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

Announcements l l Homework 8 is 11. 19, 11. 21, 11. 26, 11. 27, due now Homework 9 is 7. 1, 7. 17, 7. 20, 7. 24, 7. 27 – l l l 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 1

Awards, Scholarships, Articles l Grainger Power Engineering. Award forms are due – l l http: //www. power. ece. uiuc. edu/Grainger. html 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. 2

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 3

Economic Dispatch - Summary l l 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 – – l l 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 4

Optimal Power Flow l l l 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. 5

Electricity Markets l l l 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). 6

“Ideal” Power Market l l l 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 – – l 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 l l 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 10

Optimal Power Flow (OPF) l l l 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 11

OPF, cont’d l Inequality constraints – – l 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 12

OPF Solution Methods l Non-linear approach using Newton’s method – l 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 Power. World Simulator 13

LP OPF Solution Method l Solution iterates between – – solving a full ac power flow solution l enforces real/reactive power balance at each bus l enforces generator reactive limits l system controls are assumed fixed l takes into account non-linearities solving a primal LP l changes system controls to enforce linearized constraints while minimizing cost 14

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) 15

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. 16

Three Bus (B 3) Example l l 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 – – – l 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 17

B 3 with Line Limits NOT Enforced Line from Bus 1 to Bus 3 is overloaded; all buses have same marginal cost 18

B 3 with Line Limits Enforced LP OPF redispatches to remove violation. Bus marginal costs are now different. 19

Verify Bus 3 Marginal Cost One additional MW of load at bus 3 raised total cost by 14 $/hr, as G 2 went up by 2 MW and G 1 went down by 1 MW 20

Why is bus 3 LMP = $14 /MWh l 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, 2/3 MW would go from 2 to 3, while 1/3 MW would go from 2 to 1 to 3. 21

Why is bus 3 LMP $ 14 / MWh, cont’d l l 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 – – l Pg 1 + Pg 2 = 1 MW 2/3 Pg 1 + 1/3 Pg 2 = 0; (no more flow on 1 -3) Solving requires we up Pg 2 by 2 MW and drop Pg 1 by 1 MW -- a net increase of $14. 22

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? 23

Profit Maximization: 30 Bus Example 24

Typical Electricity Markets l l 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. 25

Payment l l l 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. ” 26

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

Why not pay as bid? l Two options for paying market participants – – l l 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 28

Market Experiments 29