q E [ x( )] = B! A contract is riced as an exected discounted ayoff under risk-neutral measure risk-neutral robability measure rice of a bond aying $1 at time 1 Oren - October 9,

8 The Lagrange multilier is determined so that the constraint is satisfied [ ' ( r ) q + x*( ) ) ] E U Mean-variance utility function: g( ) = * f ( ) 1 E[ U ( Y )] = E[ Y ] avar( Y ) g( ) g( ) 1 f ( ) f ( ) Q x* ( ) = 1 + E [ E[ y(, q) ]] E[ y(, q) ] a g( ) Q Q g( ) E E f ( ) f ( ) Oren - October 9, 8 15 Bivariate lognormal distribution: (log,log q) ~ N(4,.7, 5.69,., ) under P & Q E[ ] = $7 / MWh, ( ) = $56 / MWh E[ q] = 3MWh, ( q) = 6MWh r = $1 / MWh (flat retail rate) ) ( * x 8 x Otimal exotic ayoff = =.3 =.5 =.7 =.8 E[] Mean-Var y 7 t i 6 l i b 5 a b 4 o r 3 P -5 x = =.3 =.5 =.7 =.8 Dist n of rofit -5 5 rofit: y = (r-)q + x() x 1 4 Note: For the mean-variance utility, the otimal ayoff is linear in when correlation is, Sot rice Oren - October 9,

9 Comarison of rofit distribution for mean-variance utility (=.8) Price hedge: otimal forward hedge Price and quantity hedge: otimal exotic hedge Bivariate lognormal for (,q) -4 x 1 3 y t i l i b a b o r P After rice & quantity hedge After rice hedge Before hedge Profit x 1 4 Oren - October 9, 8 17 Q m = E [log ] ) ( * x With Mean-variance utility (a =.1) 7 x E Q 63.1, if m m1 = , if m m1 =.5 [ ] = 69.8, if m m1 = 73.3, if m m1 = , if m m1 =.1 m = m1-.1 m = m1-.5 m = m1 m = m1+.5 m = m Sot rice Oren - October 9,

10 (Bigger a = more risk-averse) 1 E[ U ( Y )] = E[ Y ] avar( Y ) with mean-variance utility (m = m1+.1) x 1 4 ) ( * x a =5e-6 a =1e-5 a =5e-5 a =.1 a = Sot rice Note: if m1 = m (i.e., P=Q), a doesn t matter for the mean-variance utility. Oren - October 9, 8 19 F + x( ) = x( F) 1+ x'( F)( F) + x''( K)( K ) dk + x''( K)( K) Bond ayoff Payoff Forward ayoff Strike < F Put otion ayoff Payoff F Strike > F Payoff + Call otion ayoff dk F Forward rice Exact relication can be obtained from a long cash osition of size x(f) a long forward osition of size x (F) long ositions of size x (K) in uts struck at K, for a continuum of K which is less than F (i.e., out-of-money uts) long ositions of size x (K) in calls struck at K, for a continuum of K which is larger than F (i.e., out-of-money calls) K Strike rice K Strike rice Oren - October 9, 8 1

11 ) ( * x 8 x = =.3 =.5 =.7 =.8 x ( ) x ( ) x( ) ) ( ' * x 5 = =.3 =.5 =.7 =.8 ) ( ' ' * x uts calls = =.3 =.5 =.7 = Sot rice Sot rice 7 x F Sot rice Payoffs from discretized ortfolio f f o y a x*() Relicated when K = $1 Relicated when K = $5 Relicated when K = $ Sot rice Oren - October 9, 8 1 Objective function Utility function over rofit Q: risk-neutral robability measure zero-cost constraint (A contract is riced as an exected discounted ayoff under risk-neutral measure) (Same as Nasakkala and Keo) Oren - October 9, 8 11

12 Price and quantity dynamics Oren - October 9, 8 3 Oren - October 9, 8 4 1

13 Otimal Oren - October 9, 8 5 Oren - October 9,

14 *In reality hedging ortfolio will be determined at hedging time based on realized quantities and rices at that time. Oren - October 9, 8 7 max x() E[Y(x)] Q s.t. E [x()] = VaR (Y(x)) V where VaR (X) = í ã ã such that P{X í} = 1 ã Let s call the Otimal Solution: x*() Q: a ricing measure Y(x()) = (r-)q+x() includes multilicative term of two risk factors x() is unknown nonlinear function of a risk factor A closed form of VaR(Y(x)) cannot be obtained Oren - October 9, 8 8 Oum and Oren, July 1,

15 For a risk aversion arameter k x k 1 () = arg max E[Y(x)]- kv(y(x)) x() Q s.t. E [ x( )] = We will show how the solution to the mean-variance roblem can be used to aroximate the solution to the VaR-constrained roblem Oren - October 9, 8 9 Oum and Oren, July 1, 8 9 Suose.. There exists a continuous function h such that VaR ã (Y(x)) = h(mean(y(x)), std(y(x)),ã) with h increasing in standard deviation (std(y(x))) and nonincreasing in mean (mean(y(x))) Then x*() is on the efficient frontier of (Mean-VaR) lane and (Mean- Variance) lane Oren - October 9, 8 3 Oum and Oren, July 1,

16 The roerty holds for distributions such as normal, student-t, and Weibull For examle, for normally distributed X VaR ( x) = z std( x) E( x) The roerty is always met by Chebyshev s uer bound on the VaR: P{ x E( x) k} var( x)/ k x Px { Ex ( ) tstdx ( )} 1/ t VaR ( x) t std( x) E( x) 1 1 t Oren - October 9, 8 31 Oum and Oren, July 1, 8 31 Oren - October 9, 8 3 Oum and Oren, July 1,

17 We assume a bivariate normal dist n for log and q Under P log ~ N(4,.7 ) q~n(3,6 ) corr(log, q) =.8 Under Q: log ~ N(4.1,.7 ) Distribution of Unhedged Profit (1-)q Oren - October 9, 8 33 Oum and Oren, July 1, 8 33 x k () =(1- A B )/k-(r-)(m + E log D) + C B Oren - October 9, 8 34 Oum and Oren, July 1,

18 Once we have otimal x k (), we can calculate associated VaR by simulating and q and Find smallest k such that VaR(k) V (Smallest k => Largest Mean) Oren - October 9, 8 35 Oum and Oren, July 1, 8 35 Oren - October 9, 8 36 Oum and Oren, July 1,

19 Oren - October 9, 8 37 Oum and Oren, July 1, 8 37 Risk management is an essential element of cometitive electricity markets. The study and develoment of financial instruments can facilitate structuring and ricing of contracts. Better tools for ricing financial instruments and develoment of hedging strategy will increase the liquidity and efficiency of risk markets and enable relication of contracts through standardized and easily tradable instruments Financial instruments can facilitate market design objectives such as mitigating risk exosure created by functional unbundling, containing market ower, romoting demand resonse and ensuring generation adequacy. Oren - October 9,