1. EE 369 POWER SYSTEM ANALYSIS
Lecture 17
Optimal Power Flow, LMPs
Tom Overbye and Ross Baldick

2. Announcements Read Chapter 7.
Homework 12 is 6.43, 6.48, 6.59, 6.61, 12.19, 12.22, 12.20, 12.24, 12.26, 12.28, 12.29; due Tuesday Nov. 25.
Homework 13 is 12.21, 12.25, 12.27, 7.1, 7.3, 7.4, 7.5, 7.6, 7.9, 7.12, 7.16; due Thursday, December 4.

3. 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).
OPF is used as basis for real-time pricing in major US electricity markets such as MISO, PJM, CA, and ERCOT (from December 2010).

4. Electricity Markets
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 have been widely adopted (options, forwards, hedges, swaps).

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

6. “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.

7. Two Bus ED Example

8. 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

9. 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.

10. Pricing Electricity
LMP indicates the additional cost to supply an additional amount of electricity to bus.
Some electric markets price wholesale energy at LMP:
ERCOT began this in December 2010.
In 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).

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 (LP):
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 an 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)
This would be price paid by load and paid to the generators.

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.
Similarly, prices paid by load and paid to generators will differ bus by bus.

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
(but not allowed to
dispatch to overload
line!)

19. B3 with Line Limits Enforced
LP OPF redispatches
to remove violation.
Bus marginal
costs are now
different.
Prices will be different
at each bus.

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:
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.

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
breaker opens?

24. 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.

25. 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 average of LMPs.
LMPs differ across the system due to transmission system “congestion.”

26. LMPs at 8:55 AM on one day in Midwest.
Source:

27. LMPs at 9:30 AM on same day

28. MISO LMP Contours – 10/30/08

29. 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.