0. ECE 476 POWER SYSTEM ANALYSIS
Lecture 18
Optimal Power Flow, LMPs
Professor Tom Overbye
Department of Electrical and Computer Engineering

1. Announcements Homework 8 is 11.19, 11.21, 11.26, 11.27, due now
you do not need to turn it in, but must do it before exam
Second exam is Tuesday Nov 13 in class
Design Project 2 from the book (page 345 to 348) was due on Nov 15, but I have given you an extension to Nov 29. The Nov 29 date is firm!
Be reading Chapter 7

2. Awards, Scholarships, Articles
Grainger Power EngineeringAward forms are due
Scholarship forms should be turned in ASAP, require a 3.0 minimum GPA
Make sure you read the articles at the beginning of the Chapters 11.

3. 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

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 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. “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

8. Two Bus ED Example

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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)

16. 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.

17. 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

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

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

20. 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

21. 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.

22. 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.

23. 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?

24. Profit Maximization: 30 Bus Example

25. 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.

26. 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.”

27. MISO LMP Contours – 11/06/07
LMP at one location was actually negative ($-302/MWh)!

28. 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

29. Market Experiments