EE 369 POWER SYSTEM ANALYSIS

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EE 369POWER SYSTEM ANALYSIS

• Lecture 17
• Optimal Power Flow, LMPs
• Tom Overbye and Ross Baldick

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Announcements

• Read Chapter 7 (sections 7.1 to 7.3).
• Homework 12 is 6.62, 6.63, 6.67 (calculate
  economic dispatch for values of load from 55 MW
  to 350 MW) due Tuesday, 11/29.
• Class review and course evaluation on Tuesday,
  11/29.
• Midterm III on Thursday, 12/1, including material
  through Homework 12.

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Electricity Markets

• Over last 20 years electricity markets have moved
  from bilateral contracts between utilities to
  also include centralized markets operated by
  Independent System Operators/Regional
  Transmission Operators
• Day-ahead market that establishes unit commitment
  and forward financial positions,
• Real-time market, run every 5 or 15 minutes that
  arranges for physical dispatch, the spot
  market.
• Basic engine for operating centralized markets
  is Optimal Power Flow (OPF).

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Electricity Markets

• OPF is used as basis for day-ahead and real-time
  dispatch pricing in US ISO/RTO electricity
  markets
• MISO, PJM, ISO-NE, NYISO, SPP, CA, and ERCOT.
• Electricity (MWh) is treated as a commodity (like
  corn, coffee, natural gas) but with the extent of
  the market limited by transmission system
  constraints.
• Tools of commodity trading have been widely
  adopted (options, forwards, hedges, swaps).

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Electricity Futures Example
Source Wall Street Journal Online, 10/30/08
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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.

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Two Bus ED Example
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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.
Current generator operating point
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Real Power Markets

• Different operating regions impose constraints
  may limit ability to achieve economic dispatch
  globally.
• Transmission system imposes constraints on the
  market
• Marginal costs differ at different buses.
• Optimal dispatch solution requires solution by an
  optimal power flow
• Charging for energy based on marginal costs at
  different buses is called locational marginal
  pricing (LMP) or nodal pricing.

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Pricing Electricity

• LMP indicates the additional cost to supply an
  additional amount of electricity to bus.
• All North American ISO/RTO electricicity markets
  price wholesale energy at LMP.
• If there were no transmission limitations then
  the LMPs would be the same at all buses
• Equal to value of lambda from economic dispatch.
• Transmission constraints result in differing LMPs
  at buses.
• Determination of LMPs requires the solution of an
  Optimal Power Flow (OPF).

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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

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OPF, contd

• 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

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OPF Solution Methods

• Non-linear approach using Newtons method
• handles marginal losses well, but is relatively
  slow and has problems determining binding
  constraints
• Linear Programming (LP)
• fast and efficient in determining binding
  constraints, but can have difficulty with
  marginal losses.
• used in PowerWorld Simulator

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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 an LP
• changes system controls to enforce linearized
  constraints while minimizing cost

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Two Bus with Unconstrained Line
With no overloads the OPF matches the
economic dispatch
Transmission line is not overloaded
Marginal cost of supplying power to each bus
(locational marginal costs) This would be price
paid by load and paid to the generators.
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Two Bus with Constrained Line
With the line loaded to its limit, additional
load at Bus A must be supplied locally, causing
the marginal costs to diverge. Similarly,
prices paid by load and paid to generators will
differ bus by bus. (In practice, some markets
such as ERCOT charge zonal averaged price to
load.)
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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.

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B3 with Line Limits NOT Enforced
Line from Bus 1 to Bus 3 is over- loaded all
buses have same marginal cost (but not allowed
to dispatch to overload line!)
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B3 with Line Limits Enforced
LP OPF redispatches to remove violation. Bus
marginal costs are now different. Prices will
be different at each bus.
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Verify Bus 3 Marginal Cost
One additional MW of load at bus 3 raised total
cost by 14 /hr, as G2 went up by 2 MW and
G1 went down by 1MW.
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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.

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Why is bus 3 LMP 14 / MWh, contd

• 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
• Extra production of 1MW Pg1 Pg2 1 MW
• No more flow on line 1 to 3 2/3 Pg1 1/3 Pg2
  0
• Solving requires we increase Pg2 by 2 MW and
  decrease Pg1 by 1 MW for a net increase of
  14/h for the 1 MW increase.
• That is, the marginal cost of delivering power to
  bus 3 is 14/MWh.

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Both lines into Bus 3 Congested
For bus 3 loads above 200 MW, the load must
be supplied locally. Then what if the bus 3
generator breaker opens?
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Typical Electricity Markets

• Electricity markets trade various 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 (offer roughly represents
  marginal costs) OPF is used to determine who
  gets dispatched based upon forecasted conditions.
  Results are financially binding either
  generate or pay for someone else.
• Real-time Modifies the conditions from the day
  ahead market based upon real-time conditions.

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Payment

• Generators are not paid their offer, rather they
  are paid the LMP at their bus, while the loads
  pay the LMP
• In most systems, loads are charged based on a
  zonal weighted average of LMPs.
• At the residential/small commercial level the LMP
  costs are usually not passed on directly to the
  end consumer. Rather, these consumers typically
  pay a fixed rate that reflects time and
  geographical average of LMPs.
• LMPs differ across the system due to transmission
  system congestion.

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LMPs at 855 AM on one day in Midwest.
Source www.midwestmarket.org
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LMPs at 930 AM on same day
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MISO LMP Contours 10/30/08
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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 involves setting a price and
  emitter of CO2 pays based upon how much CO2 is
  emitted.
• 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.