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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 02-17836 & 05-23620 Control Techniques for Complex Networks
DOE Support: http://www.sc.doe.gov/grants/FAPN08-13.html
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
OREGON

 California’s 25,000
 Mile Electron Highway                            COAL
                                                            LEGEND


                                                  GEOTHERMAL
                                                  HYDROELECTRIC
                                                  NUCLEAR
                                                  OIL/GAS
                                                  BIOMASS
                                                  MUNICIPAL SOLID
                                                  WASTE (MSW)
                                                  SOLAR
                                                  WIND AREAS



                             SAN
                             FRANCISCO




What is the value
of improved transmission?                                             NEVADA



More responsive ancillary service?

How does a centralized planner
optimize capacity?
                                                                     MEXICO


Is there an efficient decentralized solution?
Meeting Calendar   |   OASIS   |   Employment   |   Site Map   |   Contact Us

                                     HOME   |                                       Search
                                                                                                                                http://www.caiso.com/outlook/outlook.html
                                                                                                                                September 28, 2008
                                                               40,000 MW
                                                                                                                                                                           4,000
                                                                                                                                                                           Megawatts



Available Resources
The current forecast of generating and import
resources available to serve the demand for energy 30,000 MW
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                                      20,000 MW                     Day Ahead Demand Forecast           Revised Demand Forecast   Actual Demand      Hour Beginning
                                                                                                Available Resources Forecast



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.
Meeting Calendar   |   OASIS   |   Employment   |   Site Map   |   Contact Us

                  HOME   |                                       Search                                      http://www.caiso.com



               Emergency Notices
Generating
 Reserves
7.0%           Stage 1
              Stage 1                   Generating reserves less than requirements
             Emergency
             Emergency                  (Continuously recalculated. Between 6.0% & 7.0%)

6.0%

5.0%
               Stage 2
              Stage 2                   Generating reserves less than 5.0%
4.0%         Emergency
             Emergency
3.0%

               Stage 3
              Stage 3                   Generating reserves less than largest contingency
2.0%
             Emergency
             Emergency                  (Continuously recalculated. Between 1.5% & 3.0%)
1.0%

0.0%
One Hot Week in Urbana ...



                          Purchase Price $/MWh
 5000


 4000


 3000


 2000


 1000


   0
                    13
                         17
                              21




                                               13
                                                    17
                                                         21




                                                                          13
                                                                               17
                                                                                    21




                                                                                                     13
                                                                                                          17
                                                                                                               21




                                                                                                                                 13
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                                                                                                                                           21
        1
            5
                9




                                   1
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                                           9




                                                              1
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                                                                                         1
                                                                                             5
                                                                                                 9




                                                                                                                    1
                                                                                                                        5
                                                                                                                            9
                Monday                     Tuesday                    Wednesday                  Thursday                       Friday
Southern California, July 8-15, 1998 ...


             Spinning Reserve Prices                        PX Prices
                                                                 70
250
                                                                 60
200
                                                                 50

150                                                              40

                                                                 30
100
                                                                 20
50
                                                                 10

 0
      Weds    Thurs     Fri      Sat   Sun   Mon   Tues   Weds
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".
APX Europe, June 2003



      Prices (Eur/MWh)                               Week 25
400                                                  Week 26
350

300

250

200

150

100

 50

 0
       Mon         Tues   Weds   Thurs   Fri   Sat         Sun
APX Europe, October 2005


4000

3000                                                 Previous week
2000                                                 Current week (10/24/05)
1000
                                 Volume (MWh)
   0


 800
                                 Price (Euro)
 600

 400

 200

   0
        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
Australia January 16 2007



                            Victoria




                                       0




                            Tasmania
Australia January 16 2007

                        Victoria
 Volume (MWh)



                8,000


                6,000
                                   + 10,000




                                             Price (Aus $/MWh)
                4,000




                                   0
                                   - 500
                1,400
 Volume (MWh)




                        Tasmania
                1,200

                1,000




                                             Price (Aus $/MWh)
                                   + 500
                 800
                                   0

                                   - 500

                                   - 1,000
16
                  3




         I
Centralized Control
Dynamic model
                               On-line capacity
                               Forecast
Reserve options for services   Actual demand
based on forecast statistics   Revised forecast




Centered demand:

D(t) = demand - forecast




                                                  t
Dynamic Single-Commodity Model

Stochastic model: G Goods available at time t
                  D Normalized demand

Excess/shortfall:   Q(t) = G(t) − D(t)


Normalized cost as a function of Q:


     c−
                     Excess production
                                         c+
 Shortfall
                                                q

                                                    Scarf, Arrow, Bellman, ...
Dynamic Single-Commodity Model

Stochastic model: G Goods available at time t
                  D Normalized demand

Excess/shortfall:    Q(t) = G(t) − D(t)


Generation is rate-constrained:
                          -ζ-

 q

                                                        Q(t)

                     ζ+

                                                           t
                                            High cost
Dynamic Single-Commodity Model

Average cost: pQ density when q ∗ = 0
                              ¯

       E[c(Q)] =   c( q − ¯ ) pQ (dq)
                          q

                                                 q∗
                                                 ¯
             c−                  pQ (q − q ∗ )
                                         ¯

                                                      c+
                                                           q



Optimal hedging-point:    q ∗ solves
                          ¯

         − c− P{ Q ≤ 0} + c+P{ Q ≥ 0} = 0
Dynamic Single-Commodity Model

Average cost: pQ density when q ∗ = 0
                              ¯                   RBM model:

       E[c(Q)] =   c( q − ¯ ) pQ (dq)
                          q                           pQ exponential

                                                 q∗
                                                 ¯
             c−
                                 pQ (q − q ∗ )
                                         ¯

                                                            c+
                                                                  q



Optimal hedging-point:    q ∗ solves
                          ¯
                                                    ∗ = 1 σ2      c−
                                                  q
                                                  ¯     2 ζ + log c+
         − c−P{ Q ≤ 0} + c+P{ Q ≥ 0} = 0
                                                                  Wein '92
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:
                d a                                   d
      −ζ a− ≤      G (t) ≤ ζ a+             −ζ − ≤      G(t) ≤ ζ +
                dt                                   dt
Ancillary service
                             a        G(t )
                            G (t )
                                                                K



Policy: hedging policy with multiple thresholds

       Q(t) = G(t) + G a(t) − D (t)            -ζ-   Downward trend: Ga(t) = 0
                                                                    d
                                                                      G (t) = - ζ -
                                                                   dt
  q



                                              ζ+

  q
                                              ζ ++ ζ a+                               t
                                                                       Blackout
Q(t)
Diffusion model & control                X(t) =
                                                   G a(t)


Relaxations: instantaneous ramp-down rates:

                d                      d a
            −∞ ≤ G (t) ≤ ζ +,      −∞ ≤ G (t) ≤ ζ a+.
                dt                     dt

Cost structure:

           c(X(t)) = c1G (t) + c2G a(t) + c3|Q(t)|1{Q(t) < 0}



Control: design hedging points to minimize average-cost,

                    min Eπ [c(Q(t))] .
Diffusion model & control

 Markov model:            Hedging-point policy:

         Q(t)
X(t) =
         G a(t)



                          Ancillary service
                          is ramped-up when
                          excess capacity falls
                          below q2
                                 ¯




                     q2
                     ¯                  q1
                                        ¯
Markov model & control

 Markov model:             Hedging-point policy:

         Q(t)              Optimal parameters:
X(t) =                                                   (proof)
         G a(t)

                             ∗ − q ∗ = 1 log c2
                            q1 ¯2
                            ¯
                                       γ1    c1

                                 ∗ = 1 log c3
                                q2
                                ¯
                                     γ0    c2

                                       ζ + +ζ a+           ζ+
                                γ0 = 2           ,   γ1 = 2 2 .
                                           σD
                                            2              σD


                   q2
                   ¯                       q1
                                           ¯
Simulation

Discrete Markov model:                 Optimal hedging-points
                                       for RBM:
Q(k + 1) − Q(k)
                                             q1 − q2 = 14.978
                                             ¯    ¯
          = ζ(k) + ζ a(k) + E(k + 1)
                                                 q2 = 2.996
                                                 ¯
E(k) i.i.d. Bernoulli.

ζ(k), ζ a(k) allocation increments.
Simulation

Discrete Markov model:                            Optimal hedging-points
                                                  for RBM:
Q(k + 1) − Q(k)
                                                             q1 − q2 = 14.978
                                                             ¯    ¯
                = ζ(k) + ζ a(k) + E(k + 1)
                                                                     ¯
                                                                     q2 = 2.996

      Average cost: CRW                           Average cost: Diffusion model


 24                                          24


 22                                          22


 20                                          20


 18                                          18




                                                                                  7
        16                                        ¯    ¯
                                                  q1 − q2
      ¯    ¯
      q1 − q2               3   ¯
                                q2                                          ¯
                                                                            q2
II
Relaxations



              (skip to market)
Texas model

                D1            E1             Gp
                                              1




                     Line 1
           Ga
            2
                                            Line 2   Ga
                                                      3




  D2                                                      D3
                                   Line 3
           E2                                        E3


  Resource pooling from San Antonio to Houston?
Aggregate model
                                                    Line 1            Line 2




                                                             Line 3




QA(t) = extraction - demand   =     Ei(t) − Di(t)

 a                                    a
GA(t) = aggregate ancillary   =     Gi (t)

Assume Brownian demand, rate constraints as before

Provided there are no transmission constraints,



             a
  XA = (QA ,GA ) ≡ single producer/consumer model
Effective cost ¯(xA, d)
               c

                                                    Line 1            Line 2




 Given demand and aggregate state                            Line 3


 find the cheapest consistent
 network configuration subject to transmission constraints
            p p               −
min       (ci gi + cagi + cboqi )
                    i
                      a
                           i

s.t.      qA =      (ei − di)         consistency
           a
          gA =       a
                    gi
                      p    a
           0 =      (gi + gi − ei )   extraction = generation
           q = e−d                    vector reserves
           f = ∆p                     power flow equations
            f ∈F                      transmission constraints
Effective cost ¯(xA, d)
                      c

                                                                                         Line 1            Line 2




                                                                                                  Line 3



        a
       gA

50




40                        xA
                           1



                                                                               What do these
30

                          xA
                           2                 +                                 aggregate states say
                                             X        R                        about the network?
20




10                        xA
                           3
                                                                xA
                                                                 4



                                                                               qA
0
- 50        - 40   - 30        - 20   - 10   0   10   20   30        40   50
Effective cost ¯(xA, d)
                      c

                                                                                                             Line 1            Line 2




                                                                                                                      Line 3



        a
       gA

50
                                                           City 1 in blackout:
                                                            q1 = −46,
                                                                  46,              q2 = 3.0564,
                                                                                        3.0564,        q3 = 12.9436
                                                                                                            12
40                        xA
                           1

                                                           Insufficient primary generation:
                                                                                    a                   a
                                                            g1 = −71,              g2 = 33.0564,
                                                                                        33 0564,       g3 = 6.9436
                                                                                                            6.
30

                                                           Transmission constraints binding:
20                                                          f1 =        13,
                                                                        13,        f2 = −5,       f 3 = −8


10


                                                                                       qA
0
- 50        - 40   - 30        - 20   - 10   0   10   20   30      40         50
Inventory model




                                                         demand 1




demand 2




    Controlled work-release, controlled routing,
                                           uncertain demand.
Inventory model:
Workload relaxation


 w2
      Resource 1                +
            is idle         W

                                      −(1 − ρ)


                                                 Asymptotes:
                                                        1 σ2      c−
                      Resource 2                  q ∗ = 2 ζ + log c+
                                                  ¯
                            is idle
                                         w1
III
Decentralized Control
Supply & Demand


Cost of generation depends on source

     Price

                                                                        Supply Curve


                                                Gas Turbine ($20-$30)

                              Coal ($10 -$15)

               Nuclear ($6)


                                                                           Quantity
Supply & Demand


Demand for power is not flexible

               High Priority Customers $5,000/MW?
     Price                                          Customers with interruptible services




                                                                                            Demand Curve

                                                                                               Quantity
Supply & Demand


Equilibrium Price & Quantity: Intersection of supply & demand curves

                 High Priority Customers
     Price                                           Customers with interruptible services


                                                                                             Supply Curve
                       Consumer Surplus
   Equilibrium
   Price
   $20 - $30           Supplier Surplus                      Gas Turbine ($20-$30)

                                   Coal ($10 -$15)

                 Nuclear ($6)
                                                                                             Demand Curve

                                                                     Equilibrium
                                                                     Quantity 45,000MW
                                                                                                Quantity
Supply & Demand


Equilibrium Price & Quantity: Intersection of supply & demand curves

                      High Priority Customers
     Price
   5000                                                             Customers with interruptible services

 Where do ramp constraints appear?
 4000                                               Supply Curve
 Variability? Surplus
 Equilibrium
             Consumer

 Where is hedging?
 3000
 Price
 $20 - $30   Supplier Surplus Gas Turbine ($20-$30)
   2000
                                             Coal ($10 -$15)
   1000
                      Nuclear ($6)
                                                                                                                          Demand Curve

                                                                                           Equilibrium                        Quantity
                                                 13
                                                      17
                                                           21




                                                                            13
                                                                                 17
                                                                                      21




                                                                                                       13
                                                                                                            17
                                                                                                                 21




                                                                                                                                  13
                                                                                                                                       17
                                                                                                                                            21
                      13
                           17
                                21




                                                                                                                      1
                                                                                                                          5
                                                                                                                              9
          1
              5
                  9




                                     1
                                         5
                                             9




                                                                1
                                                                    5
                                                                        9




                                                                                           1
                                                                                               5
                                                                                                   9
                                                                                           Quantity 45,000MW
www.apx.nl

          Main Page                      Market Results




Welcome to APX!
APX is the first electronic energy trading                Prices (Eur/MWh)
                                                           400
platform in continental Europe. The daily spot             350
                                                                                                        Week 25
market has been operational since May 1999.                300
                                                                                                        Week 26
                                                           250
The spot market enables distributors,                      200
producers, traders, brokers and industrial                 150
end-users to buy and sell electricity on a                 100
                                                            50
day-ahead basis.                                             0

                                                          Volumes (MWh)
The APX-index will be published daily around          2500 MWh

12h00 (GMT +01:00) to provide transparency            2000 MWh
in the market. Prices can be used as a
                                                      1500 MWh
benchmark.
                                                      1000 MWh

                                                          500 MWh
                                                                    Mon   Tue   Wed   Thu   Fri   Sat    Sun
Second Welfare Theorem


Each player independently optimizes ...

Consumer: value of consumption
  5000
             minus prices paid
  4000
             minus disaster
  WD (t) := v min D(t),G p (t) + Ga (t) − pp Gp (t) + pa Ga (t) + cbo Q− (t)
   3000


   2000
Supplier:     profits from two sources of generation
   1000
  WS (t) := pp − cp Gp (t) + pa − ca )Ga (t)
                                                                            13
                                                                                 17
                                                                                      21




                                                                                                       13
                                                                                                            17
                                                                                                                 21




                                                                                                                                  13
                                                                                                                                       17
                                                                                                                                            21
                                                 13
                                                      17
                                                           21
                      13
                           17
                                21




                                                                                           1
                                                                                               5
                                                                                                   9




                                                                                                                      1
                                                                                                                          5
                                                                                                                              9
                                                                1
                                                                    5
                                                                        9
          1
              5
                  9




                                     1
                                         5
                                             9
Second Welfare Theorem


Is there an equilibrium price functional?

Is the equilibrium efficient??
  5000


  4000


  3000


  2000


  1000
                     13
                          17
                               21




                                                13
                                                     17
                                                          21




                                                                           13
                                                                                17
                                                                                     21




                                                                                                      13
                                                                                                           17
                                                                                                                21




                                                                                                                                 13
                                                                                                                                      17
                                                                                                                                           21
         1
             5
                 9




                                    1
                                        5
                                            9




                                                               1
                                                                   5
                                                                       9




                                                                                          1
                                                                                              5
                                                                                                  9




                                                                                                                     1
                                                                                                                         5
                                                                                                                             9
Second Welfare Theorem


Is there an equilibrium price functional?

Is the equilibrium efficient??
  5000


  4000
                           Yes to all !

  3000


  2000   p (r , d) = (v + c )I{r < 0}
                 e         e                                                         bo                         e

  1000
                     13
                          17
                               21




                                                13
                                                     17
                                                          21




                                                                           13
                                                                                17
                                                                                     21




                                                                                                      13
                                                                                                           17
                                                                                                                21




                                                                                                                                 13
                                                                                                                                      17
                                                                                                                                           21
         1
             5
                 9




                                    1
                                        5
                                            9




                                                               1
                                                                   5
                                                                       9




                                                                                          1
                                                                                              5
                                                                                                  9




                                                                                                                     1
                                                                                                                         5
                                                                                                                             9
                                    cbo cost of insufficient service                                                 reserve               Q(t) = q
                                    v value of consumption                                                           demand D(t) = d
Efficient Equilibrium
                        Prices
                        Normalized demand
                        Reserve


 10000


  8000


  6000


  4000


  2000


     0
Southern California, July 8-15, 1998 ...


             Spinning Reserve Prices                        PX Prices
                                                                 70
250
                                                                 60
200
                                                                 50

150                                                              40

                                                                 30
100
                                                                 20
50
                                                                 10

 0
      Weds    Thurs     Fri      Sat   Sun   Mon   Tues   Weds
Conclusions
                                                       Spinning Reserve Prices                        PX Prices
                                                                                                           70
                                          250
                                                                                                           60
                                          200
                                                                                                           50

                                          150                                                              40

                                                                                                           30
                                          100
                                                                                                           20
                                          50
                                                                                                           10

                                           0
                                                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?
Extensions and future work        10000


                                   8000


                                   6000


                                   4000


                                   2000


                                      0




 Complex models:
            Workload or aggregate relaxations


 Price caps: No!

 Responsive demand: Yes!

 Is ENRON off the hook: ?
Extensions and future work                10000


                                           8000


                                           6000


                                           4000


                                           2000


                                              0




 Complex models:
            Workload or aggregate relaxations


 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
Epilogue                           10000


                                    8000


                                    6000


                                    4000


                                    2000


                                       0




 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
Epilogue                           10000


                                    8000


                                    6000


                                    4000


                                    2000


                                       0




 Justification:   1. Economic systems are complex
                  2. Regulators cannot be trusted
Epilogue                             10000


                                      8000


                                      6000


                                      4000


                                      2000


                                         0




 Justification:   1. Economic systems are complex
                  2. Regulators cannot be trusted
 Airplanes, highway networks, cell phones... all complex
Epilogue                             10000


                                      8000


                                      6000


                                      4000


                                      2000


                                         0




 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.
References
                                                                q1 − q2
                                                                ¯    ¯    16

                                                                                     3   q2
                                                                                         ¯
                                                                          1     c
                                                              ¯∗ ¯∗
                                                              q1 − q2 =      log 2
                                                                          γ1    c1
                                                                          1     c
                                                                   ¯∗
                                                                   q2 =      log 3
                                                                          γ0    c2




•  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. De-
cember 2003

•  I.-K. Cho and S. P. Meyn. Efficiency and marginal cost pricing in dy-
namic 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 disserta-
tion, ECE UIUC 2008
Poisson’s Equation


First reflection times,

  τp :=inf{t ≥ 0 : Q(t) = q p },
                          ¯           τa :=inf{t ≥ 0 : Q(t) ≥ q a }
                                                              ¯

                              τp
           h(x) = Ex               c(X(s)) − φ ds
                          0




                                                                      return
Poisson’s Equation


First reflection times,

  τp :=inf{t ≥ 0 : Q(t) = q p },
                          ¯                 τa :=inf{t ≥ 0 : Q(t) ≥ q a }
                                                                    ¯

                              τp
           h(x) = Ex               c(X(s)) − φ ds
                          0


Solves martingale problem,

                                       t
       M (t) = h(X(t)) +                   c(X(s)) − φ ds
                                   0
Poisson’s Equation

         Q(t)
X(t) =
         Ga(t)
                      h(X(τa ))




                                            τa
                 h(x) = Ex h(X(τa ))+            c(X(s))−φ ds
                                        0




                      qa
                      ¯                          qp
                                                 ¯
Derivative Representations



                               1
                      ∇h(x),   1   = 0,        x = X(τa )



                                              τa
                h(x) = Ex h(X(τa ))+               c(X(s))−φ ds
                                          0




                    qa
                    ¯                              qp
                                                   ¯
Derivative Representations                                  1
                                   λa (x) = ∇c(x),          1

                                              = ca − I{q ≤ 0}cbo

                               1
                      ∇h(x),   1   = 0,        x = X(τa )



                                              τa
                h(x) = Ex h(X(τa ))+               c(X(s))−φ ds
                                          0




                    qa
                    ¯                              qp
                                                   ¯
Derivative Representations


                           τa
           1
  ∇h(x),   1   = Ex             λa (X(t)) dt
                       0
                                             τa
               = ca E[τa ] − cbo Ex               I{Q(t) ≤ 0} dt
                                         0



 Computable based
 on one-dimensional height (ladder) process,

                H a (t) = q a − Q(t)
                          ¯
Dynamic Programming Equations


If q p = q p∗ and ¯ a = ¯ a∗
   ¯     ¯        q     q

Then h solves the dynamic programming equations,



1. Poisson's equation
              1
2. ∇h(x),     1   < 0,    x ∈ Ra



                                   ¯ a∗
                                   q                           q p∗
                                           1
                               3. ∇h(x),   0   < 0,   x ∈ Rp          return

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Efficiency and Marginal Cost Pricing in Dynamic Competitive Markets with Friction

  • 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 02-17836 & 05-23620 Control Techniques for Complex Networks DOE Support: http://www.sc.doe.gov/grants/FAPN08-13.html 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. OREGON California’s 25,000 Mile Electron Highway COAL LEGEND GEOTHERMAL HYDROELECTRIC NUCLEAR OIL/GAS BIOMASS MUNICIPAL SOLID WASTE (MSW) SOLAR WIND AREAS SAN FRANCISCO What is the value of improved transmission? NEVADA More responsive ancillary service? How does a centralized planner optimize capacity? MEXICO Is there an efficient decentralized solution?
  • 3. Meeting Calendar | OASIS | Employment | Site Map | Contact Us HOME | Search http://www.caiso.com/outlook/outlook.html September 28, 2008 40,000 MW 4,000 Megawatts Available Resources The current forecast of generating and import resources available to serve the demand for energy 30,000 MW 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 20,000 MW Day Ahead Demand Forecast Revised Demand Forecast Actual Demand Hour Beginning Available Resources Forecast 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. Meeting Calendar | OASIS | Employment | Site Map | Contact Us HOME | Search http://www.caiso.com Emergency Notices Generating Reserves 7.0% Stage 1 Stage 1 Generating reserves less than requirements Emergency Emergency (Continuously recalculated. Between 6.0% & 7.0%) 6.0% 5.0% Stage 2 Stage 2 Generating reserves less than 5.0% 4.0% Emergency Emergency 3.0% Stage 3 Stage 3 Generating reserves less than largest contingency 2.0% Emergency Emergency (Continuously recalculated. Between 1.5% & 3.0%) 1.0% 0.0%
  • 5. One Hot Week in Urbana ... Purchase Price $/MWh 5000 4000 3000 2000 1000 0 13 17 21 13 17 21 13 17 21 13 17 21 13 17 21 1 5 9 1 5 9 1 5 9 1 5 9 1 5 9 Monday Tuesday Wednesday Thursday Friday
  • 6. Southern California, July 8-15, 1998 ... Spinning Reserve Prices PX Prices 70 250 60 200 50 150 40 30 100 20 50 10 0 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 2003 Prices (Eur/MWh) Week 25 400 Week 26 350 300 250 200 150 100 50 0 Mon Tues Weds Thurs Fri Sat Sun
  • 9. APX Europe, October 2005 4000 3000 Previous week 2000 Current week (10/24/05) 1000 Volume (MWh) 0 800 Price (Euro) 600 400 200 0 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 16 2007 Victoria 0 Tasmania
  • 14. Australia January 16 2007 Victoria Volume (MWh) 8,000 6,000 + 10,000 Price (Aus $/MWh) 4,000 0 - 500 1,400 Volume (MWh) Tasmania 1,200 1,000 Price (Aus $/MWh) + 500 800 0 - 500 - 1,000
  • 15. 16 3 I Centralized Control
  • 16. Dynamic model On-line capacity Forecast Reserve options for services Actual demand based on forecast statistics Revised forecast Centered demand: D(t) = demand - forecast t
  • 17. Dynamic Single-Commodity Model Stochastic model: G Goods available at time t D Normalized demand Excess/shortfall: Q(t) = G(t) − D(t) Normalized cost as a function of Q: c− Excess production c+ Shortfall q Scarf, Arrow, Bellman, ...
  • 18. Dynamic Single-Commodity Model Stochastic model: G Goods available at time t D Normalized demand Excess/shortfall: Q(t) = G(t) − D(t) Generation is rate-constrained: -ζ- q Q(t) ζ+ t High cost
  • 19. Dynamic Single-Commodity Model Average cost: pQ density when q ∗ = 0 ¯ E[c(Q)] = c( q − ¯ ) pQ (dq) q q∗ ¯ c− pQ (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: pQ density when q ∗ = 0 ¯ RBM model: E[c(Q)] = c( q − ¯ ) pQ (dq) q pQ exponential q∗ ¯ c− pQ (q − q ∗ ) ¯ c+ q Optimal hedging-point: q ∗ solves ¯ ∗ = 1 σ2 c− q ¯ 2 ζ + log c+ − c−P{ Q ≤ 0} + c+P{ Q ≥ 0} = 0 Wein '92
  • 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: d a d −ζ a− ≤ G (t) ≤ ζ a+ −ζ − ≤ G(t) ≤ ζ + dt dt
  • 25. Ancillary service a G(t ) G (t ) K Policy: hedging policy with multiple thresholds Q(t) = G(t) + G a(t) − D (t) -ζ- Downward trend: Ga(t) = 0 d G (t) = - ζ - dt q ζ+ q ζ ++ ζ a+ t Blackout
  • 26. Q(t) Diffusion model & control X(t) = G a(t) Relaxations: instantaneous ramp-down rates: d d a −∞ ≤ G (t) ≤ ζ +, −∞ ≤ G (t) ≤ ζ a+. dt dt Cost structure: c(X(t)) = c1G (t) + c2G a(t) + c3|Q(t)|1{Q(t) < 0} Control: design hedging points to minimize average-cost, min Eπ [c(Q(t))] .
  • 27. Diffusion model & control Markov model: Hedging-point policy: Q(t) X(t) = G a(t) Ancillary service is ramped-up when excess capacity falls below q2 ¯ q2 ¯ q1 ¯
  • 28. Markov model & control Markov model: Hedging-point policy: Q(t) Optimal parameters: X(t) = (proof) G a(t) ∗ − q ∗ = 1 log c2 q1 ¯2 ¯ γ1 c1 ∗ = 1 log c3 q2 ¯ γ0 c2 ζ + +ζ a+ ζ+ γ0 = 2 , γ1 = 2 2 . σD 2 σD q2 ¯ q1 ¯
  • 29. Simulation Discrete Markov model: Optimal hedging-points for RBM: Q(k + 1) − Q(k) q1 − q2 = 14.978 ¯ ¯ = ζ(k) + ζ a(k) + E(k + 1) q2 = 2.996 ¯ E(k) i.i.d. Bernoulli. ζ(k), ζ a(k) allocation increments.
  • 30. Simulation Discrete Markov model: Optimal hedging-points for RBM: Q(k + 1) − Q(k) q1 − q2 = 14.978 ¯ ¯ = ζ(k) + ζ a(k) + E(k + 1) ¯ q2 = 2.996 Average cost: CRW Average cost: Diffusion model 24 24 22 22 20 20 18 18 7 16 ¯ ¯ q1 − q2 ¯ ¯ q1 − q2 3 ¯ q2 ¯ q2
  • 31. II Relaxations (skip to market)
  • 32. Texas model D1 E1 Gp 1 Line 1 Ga 2 Line 2 Ga 3 D2 D3 Line 3 E2 E3 Resource pooling from San Antonio to Houston?
  • 33. Aggregate model Line 1 Line 2 Line 3 QA(t) = extraction - demand = Ei(t) − Di(t) a a GA(t) = aggregate ancillary = Gi (t) Assume Brownian demand, rate constraints as before Provided there are no transmission constraints, a XA = (QA ,GA ) ≡ single producer/consumer model
  • 34. Effective cost ¯(xA, d) c Line 1 Line 2 Given demand and aggregate state Line 3 find the cheapest consistent network configuration subject to transmission constraints p p − min (ci gi + cagi + cboqi ) i a i s.t. qA = (ei − di) consistency a gA = a gi p a 0 = (gi + gi − ei ) extraction = generation q = e−d vector reserves f = ∆p power flow equations f ∈F transmission constraints
  • 35. Effective cost ¯(xA, d) c Line 1 Line 2 Line 3 a gA 50 40 xA 1 What do these 30 xA 2 + aggregate states say X R about the network? 20 10 xA 3 xA 4 qA 0 - 50 - 40 - 30 - 20 - 10 0 10 20 30 40 50
  • 36. Effective cost ¯(xA, d) c Line 1 Line 2 Line 3 a gA 50 City 1 in blackout: q1 = −46, 46, q2 = 3.0564, 3.0564, q3 = 12.9436 12 40 xA 1 Insufficient primary generation: a a g1 = −71, g2 = 33.0564, 33 0564, g3 = 6.9436 6. 30 Transmission constraints binding: 20 f1 = 13, 13, f2 = −5, f 3 = −8 10 qA 0 - 50 - 40 - 30 - 20 - 10 0 10 20 30 40 50
  • 37. Inventory model demand 1 demand 2 Controlled work-release, controlled routing, uncertain demand.
  • 38. Inventory model: Workload relaxation w2 Resource 1 + is idle W −(1 − ρ) Asymptotes: 1 σ2 c− Resource 2 q ∗ = 2 ζ + log c+ ¯ is idle w1
  • 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 High Priority Customers $5,000/MW? Price Customers with interruptible services Demand Curve Quantity
  • 42. Supply & Demand Equilibrium Price & Quantity: Intersection of supply & demand curves High Priority Customers Price Customers with interruptible services Supply Curve Consumer Surplus Equilibrium Price $20 - $30 Supplier Surplus Gas Turbine ($20-$30) Coal ($10 -$15) Nuclear ($6) Demand Curve Equilibrium Quantity 45,000MW Quantity
  • 43. Supply & Demand Equilibrium Price & Quantity: Intersection of supply & demand curves High Priority Customers Price 5000 Customers with interruptible services Where do ramp constraints appear? 4000 Supply Curve Variability? Surplus Equilibrium Consumer Where is hedging? 3000 Price $20 - $30 Supplier Surplus Gas Turbine ($20-$30) 2000 Coal ($10 -$15) 1000 Nuclear ($6) Demand Curve Equilibrium Quantity 13 17 21 13 17 21 13 17 21 13 17 21 13 17 21 1 5 9 1 5 9 1 5 9 1 5 9 1 5 9 Quantity 45,000MW
  • 44. www.apx.nl Main Page Market Results Welcome to APX! APX is the first electronic energy trading Prices (Eur/MWh) 400 platform in continental Europe. The daily spot 350 Week 25 market has been operational since May 1999. 300 Week 26 250 The spot market enables distributors, 200 producers, traders, brokers and industrial 150 end-users to buy and sell electricity on a 100 50 day-ahead basis. 0 Volumes (MWh) The APX-index will be published daily around 2500 MWh 12h00 (GMT +01:00) to provide transparency 2000 MWh in the market. Prices can be used as a 1500 MWh benchmark. 1000 MWh 500 MWh Mon Tue Wed Thu Fri Sat Sun
  • 45. Second Welfare Theorem Each player independently optimizes ... Consumer: value of consumption 5000 minus prices paid 4000 minus disaster WD (t) := v min D(t),G p (t) + Ga (t) − pp Gp (t) + pa Ga (t) + cbo Q− (t) 3000 2000 Supplier: profits from two sources of generation 1000 WS (t) := pp − cp Gp (t) + pa − ca )Ga (t) 13 17 21 13 17 21 13 17 21 13 17 21 13 17 21 1 5 9 1 5 9 1 5 9 1 5 9 1 5 9
  • 46. Second Welfare Theorem Is there an equilibrium price functional? Is the equilibrium efficient?? 5000 4000 3000 2000 1000 13 17 21 13 17 21 13 17 21 13 17 21 13 17 21 1 5 9 1 5 9 1 5 9 1 5 9 1 5 9
  • 47. Second Welfare Theorem Is there an equilibrium price functional? Is the equilibrium efficient?? 5000 4000 Yes to all ! 3000 2000 p (r , d) = (v + c )I{r < 0} e e bo e 1000 13 17 21 13 17 21 13 17 21 13 17 21 13 17 21 1 5 9 1 5 9 1 5 9 1 5 9 1 5 9 cbo cost of insufficient service reserve Q(t) = q v value of consumption demand D(t) = d
  • 48. Efficient Equilibrium Prices Normalized demand Reserve 10000 8000 6000 4000 2000 0
  • 49. Southern California, July 8-15, 1998 ... Spinning Reserve Prices PX Prices 70 250 60 200 50 150 40 30 100 20 50 10 0 Weds Thurs Fri Sat Sun Mon Tues Weds
  • 50. Conclusions Spinning Reserve Prices PX Prices 70 250 60 200 50 150 40 30 100 20 50 10 0 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 10000 8000 6000 4000 2000 0 Complex models: Workload or aggregate relaxations Price caps: No! Responsive demand: Yes! Is ENRON off the hook: ?
  • 52. Extensions and future work 10000 8000 6000 4000 2000 0 Complex models: Workload or aggregate relaxations 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 10000 8000 6000 4000 2000 0 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 10000 8000 6000 4000 2000 0 Justification: 1. Economic systems are complex 2. Regulators cannot be trusted
  • 55. Epilogue 10000 8000 6000 4000 2000 0 Justification: 1. Economic systems are complex 2. Regulators cannot be trusted Airplanes, highway networks, cell phones... all complex
  • 56. Epilogue 10000 8000 6000 4000 2000 0 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 q1 − q2 ¯ ¯ 16 3 q2 ¯ 1 c ¯∗ ¯∗ q1 − q2 = log 2 γ1 c1 1 c ¯∗ q2 = log 3 γ0 c2 • 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. De- cember 2003 • I.-K. Cho and S. P. Meyn. Efficiency and marginal cost pricing in dy- namic 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 disserta- tion, ECE UIUC 2008
  • 58. Poisson’s Equation First reflection times, τp :=inf{t ≥ 0 : Q(t) = q p }, ¯ τa :=inf{t ≥ 0 : Q(t) ≥ q a } ¯ τp h(x) = Ex c(X(s)) − φ ds 0 return
  • 59. Poisson’s Equation First reflection times, τp :=inf{t ≥ 0 : Q(t) = q p }, ¯ τa :=inf{t ≥ 0 : Q(t) ≥ q a } ¯ τp h(x) = Ex c(X(s)) − φ ds 0 Solves martingale problem, t M (t) = h(X(t)) + c(X(s)) − φ ds 0
  • 60. Poisson’s Equation Q(t) X(t) = Ga(t) h(X(τa )) τa h(x) = Ex h(X(τa ))+ c(X(s))−φ ds 0 qa ¯ qp ¯
  • 61. Derivative Representations 1 ∇h(x), 1 = 0, x = X(τa ) τa h(x) = Ex h(X(τa ))+ c(X(s))−φ ds 0 qa ¯ qp ¯
  • 62. Derivative Representations 1 λa (x) = ∇c(x), 1 = ca − I{q ≤ 0}cbo 1 ∇h(x), 1 = 0, x = X(τa ) τa h(x) = Ex h(X(τa ))+ c(X(s))−φ ds 0 qa ¯ qp ¯
  • 63. Derivative Representations τa 1 ∇h(x), 1 = Ex λa (X(t)) dt 0 τa = ca E[τa ] − cbo Ex I{Q(t) ≤ 0} dt 0 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 ¯ a = ¯ a∗ ¯ ¯ q q Then h solves the dynamic programming equations, 1. Poisson's equation 1 2. ∇h(x), 1 < 0, x ∈ Ra ¯ a∗ q q p∗ 1 3. ∇h(x), 0 < 0, x ∈ Rp return