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					         ECE 476
POWER SYSTEM ANALYSIS


         Lecture 17
   Optimal Power Flow, LMPs




    Professor Tom Overbye
    Department of Electrical and
      Computer Engineering
Announcements

   Homework 7 is due now.
   Homework 8 is 7.1, 7.17, 7.20, 7.24, 7.27
       Should be done before second exam; not turned in
   Be reading Chapter 7
   Design Project is assigned today (see website for
    details). Due date is Nov 20.
   Exam 2 is Thursday Nov 13 in class.
   Grainger Power Engineering Award Applications
    Due Nov 1. See below for details:
    http://energy.ece.uiuc.edu/Grainger.html
                                                           1
Back of Envelope Values

   Often times incremental costs can be approximated
    by a constant value:
    –   $/MWhr = fuelcost * heatrate + variable O&M
    –   Typical heatrate for a coal plant is 10, modern
        combustion turbine is 10, combined cycle plant is 7 to 8,
        older combustion turbine 15.
    –   Fuel costs ($/MBtu) are quite variable, with current
        values around 2 for coal, 7 for natural gas, 0.5 for
        nuclear, probably 10 for fuel oil.
    –   Hydro costs tend to be quite low, but are fuel (water)
        constrained

                                                                2
Aside: Levelized Cost of Generation
      Technology                              $/MWh (2007 Dollars) (IOU)
      Advanced Nuclear                                      104
      Wind – Class 5                                        67
      Solar – Photovoltaic                                  686
      Solar – Concentrating                                 434
      Solar – Parabolic Trough                              281
      Ocean Wave (Pilot)                                    838
      Small Scale Hydro                                     118
      Geothermal                                            63
  Keep in mind these numbers involve LOTs of assumptions
  that can drastically affect the value, and that many
  technology costs are site dependent.
  Source: California Energy Commission:
  http://energyalmanac.ca.gov/electricity/levelized_costs.html
                                                                           3
Area Supply Curve

The area supply curve shows the cost to produce the
next MW of electricity, assuming area is economically
dispatched   10.00




              7.50


Supply
curve for     5.00

thirty bus
system        2.50




              0.00
                     0   100                200             300   400
                               Total Area Generation (MW)




                                                                        4
Economic Dispatch - Summary

   Economic dispatch determines the best way to
    minimize the current generator operating costs
   The lambda-iteration method is a good approach for
    solving the economic dispatch problem
    –   generator limits are easily handled
    –   penalty factors are used to consider the impact of losses
   Economic dispatch is not concerned with
    determining which units to turn on/off (this is the
    unit commitment problem)
   Economic dispatch ignores the transmission system
    limitations
                                                                    5
Optimal Power Flow

   The goal of an optimal power flow (OPF) is to
    determine the “best” way to instantaneously operate
    a power system.
   Usually “best” = minimizing operating cost.
   OPF considers the impact of the transmission system
   OPF is used as basis for real-time pricing in major
    US electricity markets such as MISO and PJM.
   ECE 476 introduces the OPF problem and provides
    some demonstrations.


                                                     6
Electricity Markets

   Over last ten years electricity markets have moved
    from bilateral contracts between utilities to also
    include spot markets (day ahead and real-time).
   Electricity (MWh) is now being treated as a
    commodity (like corn, coffee, natural gas) with the
    size of the market transmission system dependent.
   Tools of commodity trading are being widely
    adopted (options, forwards, hedges, swaps).



                                                          7
Electricity Futures Example




   Source: Wall Street Journal Online, 10/30/08
                                                  8
“Ideal” Power Market

   Ideal power market is analogous to a lake.
    Generators supply energy to lake and loads remove
    energy.
   Ideal power market has no transmission constraints
   Single marginal cost associated with enforcing
    constraint that supply = demand
    –   buy from the least cost unit that is not at a limit
    –   this price is the marginal cost
   This solution is identical to the economic dispatch
    problem solution

                                                              9
Two Bus ED Example

                      Total Hourly Cost : 8459 $/hr
                            Area Lambda : 13.02




   Bus A                                                Bus B


                          300.0 MW                      300.0 MW
           199.6 MW                          400.4 MW
           AGC ON                             AGC ON




                                                                   10
Market Marginal (Incremental) Cost
Below are some graphs associated with this two bus
system. The graph on left shows the marginal cost for each
of the generators. The graph on the right shows the
system supply curve, assuming the system is optimally
dispatched.
 16.00                                           16.00


 15.00                                           15.00


 14.00                                           14.00


 13.00                                           13.00


 12.00                                           12.00
         0   175        350          525   700           0   350            700        1050     1400
                   Generator Power (MW)                            Total Area Generation (MW)




             Current generator operating point
                                                                                                 11
Real Power Markets

   Different operating regions impose constraints --
    total demand in region must equal total supply
   Transmission system imposes constraints on the
    market
   Marginal costs become localized
   Requires solution by an optimal power flow




                                                        12
Optimal Power Flow (OPF)

   OPF functionally combines the power flow with
    economic dispatch
   Minimize cost function, such as operating cost,
    taking into account realistic equality and inequality
    constraints
   Equality constraints
    –   bus real and reactive power balance
    –   generator voltage setpoints
    –   area MW interchange



                                                            13
OPF, cont’d

   Inequality constraints
    –   transmission line/transformer/interface flow limits
    –   generator MW limits
    –   generator reactive power capability curves
    –   bus voltage magnitudes (not yet implemented in
        Simulator OPF)
   Available Controls
    –   generator MW outputs
    –   transformer taps and phase angles



                                                              14
OPF Solution Methods

   Non-linear approach using Newton’s method
    –   handles marginal losses well, but is relatively slow and
        has problems determining binding constraints
   Linear Programming
    –   fast and efficient in determining binding constraints, but
        can have difficulty with marginal losses.
    –   used in PowerWorld Simulator




                                                                     15
LP OPF Solution Method

   Solution iterates between
    –   solving a full ac power flow solution
          enforces real/reactive power balance at each bus
          enforces generator reactive limits
          system controls are assumed fixed
          takes into account non-linearities
    –   solving a primal LP
          changes system controls to enforce linearized
           constraints while minimizing cost



                                                              16
    Two Bus with Unconstrained Line

With no
overloads the                                              Transmission
                         Total Hourly Cost : 8459 $/hr
OPF matches                    Area Lambda : 13.01         line is not
the economic                                               overloaded
dispatch
      Bus A                   13.01 $/MWh    Bus B         13.01 $/MWh


                             300.0 MW                      300.0 MW
              197.0 MW                          403.0 MW
              AGC ON                             AGC ON




                         Marginal cost of supplying
                         power to each bus
                         (locational marginal costs)
                                                                          17
 Two Bus with Constrained Line


                        Total Hourly Cost : 9513 $/hr
                              Area Lambda : 13.26




     Bus A                   13.43 $/MWh    Bus B         13.08 $/MWh


                            380.0 MW                      300.0 MW
             260.9 MW                          419.1 MW
             AGC ON                             AGC ON


With the line loaded to its limit, additional load at Bus A
must be supplied locally, causing the marginal costs to
diverge.
                                                                        18
Three Bus (B3) Example

   Consider a three bus case (bus 1 is system slack),
    with all buses connected through 0.1 pu reactance
    lines, each with a 100 MVA limit
   Let the generator marginal costs be
    –   Bus 1: 10 $ / MWhr; Range = 0 to 400 MW
    –   Bus 2: 12 $ / MWhr; Range = 0 to 400 MW
    –   Bus 3: 20 $ / MWhr; Range = 0 to 400 MW
   Assume a single 180 MW load at bus 2



                                                         19
B3 with Line Limits NOT Enforced

            60 MW                   60 MW
    Bus 2                                     Bus 1
                                              10.00 $/MWh


0.0 MW   10.00 $/MWh
                              120 MW           180.0 MW
                                       120%
  0 MW
         60 MW
                           120%   120 MW
Total Cost 60 MW                              Line from Bus 1
1800 $/hr
            Bus 3             10.00 $/MWh     to Bus 3 is over-
                              180 MW          loaded; all buses
                    0 MW                      have same
                                              marginal cost
                                                              20
B3 with Line Limits Enforced

            20 MW                   20 MW
    Bus 2                                     Bus 1
                                               10.00 $/MWh


60.0 MW 12.00 $/MWh
                              100 MW            120.0 MW
                                       100%
  0 MW
         80 MW
                           100%   100 MW
Total Cost 80 MW
1920 $/hr                                   LP OPF redispatches
            Bus 3             14.00     $/MWh
                                            to remove violation.
                              180 MW
                                            Bus marginal
                    0 MW                    costs are now
                                            different.
                                                               21
Verify Bus 3 Marginal Cost

            19 MW                   19 MW
    Bus 2                                     Bus 1
                                               10.00 $/MWh


62.0 MW 12.00 $/MWh
                              100 MW            119.0 MW
            81%                        100%
  0 MW
         81 MW
                    81%    100%   100 MW    One additional MW
Total Cost 81 MW
1934 $/hr                                   of load at bus 3
            Bus 3             14.00     $/MWh
                                            raised total cost by
                              181 MW
                                            14 $/hr, as G2 went
                    0 MW                    up by 2 MW and G1
                                            went down by 1MW
                                                               22
Why is bus 3 LMP = $14 /MWh

   All lines have equal impedance. Power flow in a
    simple network distributes inversely to impedance
    of path.
    –   For bus 1 to supply 1 MW to bus 3, 2/3 MW would take
        direct path from 1 to 3, while 1/3 MW would “loop
        around” from 1 to 2 to 3.
    –   Likewise, for bus 2 to supply 1 MW to bus 3, 2/3MW
        would go from 2 to 3, while 1/3 MW would go from 2 to
        1to 3.




                                                            23
Why is bus 3 LMP $ 14 / MWh, cont’d

   With the line from 1 to 3 limited, no additional
    power flows are allowed on it.
   To supply 1 more MW to bus 3 we need
    –   Pg1 + Pg2 = 1 MW
    –   2/3 Pg1 + 1/3 Pg2 = 0; (no more flow on 1-3)
   Solving requires we up Pg2 by 2 MW and drop Pg1
    by 1 MW -- a net increase of $14.




                                                       24
Both lines into Bus 3 Congested

              0 MW                       0 MW
   Bus 2                                        Bus 1
                                                10.00 $/MWh


100.0 MW12.00 $/MWh
                                100 MW           100.0 MW
            100%                         100%
  0 MW
         100 MW                            For bus 3 loads
                     100%    100% 100 MW
Total Cost MW
          100
                                           above 200 MW,
2280 $/hr
             Bus 3                         the
                                 20.00 $/MWh load must be
                                204 MW     supplied locally.
                                           Then what if the
                      4 MW
                                           bus 3 generator
                                           opens?
                                                               25
Profit Maximization: 30 Bus Example

                                             52.45 MW                            69.58 MW
                                                   slack

                                                                                                                                                                                                                 A

                                         1                                           2                                                                                                                          84%
                                                                                                                                                                                                                MVA
                                                                                                                                                                                                                                           18
                                                                     A                                                                                                              15
                                                                    68%


                                              A
                                                                     MVA




                                                                                                    A            A        A
                                                                                                                                            1.000                            A
                                                                                                                                                                                                                                      A

                                                                                                                                                                                                                                     62%
                                                                                                                                                                                                                                     MVA
                                                                                                                                                                             MVA
                                             MVA                                                                                                                                                                                           19
                                                                                                  MVA        MVA          MVA
                                                                                                                                                                                   14           A
                28                                                       A                                                                                                                     67%

                                                                     MVA
                                                                                                                                                                                               MVA
                                                                                                                                                                                                                35.00 MW

       A
                         A


                         MVA
                                         3                      A
                                                                                     4

                                                                                                                                             A
                                                                                                                                                       16 MW
                                                                                                                                                                             A


                                                                                                                                                                             MVA
                                                                                                                                                                                                    A
                                                                                                                                                                                                                                                      Gen 13 LMP
                                                            MVA
                                                                                                                                                                                                56%
                                                                                                                                            82%
       MVA
                                                                                                                                                                                                    MVA




                8
                                                                             7                                                  5           MVA
                                                                                                                                                                                          12                          13                        A


                                                                                                                                                                                                                                                MVA
                                                                                                                                                                                                                                                      33.46 $/MWh
                                                     6                                                                                                                       A
                          A                                                                                                                                                                               A
                                                                                 A                                                                                           MVA

                         MVA                                                                                                                                                                              MVA
                                                                             MVA
                     9                                                                                                                                                                    16                               17
                                              11
                                                                                                            11 MW
                              A


                             MVA
                                                                                                        19 MW
                                                             A
                                                                                                                                                       11 MW
                                                            MVA                                                                                                         A


                                                                                                                                                                       MVA

       10 MW

                                                                                                                                                                        A
                                                                                                                                                                                                                            20
                                                                                                                                                                       66%
                                                                                                                                                                       MVA
                                                                                                                                                               21 MW
                                              26                                                                     10
 A                                                         24.00 MW                                     A


 MVA                                                                                                MVA                                                                                                                         23
                                                                                                             A

                                                                                                            73%
                                                                                                            MVA



                                             25
                                                                    22                                                21                          24
                                   A                                                                                                                           2 MW
                                  52%                                                                                                                                               A                                  16.00 MW
                                   MVA

                                                                                                                                                                                    MVA

             40.00 MW                                                                                                            A

                                                                                                                                52%
                                                                                              A                                 MVA


                                                                                         87%
                                                                                          MVA

                                                                                         29                                                       30
                    27
                                                            A
                                                                                                                                      A

                                                            MVA
                                                                                              A                                       MVA


                                                                                              MVA




                                                                                                                                                                                                                                                                    26
Typical Electricity Markets

   Electricity markets trade a number of different
    commodities, with MWh being the most important
   A typical market has two settlement periods: day
    ahead and real-time
    –   Day Ahead: Generators (and possibly loads) submit
        offers for the next day; OPF is used to determine who
        gets dispatched based upon forecasted conditions.
        Results are financially binding
    –   Real-time: Modifies the day ahead market based upon
        real-time conditions.


                                                                27
Payment

   Generators are not paid their offer, rather they are
    paid the LMP at their bus, the loads pay the LMP.
   At the residential/commercial level the LMP costs
    are usually not passed on directly to the end
    consumer. Rather, they these consumers typically
    pay a fixed rate.
   LMPs may differ across a system due to
    transmission system “congestion.”



                                                           28
MISO LMP Contours – 10/30/08




                               29
Why not pay as bid?

   Two options for paying market participants
    –   Pay as bid
    –   Pay last accepted offer
   What would be potential advantages/disadvantages
    of both?
   Talk about supply and demand curves, scarcity,
    withholding, market power




                                                       30
Market Experiments




                     31
Limiting Carbon Dioxide Emissions

• There is growing concern about the need to limit
    carbon dioxide emissions.
•   The two main approaches are 1) a carbon tax, or 2)
    a cap-and-trade system (emissions trading)
    • The tax approach is straightforward – pay a fixed rate
        based upon how the amount of CO2 is emitted. But there
        is a need to differentiate between carbon and CO2
        (related by 12/44).
    •   A cap-and-trade system limits emissions by requiring
        permits (allowances) to emit CO2. The government sets
        the number of allowances, allocates them initially, and
        then private markets set their prices and allow trade.
                                                               32

				
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