EE%20369%20POWER%20SYSTEM%20ANALYSIS

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EE 369POWER SYSTEM ANALYSIS

• Lecture 16
• Economic Dispatch
• Tom Overbye and Ross Baldick

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Announcements

• Read Chapters 6 (section 6.12) and 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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Economic Dispatch Formulation

• The goal of economic dispatch is to determine the
  generation dispatch that minimizes the
  instantaneous operating cost, subject to the
  constraint that total generation total load
  losses

Initially we'll ignore generator limits and
the losses
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Unconstrained Minimization

• This is a minimization problem with a single
  equality constraint
• For an unconstrained minimization a necessary
  (but not sufficient) condition for a minimum is
  the gradient of the function must be zero,
• The gradient generalizes the first derivative for
  multi-variable problems

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Minimization with Equality Constraint

• When the minimization is constrained with an
  equality constraint we can solve the problem
  using the method of Lagrange Multipliers
• Key idea is to represent a constrained
  minimization problem as an unconstrained problem.

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Economic Dispatch Lagrangian
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Economic Dispatch Example
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Economic Dispatch Example, contd
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Economic dispatch example, contd

• At the solution, both generators have the same
  marginal (or incremental) cost, and this common
  marginal cost is equal to ?.
• Intuition behind solution
• If marginal costs of generators were different,
  then by decreasing production at higher marginal
  cost generator, and increasing production at
  lower marginal cost generator we could lower
  overall costs.
• Generalizes to any number of generators.
• If demand changes, then change in total costs can
  be estimated from ?.

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Economic dispatch example, contd

• Another way to solve the equations is to
• Rearrange the first two equations to solve for
  PG1 and PG2 in terms of ?,
• Plug into third equation and solve for ?,
• Use the solved value of ? to evaluate PG1 and
  PG2.
• This works even when relationship between
  generation levels and ? is more complicated
• Equations are more complicated than linear when
  there are maximum and minimum generation limits
  or we consider losses.

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Lambda-Iteration Solution Method

• Discussion on previous page leads to
  lambda-iteration method
• this method requires a unique mapping from a
  value of lambda (marginal cost) to each
  generators MW output
• for any choice of lambda (common marginal cost),
  the generators collectively produce a total MW
  output,
• the method then starts with values of lambda
  below and above the optimal value (corresponding
  to too little and too much total output), and
  then iteratively brackets the optimal value.

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Lambda-Iteration Algorithm
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Lambda-Iteration Graphical View
In the graph shown below for each value of lambda
there is a unique PGi for each generator. This
relationship is the PGi(?) function.
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Lambda-Iteration Example
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Lambda-Iteration Example, contd
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Lambda-Iteration Example, contd
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Lambda-Iteration Example, contd
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Thirty Bus ED Example
Case is economically dispatched (without
considering the incremental impact of the system
losses).
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Generator MW Limits

• Generators have limits on the minimum and maximum
  amount of power they can produce
• Typically the minimum limit is not zero.
• Because of varying system economics usually many
  generators in a system are operated at their
  maximum MW limits
• Baseload generators are at their maximum limits
  except during the off-peak.

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Lambda-Iteration with Gen Limits
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Lambda-Iteration Gen Limit Example
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Lambda-Iteration Limit Example,contd
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Back of Envelope Values

• /MWhr fuelcost heatrate variable OM
• Typical incremental costs can be roughly
  approximated
• Typical heatrate for a coal plant is 10, modern
  combustion turbine is 10, combined cycle plant is
  6 to 8, older combustion turbine 15.
• Fuel costs (/MBtu) are quite variable, with
  current values around 2 for coal, 3 to 5 for
  natural gas, 0.5 for nuclear, probably 10 for
  fuel oil.
• Hydro costs tend to be quite low, but are fuel
  (water) constrained
• Wind and solar costs are zero.

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Inclusion of Transmission Losses

• The losses on the transmission system are a
  function of the generation dispatch.
• In general, using generators closer to the load
  results in lower losses
• This impact on losses should be included when
  doing the economic dispatch
• Losses can be included by slightly rewriting the
  Lagrangian to include losses PL

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Impact of Transmission Losses
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Impact of Transmission Losses
The penalty factor at the slack bus is always
unity!
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Impact of Transmission Losses
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Calculation of Penalty Factors
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Two Bus Penalty Factor Example
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Thirty Bus ED Example
Now consider losses. Because of the penalty
factors the generator incremental costs are no
longer identical.
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Area Supply Curve
The area supply curve shows the cost to produce
the next MW of electricity, assuming area is
economically dispatched
Supply curve for thirty bus system
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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).
• Basic form of economic dispatch ignores the
  transmission system limitations.

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Security Constrained EDor Optimal Power Flow

• Transmission constraints often limit ability to
  use lower cost power.
• Such limits require deviations from what would
  otherwise be minimum cost dispatch in order to
  maintain system security.
• Need to solve or approximate power flow in order
  to consider transmission constraints.

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Security Constrained EDor Optimal Power Flow

• The goal of a security constrained ED or optimal
  power flow (OPF) is to determine the best way
  to instantaneously operate a power system,
  considering transmission limits.
• Usually best minimizing operating cost, while
  keeping flows on transmission below limits.
• In three bus case the generation at bus 3 must be
  limited to avoid overloading the line from bus 3
  to bus 2.

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Security Constrained Dispatch
Need to dispatch to keep line from bus 3 to bus
2 from overloading
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Multi-Area Operation

• In multi-area system, rules have been
  established regarding transactions on tie-lines
• In Eastern interconnection, in principle, up to
  nominal thermal interconnection capacity,
• In Western interconnection there are more
  complicated rules
• The actual power that flows through the entire
  network depends on the impedance of the
  transmission lines, and ultimately determine what
  are acceptable patterns of dispatch
• Can result in need to curtail transactions that
  otherwise satisfy rules.
• Economically uncompensated flow through other
  areas is known as parallel path or loop
  flows.
• Since ERCOT is one area, all of the flows on AC
  lines are inside ERCOT and there is no
  uncompensated flow on AC lines.

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Seven Bus Case One-line
System has three areas
Top area has five buses
No net interchange between Any areas.
Left area has one bus
Right area has one bus
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Seven Bus Case Area View
Actual flow between areas
System has 40 MW of Loop Flow
Scheduled flow
Loop flow can result in higher losses
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Seven Bus - Loop Flow?
Note that Tops Losses have increased from
7.09MW to 9.44 MW
Transaction has actually decreased the loop flow
100 MW Transaction between Left and Right