Optimization and the Price of Anarchy in a Dynamic Newsboy Model

Optimization and the Price of Anarchy in a Dynamic Newsboy Model Sean Meyn Prices Normalized demand Reserve Department of ECE and the Coordinated Science Laboratory University of Illinois Joint work with M. Chen, S. Neumayer, and I-K. Cho NSF support: ECS 02-17836, ITR 00-85929 and CCF 00-49089

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

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

HOME Meeting Calendar OASIS Employment Site Map Contact Us Search Available Resources The current forecast of generating and import resources available to serve the demand for energy within the California ISO service area Forecast Demand Forecast of the demand expected today. The procurement of energy resources for the day is based on this forecast Actual Demand Today's actual system demand with historical trend Revised Demand Forecast The current forecast of the system demand expected throughout the remainder of the day. This forecast is updated hourly.

HOME Meeting Calendar OASIS Employment Site Map Contact Us Search http://www.caiso.com 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%)

One Hot Week in Urbana... 5000 Purchase Price $/MWh 4000 3000 2000 1000 0 1 5 9 13 17 21 1 5 9 13 17 21 1 5 9 13 17 21 1 5 9 13 17 21 1 5 9 13 17 21 Monday Tuesday Wednesday Thursday Friday

Southern California, July 8-15, 1998... Spinning Reserve Prices PX Prices 250 200 150 100 50 70 60 50 40 30 20 10 0 Weds Thurs Fri Sat Sun Mon Tues Weds

APX Europe, June 2003 400 Prices (Eur/MWh) Week 25 Week 26 350 300 250 200 150 100 50 0 Mon Tues Weds Thurs Fri Sat Sun

APX Europe, October 2005 4000 3000 2000 1000 0 Volume (MWh) Previous week Current week (10/24/05) 800 600 400 200 0 Price (Euro) vr za zo ma di wo do

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

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

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

16 3 I Centralized Control

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

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

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

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

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 +

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

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

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

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 +

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

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))].

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

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

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 =14.978 q 2 =2.996 ζ(k), ζ a (k) allocation increments.

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 =14.978 q 2 =2.996 Average cost: CRW Average cost: Diffusion model 24 24 22 22 20 20 18 18 18 17 5 16 15 q2 q 1 q 2 q 2 q 1 q 2 3 7

II Relaxations

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?

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

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

Effective cost c(x A,d) Line 1 Line 2 Line 3 g a A 50 40 1 x A 30 20 2 x A X + R What do these aggregate states say about the network? 10 3 4 x A x A q A 0-50 - 40-30 - 20-10 0 10 20 30 40 50

Effective cost c(x A,d) Line 1 Line 2 Line 3 50 40 30 20 g a A 1 x A City 1 in blackout: q 1 = 46,, q 2 =3.0564,, q 3 =12.9436 Insufficient primary generation: g 1 = 71,, g2 a =33.0564,, g3 a =6.9436 Transmission constraints binding: f 1 = 13,, f 2 = 5, f 3 = 8 10 q A 0-50 - 40-30 - 20-10 0 10 20 30 40 50

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

III Decentralized Control

www.apx.nl 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 1999. 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) 400 350 300 250 200 150 100 50 0 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

What is the value of rope? Available at Menards: $.99/yard

What is the value of rope? Available at Menards: $.99/yard Would you pay $9,999 / yard?

What is the value of rope? The value of rope is a dynamic quantity Would you pay $9,999 / yard?

Second Welfare Theorem Is there an equilibrium price functional? Is the equilibrium efficient?? 5000 4000 3000 2000 1000 1 5 9 13 17 21 1 5 9 13 17 21 1 5 9 13 17 21 1 5 9 13 17 21 1 5 9 13 17 21

Second Welfare Theorem Is there an equilibrium price functional? Is the equilibrium efficient?? 5000 4000 Yes to all! 3000 p p 2000 (q, d) =p a k (q, d) =(c bo +v 1000 ) max(d, 0) max( q, 0) d + q 1 5 9 13 17 21 c v 1 bo 5 cost of insufficient service 9 13 17 21 1 5 9 13 17 21 1 5 9 13 17 21 reserve 1 5 9 13 17 Q(t) =q value of consumption demand D(t) =d 21

Efficient Equilibrium Prices Normalized demand Reserve 10000 8000 6000 4000 2000 0

One Hot Week in Urbana... 5000 Purchase Price $/MWh 4000 3000 2000 1000 0 1 5 9 13 17 21 1 5 9 13 17 21 1 5 9 13 17 21 1 5 9 13 17 21 1 5 9 13 17 21 Monday Tuesday Wednesday Thursday Friday

Conclusions 10000 8000 6000 4000 2000 0 The hedging point (affine) policy is average cost optimal Amazing solidarity between CRW and CBM models Deregulation is a disaster! Future work?

Extensions and future work 10000 8000 6000 4000 2000 Complex models: Workload or aggregate relaxations 0 Price caps: No! Responsive demand: Yes! Is ENRON off the hook: Ha!

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. Submitted 2005. M. Chen, C. Pandit, and S. Meyn. In search of sensitivity in network optimization. Queueing Systems, 2003 I.-K. Cho and S. Meyn. Dynamics of ancillary service prices in power distribution networks. CDC, 2003 M. Chen, R. Dubrawski, and S. Meyn. Management of demand driven production systems. IEEE TAC, 2004

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

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

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

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

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

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)

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