Optimal Power Flow and LMPs in Power System Analysis

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

2. Announcements • Homework 7 is due now. • Homework 8 is 7.1, 7.17, 7.20, 7.24, 7.27 • 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: • http://energy.ece.uiuc.edu/Grainger.html

3. 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

4. Aside: Levelized Cost of Generation Keep in mind these numbers involve LOTs of assumptionsthat can drastically affect the value, and that many technology costs are site dependent. Source: California Energy Commission: http://energyalmanac.ca.gov/electricity/levelized_costs.html

5. 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

6. 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

7. 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.

8. 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).

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

10. “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

11. Two Bus ED Example

12. 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

13. 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

14. 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

15. OPF, cont’d • Inequality constraints • 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

16. 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

17. 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

18. 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)

19. 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.

20. 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

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

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

23. 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

24. 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.

25. 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.

26. 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?

27. Profit Maximization: 30 Bus Example

28. 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.

29. 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.”

30. MISO LMP Contours – 10/30/08

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

32. Market Experiments

33. 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.