0. ECE 476 POWER SYSTEM ANALYSIS
Lecture 17
Optimal Power Flow, LMPs
Professor Tom Overbye
Department of Electrical and Computer Engineering

1. Announcements Homework 7 is due now.
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:

2. 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

3. 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:

4. 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

5. 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

6. 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.

7. 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).

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

9. “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

10. Two Bus ED Example

11. 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

12. 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

13. 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

14. OPF, cont’d Inequality constraints Available Controls
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

15. 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

16. 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

17. 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)

18. 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.

19. 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

20. B3 with Line Limits NOT Enforced
Line from Bus 1
to Bus 3 is over-
loaded; all buses
have same
marginal cost

21. B3 with Line Limits Enforced
LP OPF redispatches
to remove violation.
Bus marginal
costs are now
different.

22. 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

23. 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.

24. 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.

25. 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
opens?

26. Profit Maximization: 30 Bus Example

27. 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.

28. 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.”

29. MISO LMP Contours – 10/30/08

30. Why not pay as bid? Two options for paying market participants
Pay last accepted offer
What would be potential advantages/disadvantages of both?
Talk about supply and demand curves, scarcity, withholding, market power

31. Market Experiments

32. 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.