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ECE 476 POWER SYSTEM ANALYSIS Lecture 17 Optimal Power Flow, LMPs Professor Tom Overbye Department of Electrical and Computer Engineering 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 1 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 2 Aside: Levelized Cost of Generation Technology $/MWh (2007 Dollars) (IOU) Advanced Nuclear 104 Wind – Class 5 67 Solar – Photovoltaic 686 Solar – Concentrating 434 Solar – Parabolic Trough 281 Ocean Wave (Pilot) 838 Small Scale Hydro 118 Geothermal 63 Keep in mind these numbers involve LOTs of assumptions that 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 3 Area Supply Curve The area supply curve shows the cost to produce the next MW of electricity, assuming area is economically dispatched 10.00 7.50 Supply curve for 5.00 thirty bus system 2.50 0.00 0 100 200 300 400 Total Area Generation (MW) 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 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). 7 Electricity Futures Example Source: Wall Street Journal Online, 10/30/08 8 “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 9 Two Bus ED Example Total Hourly Cost : 8459 $/hr Area Lambda : 13.02 Bus A Bus B 300.0 MW 300.0 MW 199.6 MW 400.4 MW AGC ON AGC ON 10 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. 16.00 16.00 15.00 15.00 14.00 14.00 13.00 13.00 12.00 12.00 0 175 350 525 700 0 350 700 1050 1400 Generator Power (MW) Total Area Generation (MW) Current generator operating point 11 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 12 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 13 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 14 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 15 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 16 Two Bus with Unconstrained Line With no overloads the Transmission Total Hourly Cost : 8459 $/hr OPF matches Area Lambda : 13.01 line is not the economic overloaded dispatch Bus A 13.01 $/MWh Bus B 13.01 $/MWh 300.0 MW 300.0 MW 197.0 MW 403.0 MW AGC ON AGC ON Marginal cost of supplying power to each bus (locational marginal costs) 17 Two Bus with Constrained Line Total Hourly Cost : 9513 $/hr Area Lambda : 13.26 Bus A 13.43 $/MWh Bus B 13.08 $/MWh 380.0 MW 300.0 MW 260.9 MW 419.1 MW AGC ON AGC ON With the line loaded to its limit, additional load at Bus A must be supplied locally, causing the marginal costs to diverge. 18 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 19 B3 with Line Limits NOT Enforced 60 MW 60 MW Bus 2 Bus 1 10.00 $/MWh 0.0 MW 10.00 $/MWh 120 MW 180.0 MW 120% 0 MW 60 MW 120% 120 MW Total Cost 60 MW Line from Bus 1 1800 $/hr Bus 3 10.00 $/MWh to Bus 3 is over- 180 MW loaded; all buses 0 MW have same marginal cost 20 B3 with Line Limits Enforced 20 MW 20 MW Bus 2 Bus 1 10.00 $/MWh 60.0 MW 12.00 $/MWh 100 MW 120.0 MW 100% 0 MW 80 MW 100% 100 MW Total Cost 80 MW 1920 $/hr LP OPF redispatches Bus 3 14.00 $/MWh to remove violation. 180 MW Bus marginal 0 MW costs are now different. 21 Verify Bus 3 Marginal Cost 19 MW 19 MW Bus 2 Bus 1 10.00 $/MWh 62.0 MW 12.00 $/MWh 100 MW 119.0 MW 81% 100% 0 MW 81 MW 81% 100% 100 MW One additional MW Total Cost 81 MW 1934 $/hr of load at bus 3 Bus 3 14.00 $/MWh raised total cost by 181 MW 14 $/hr, as G2 went 0 MW up by 2 MW and G1 went down by 1MW 22 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. 23 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. 24 Both lines into Bus 3 Congested 0 MW 0 MW Bus 2 Bus 1 10.00 $/MWh 100.0 MW12.00 $/MWh 100 MW 100.0 MW 100% 100% 0 MW 100 MW For bus 3 loads 100% 100% 100 MW Total Cost MW 100 above 200 MW, 2280 $/hr Bus 3 the 20.00 $/MWh load must be 204 MW supplied locally. Then what if the 4 MW bus 3 generator opens? 25 Profit Maximization: 30 Bus Example 52.45 MW 69.58 MW slack A 1 2 84% MVA 18 A 15 68% A MVA A A A 1.000 A A 62% MVA MVA MVA 19 MVA MVA MVA 14 A 28 A 67% MVA MVA 35.00 MW A A MVA 3 A 4 A 16 MW A MVA A Gen 13 LMP MVA 56% 82% MVA MVA 8 7 5 MVA 12 13 A MVA 33.46 $/MWh 6 A A A A MVA MVA MVA MVA 9 16 17 11 11 MW A MVA 19 MW A 11 MW MVA A MVA 10 MW A 20 66% MVA 21 MW 26 10 A 24.00 MW A MVA MVA 23 A 73% MVA 25 22 21 24 A 2 MW 52% A 16.00 MW MVA MVA 40.00 MW A 52% A MVA 87% MVA 29 30 27 A A MVA A MVA MVA 26 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. 27 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.” 28 MISO LMP Contours – 10/30/08 29 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 30 Market Experiments 31 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. 32

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