Dynamics of Prices in Electric Power Networks

Transcription

1 Dynamics of Prices in Electric Power Networks Sean Meyn Prices Normalized demand Reserve Department of ECE and the Coordinated Science Laboratory University of Illinois Joint work with M. Chen and I-K. Cho NSF support: ECS & Control Techniques for Complex Networks DOE Support: Extending the Realm of Optimization for Complex Systems: Uncertainty, Competition and Dynamics PIs: Uday V. Shanbhag, Tamer Basar, Sean P. Meyn and Prashant G. Mehta

2 California s 25,000 Mile Electron Highway OREGON LEGEND COAL GEOTHERMAL HYDROELECTRIC NUCLEAR OIL/GAS BIOMASS MUNICIPAL SOLID WASTE (MSW) SOLAR WIND AREAS SAN FRANCISCO What is the value of improved transmission? More responsive ancillary service? NEVADA How does a centralized planner optimize capacity? Is there an efficient decentralized solution? MEXICO

3 HOME Meeting Calendar OASIS Employment Site Map Contact Us 40,000 MW Search September 28, ,000 Megawatts Available Resources The current forecast of generating and import resources available to serve the demand for energy within the California ISO service area 30,000 MW Forecast Demand Forecast of the demand expected today. The procurement of energy resources for the day is based on this forecast 20,000 MW Day Ahead Demand Forecast Available Resources Forecast Revised Demand Forecast Actual Demand Hour Beginning Actual Demand Today's actual system demand Revised Demand Forecast The current forecast of the system demand expected throughout the remainder of the day. This forecast is updated hourly.

4 HOME Meeting Calendar OASIS Employment Site Map Contact Us Search Emergency Notices Generating Reserves 7.0% 6.0% 5.0% 4.0% 3.0% 2.0% 1.0% 0.0% Stage 1 Emergency Stage 2 Emergency Stage 3 Emergency Generating reserves less than requirements (Continuously recalculated. Between 6.0% & 7.0%) Generating reserves less than 5.0% Generating reserves less than largest contingency (Continuously recalculated. Between 1.5% & 3.0%)

5 One Hot Week in Urbana Purchase Price $/MWh Monday Tuesday Wednesday Thursday Friday

6 Southern California, July 8-15, Spinning Reserve Prices PX Prices Weds Thurs Fri Sat Sun Mon Tues Weds

7 First Impressions: July 1998: first signs of "serious market dysfunction" in California Lessons From the California Apocalypse: Jurisdiction Over Electric Utilities Nicholas W. Fels and Frank R. Lindh Energy Law Journal, Vol 22, No. 1, 2001 FERC...authorized the ISO to "[reject]...bids in excess of whatever price levels it believes are appropriate... file additional market-monitoring reports".

8 APX Europe, June Prices (Eur/MWh) Week 25 Week Mon Tues Weds Thurs Fri Sat Sun

9 APX Europe, October Volume (MWh) Previous week Current week (10/24/05) Price (Euro) vr za zo ma di wo do

10 Ontario, November 2005 Ancillary service contract clause: Minimum overall ramp rate of 50 MW/min. Projected power prices reached $2000/MWh

11 Ontario, November 2005 Ancillary service contract clause: Minimum overall ramp rate of 50 MW/min. Projected power prices reached $2000/MWh

12 Ontario, November 2005 Ancillary service contract clause: Minimum overall ramp rate of 50 MW/min. Projected power prices reached $2000/MWh

13 Australia January Victoria 0 Tasmania

14 Australia January Volume (MWh) Volume (MWh) 8,000 6,000 4,000 1,400 1,200 1, Victoria Tasmania + 10, ,000 Price (Aus $/MWh) Price (Aus $/MWh)

15 16 3 I Centralized Control

16 Dynamic model Reserve options for services based on forecast statistics On-line capacity Forecast Actual demand Revised forecast Centered demand: D(t)=demand - forecast t

17 Dynamic Single-Commodity Model Stochastic model: Excess/shortfall: G D Goods available at time t Normalized demand Q(t) =G(t) D(t) Normalized cost as a function of Q: c Shortfall Excess production c + q

18 Dynamic Single-Commodity Model Stochastic model: Excess/shortfall: G D Goods available at time t Normalized demand Q(t) =G(t) D(t) Generation is rate-constrained: q - ζ - Q(t) ζ + High cost t

19 Dynamic Single-Commodity Model Average cost: E[c(Q)] p Q density when q = 0 = c( q q ) p Q (dq) c p Q (q q ) q c + q Optimal hedging-point: q solves c P{ Q 0} + c + P{ Q 0} =0

20 Dynamic Single-Commodity Model Average cost: E[c(Q)] p Q = density when q c( q q ) p Q (dq) = 0 RBM model: p Q exponential c p Q (q q ) q c + q Optimal hedging-point: q solves c P{ Q 0} + c + P{ Q 0} =0 q = 1 2 σ 2 ζ + log c c +

21 Ancillary service G(t) a G (t) The two goods are substitutable, but 1. primary service is available at a lower price 2. ancillary service can be ramped up more rapidly

22 Ancillary service G(t) a G (t) The two goods are substitutable, but 1. primary service is available at a lower price 2. ancillary service can be ramped up more rapidly

23 Ancillary service G(t) a G (t) The two goods are substitutable, but 1. primary service is available at a lower price 2. ancillary service can be ramped up more rapidly

24 Ancillary service a G (t) G(t ) K Excess capacity: Q(t) =G(t)+G a (t) D (t), t 0. Power flow subject to peak and rate constraints: ζ a d dt Ga (t) ζ a + ζ d dt G(t) ζ +

25 Ancillary service a G (t) G(t ) K Policy: hedging policy with multiple thresholds q 1 Q(t) =G(t)+G a (t) D (t) - ζ - Downward trend: G (t ) =0 d G (t ) =-ζ - dt a ζ + q 2 ζ + + ζ a + t Blackout

26 Diffusion model & control ( ) X(t) = ( Q(t) G a (t) Relaxations: instantaneous ramp-down rates: d dt G (t) ζ +, d dt Ga (t) ζ a +. Cost structure: c(x(t)) = c 1 G (t)+c 2 G a (t)+c 3 Q(t) 1 {Q(t) < 0} Control: design hedging points to minimize average-cost, min E π [c(q(t))].

27 Diffusion model & control Markov model: Hedging-point policy: ( ) X(t) = ( Q(t) G a (t) Ancillary service is ramped-up when excess capacity falls below q 2 q 2 q 1

28 Markov model & control Markov model: ( ) X(t) = ( Q(t) G a (t) Hedging-point policy: Optimal parameters: q 1 q 2 = 1 γ 1 log c 2 c 1 q 2 = 1 γ 0 log c 3 c 2 γ 0 =2 ζ+ +ζ a+ σ 2 D, γ 1 =2 ζ+ σd 2. q 2 q 1

29 Simulation Discrete Markov model: Q(k +1) Q(k) = ζ(k)+ζ a (k)+e(k +1) E(k) i.i.d. Bernoulli. Optimal hedging-points for RBM: q 1 q 2 = q 2 =2.996 ζ(k), ζ a (k) allocation increments.

30 Simulation Discrete Markov model: Q(k + 1) Q(k) = ζ(k) + ζ a (k) + E(k + 1) Optimal hedging-points for RBM: q 1 q 2 = q 2 = Average cost: CRW Average cost: Diffusion model q 1 q 2 q 1 q 2 3 q 2 q 2 7

31 II Relaxations (skip to market)

32 Texas model D 1 E 1 G p 1 G a 2 Line 1 Line 2 G a 3 D 2 E 3 D 3 E 2 Line 3 Resource pooling from San Antonio to Houston?

33 Aggregate model Line 1 Line 2 Line 3 Q A (t) = extraction - demand = Ei (t) D i (t) G a A (t) = aggregate ancillary = G a i (t) Assume Brownian demand, rate constraints as before Provided there are no transmission constraints, X A =(Q A G a A ), single producer/consumer model

34 Effective cost c(x A,d) Line 1 Line 2 Given demand and aggregate state Line 3 find the cheapest consistent network configuration subject to transmission constraints min s.t. (c p i gp i + ca i ga i + cbo i q i ) q A = (e i d i ) g a A = g a i 0 = (g p i + ga i e i) q = e d f = p f F consistency extraction = generation vector reserves power flow equations transmission constraints

35 Effective cost c(x A,d) Line 1 Line 2 Line 3 g a A x A x A X + R What do these aggregate states say about the network? x A x A q A

36 Effective cost c(x A,d) Line 1 Line 2 Line g a A 1 x A City 1 in blackout: q 1 = 46,, q 2 =3.0564,, q 3 = Insufficient primary generation: g 1 = 71,, g2 a = ,, g3 a = Transmission constraints binding: f 1 = 13,, f 2 = 5, f 3 = 8 10 q A

37 Inventory model demand 1 demand 2 Controlled work-release, controlled routing, uncertain demand.

38 Inventory model: Workload relaxation w 2 Resource 1 is idle W + (1 ρ) Asymptotes: Resource 2 is idle q = 1 2 σ 2 ζ + log c c + w 1

39 III Decentralized Control

40 Supply & Demand Cost of generation depends on source Price Supply Curve Gas Turbine ($20-$30) Coal ($10 -$15) Nuclear ($6) Quantity

41 Supply & Demand Demand for power is not flexible Price High Priority Customers $5,000/MW? Customers with interruptible services Demand Curve Quantity

42 Supply & Demand Equilibrium Price & Quantity: Intersection of supply & demand curves Price High Priority Customers Customers with interruptible services Equilibrium Price $20 - $30 Consumer Surplus Supplier Surplus Gas Turbine ($20-$30) Supply Curve Nuclear ($6) Coal ($10 -$15) Demand Curve Equilibrium Quantity 45,000MW Quantity

43 Supply & Demand Equilibrium Price & Quantity: Intersection of supply & demand curves 5000 Price Equilibrium Price 3000 $20 - $ High Priority Customers Customers with interruptible services Where do ramp constraints appear? 4000 Consumer Surplus Variability? Where is hedging? Supplier Surplus Coal ($10 -$15) Nuclear ($6) Gas Turbine ($20-$30) Equilibrium Quantity 45,000MW Supply Curve Demand Curve 5 Quantity

44 Main Page Market Results Welcome to APX! APX is the first electronic energy trading platform in continental Europe. The daily spot market has been operational since May The spot market enables distributors, producers, traders, brokers and industrial end-users to buy and sell electricity on a day-ahead basis. Prices (Eur/MWh) Week 25 Week 26 The APX-index will be published daily around 12h00 (GMT +01:00) to provide transparency in the market. Prices can be used as a benchmark. Volumes (MWh) 2500 MWh 2000 MWh 1500 MWh 1000 MWh 500 MWh Mon Tue Wed Thu Fri Sat Sun

45 Second Welfare Theorem Each player independently optimizes... Consumer: value of consumption minus prices paid minus disaster W D (t) := v min D(t),G p (t) + G a (t) p p G p (t) + p a G a (t) + c bo Q (t) 2000 Supplier: profits from two sources of generation 1000 W S (t) := p p c p G p (t) + p a c a )G a (t)

46 Second Welfare Theorem Is there an equilibrium price functional? Is the equilibrium efficient??

47 Second Welfare Theorem Is there an equilibrium price functional? Is the equilibrium efficient?? Yes to all! p e (r e, d) = (v + c bo ) I{ r e < 0} c v 1 bo 5 cost of insufficient service reserve Q(t) = q value of consumption demand D(t) = d 21

48 Efficient Equilibrium Prices Normalized demand Reserve

49 Southern California, July 8-15, Spinning Reserve Prices PX Prices Weds Thurs Fri Sat Sun Mon Tues Weds

50 Conclusions 250 Spinning Reserve Prices PX Prices Weds Thurs Fri Sat Sun Mon Tues Weds The hedging point (affine) policy is average cost optimal Amazing solidarity between CRW and CBM models Deregulation is a disaster! Future work?

51 Extensions and future work Complex models: Workload or aggregate relaxations 0 Price caps: No! Responsive demand: Yes! Is ENRON off the hook:?

52 Extensions and future work Complex models: Workload or aggregate relaxations 0 Price caps: No! Responsive demand: Yes! Is ENRON off the hook:? What kind of society isn't structured on greed? The problem of social organization is how to set up an arrangement under which greed will do the least harm; capitalism is that kind of system. -M. Friedman

53 Epilogue Fundamentally, there are only two ways of coordinating the economic activities of millions. One is central direction involving the use of coercion - the technique of the army and of the modern totalitarian state. The other is voluntary cooperation of individuals - the technique of the marketplace. -Milton Friedman

54 Epilogue Justification: 1. Economic systems are complex 2. Regulators cannot be trusted

55 Epilogue Justification: 1. Economic systems are complex 2. Regulators cannot be trusted Airplanes, highway networks, cell phones... all complex

56 Epilogue Justification: 1. Economic systems are complex 2. Regulators cannot be trusted Airplanes, highway networks, cell phones... all complex Complexity is only inherent in the uncontrolled system: In each of these examples, the behavior of the closed loop system is very simple, provided appropriate rules of use, and appropriate feeback mechanisms are adopted.

57 References q 1 q 2 16 q 1 q 2 = 1 γ 1 log c 2 c 1 q 2 = 1 γ 0 log c 3 c 2 3 q 2 M. Chen, I.-K. Cho, and S. Meyn. Reliability by design in a distributed power transmission network. Automatica 2006 (invited) I.-K. Cho and S. P. Meyn. The dynamics of the ancillary service prices in power networks. 42nd IEEE Conference on Decision and Control. December 2003 I.-K. Cho and S. P. Meyn. Efficiency and marginal cost pricing in dynamic competitive markets. Under revision for J. Theo. Economics. 46th IEEE Conference on Decision and Control 2006 P. Ruiz. Reserve Valuation in Electric Power Systems. PhD dissertation, ECE UIUC 2008

58 Poisson s Equation First reflection times, τ p :=inf{t 0:Q(t) = q p }, τ a :=inf{t 0:Q(t) q a } [ τ p ( ) ] h(x) =E x c(x(s)) φ ds 0

59 Poisson s Equation First reflection times, τ p :=inf{t 0:Q(t) = q p }, τ a :=inf{t 0:Q(t) q a } [ τ p ( ) ] h(x) =E x c(x(s)) φ ds 0 Solves martingale problem, M(t) =h(x(t)) + t 0 ( c(x(s)) φ ) ds

60 Poisson s Equation X(t) = ( Q(t) ) G a (t) h(x(τ a )) h(x) =E x [h(x(τ a ))+ τa 0 ( c(x(s)) φ ) ds ] q a q p

61 Derivative Representations h(x), ( 1 1) = 0, x = X(τ a ) h(x) =E x [h(x(τ a ))+ τa 0 ( c(x(s)) φ ) ds ] q a q p

62 Derivative Representations λa(x) = c(x), ( 1 1) = c a I{q 0}c bo h(x), ( 1 1) = 0, x = X(τ a ) h(x) =E x [h(x(τ a ))+ τa 0 ( c(x(s)) φ ) ds ] q a q p

63 Derivative Representations [ τa h(x), ( ) 1 1 = E x 0 ] λ a (X(t)) dt = c a E[τ a ] c bo E x [ τa 0 ] I{Q(t) 0} dt Computable based on one-dimensional height (ladder) process, H a (t) = q a Q(t)

64 Dynamic Programming Equations If q p = q p and q a = q a Then h solves the dynamic programming equations, 1. Poisson's equation 2. h(x), ( ) 1 1 < 0, x R a 3. q a h(x), ( ) 1 < 0, x R p 0 q p