0. ECE 476 POWER SYSTEM ANALYSIS
Lecture 17
Economic Dispatch, OPF, Markets
Professor Tom Overbye
Department of Electrical and Computer Engineering

1. Announcements Be reading Chapter 7
HW 7 is 12.26, 12.28, 12.29, 7.1 due October 27 in class.
US citizens and permanent residents should consider applying for a Grainger Power Engineering Awards. Due Nov 1. See for details.
The Design Project, which is worth three regular homeworks, is assigned today; it is due on Nov 17 in class. It is Design Project 2 from Chapter 6 (fifth edition of course).
For tower configuration assume a symmetric conductor spacing, with the distance in feet given by the following formula:
(Last two digits of your UIN+50)/9. Example student A has an EIN of xxx65. Then his/her spacing is (65+50)/9 = ft.

2. 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:

3. Impact of Transmission Losses

4. Impact of Transmission Losses
The penalty factor
at the slack bus is
always unity!

5. Impact of Transmission Losses

6. Calculation of Penalty Factors

7. Two Bus Penalty Factor Example

8. Thirty Bus ED Example
Because of the penalty factors the generator incremental
costs are no longer identical.

9. 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

10. 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

11. Thirty Bus ED Example
Case is economically dispatched without considering
the incremental impact of the system losses

12. 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.

13. 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).

14. Electricity Futures Example
Source: Wall Street Journal Online, 10/19/11

15. Historical Variation in Oct 11 Price
Price has dropped, following the drop in natural gas prices
Source: Wall Street Journal Online, 10/19/11

16. “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

17. Two Bus ED Example

18. 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

19. 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

20. 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

21. 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

22. 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

23. 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

24. 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)

25. 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.

26. 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

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

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

29. 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

30. 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.

31. 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.

32. 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?

33. Profit Maximization: 30 Bus Example

34. 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.

35. 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.”

36. MISO and PJM Joint LMP Contour

37. 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

38. Market Experiments