Lecture 17

EE 369POWER SYSTEM ANALYSIS

Lecture 17Optimal Power Flow, LMPs

Tom Overbye and Ross Baldick

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AnnouncementsRead Chapter 7.Homework 12 is fourth edition: 11.19,

11.20, 11.21, 11.27, 11.28; fifth edition: 12.20, 12.21, 12.26, 12.27 due Tuesday November 22 (note change of date!)

Homework 13 is (fourth edition) 11.23, 11.24, 7.1, 7.3, 7.4, 7.5; (fifth edition) 12.23, 12,24, 7.1, 7.3, 7.4, 7.5; due Thursday, December 1.

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Optimal Power FlowThe goal of an optimal power flow (OPF) is to

determine the “best” way to instantaneously operate a power system.

Usually “best” = minimizing operating cost, while keeping flows on transmission below limits.

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, PJM, CA, and ERCOT (from December 2010).

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

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Electricity Futures Example

Source: Wall Street Journal Online, 10/30/08 5

“Ideal” Power MarketIdeal power market is analogous to a lake.

Generators supply energy to lake and loads remove energy.

Ideal power market has no transmission constraintsSingle 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.

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Two Bus ED Example

Total Hourly Cost :

Bus A Bus B

300.0 MWMW

199.6 MWMW 400.4 MWMW300.0 MWMW

8459 $/hr Area Lambda : 13.02

AGC ON AGC ON

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Market Marginal (Incremental) Cost

0 175 350 525 700Generator Power (MW)

12.00

13.00

14.00

15.00

16.00

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

0 350 700 1050 1400Total Area Generation (MW)

12.00

13.00

14.00

15.00

16.00

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Real Power MarketsDifferent operating regions impose constraints – may

limit ability to achieve economic dispatch “globally.”Transmission system imposes constraints on the

marketMarginal costs become localizedRequires solution by an optimal power flowCharging for energy based on localized marginal

costs is called “locational marginal pricing” or “nodal” pricing.

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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

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OPF, cont’dInequality 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

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OPF Solution Methods

Non-linear approach using Newton’s method:– handles marginal losses well, but is relatively slow

and has problems determining binding constraintsLinear Programming (LP):

– fast and efficient in determining binding constraints, but can have difficulty with marginal losses.

– used in PowerWorld Simulator

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LP OPF Solution Method

Solution iterates between:– solving a full ac power flow solution

enforces real/reactive power balance at each busenforces generator reactive limitssystem controls are assumed fixed takes into account non-linearities

– solving an LPchanges system controls to enforce linearized

constraints while minimizing cost

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Two Bus with Unconstrained Line

Total Hourly Cost :

Bus A Bus B

300.0 MWMW

197.0 MWMW 403.0 MWMW300.0 MWMW

8459 $/hr Area Lambda : 13.01

AGC ON AGC ON

13.01 $/MWh 13.01 $/MWh

Transmission line is not overloaded

With no overloads theOPF matchesthe economicdispatch

Marginal cost of supplyingpower to each bus (locational marginal costs)This would be price paid by load and paid to the generators. 14

Two Bus with Constrained Line

Total Hourly Cost :

Bus A Bus B

380.0 MWMW

260.9 MWMW 419.1 MWMW300.0 MWMW

9513 $/hr Area Lambda : 13.26

AGC ON AGC ON

13.43 $/MWh 13.08 $/MWh

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.

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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.

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Bus 2 Bus 1

Bus 3

Total Cost

0.0 MW

0 MW

180 MW

10.00 $/MWh

60 MW 60 MW

60 MW

60 MW120 MW

120 MW

10.00 $/MWh

10.00 $/MWh

180.0 MW

0 MW

1800 $/hr

120%

120%

B3 with Line Limits NOT Enforced

Line from Bus 1to Bus 3 is over-loaded; all buseshave same marginal cost(but not allowed to dispatch to overloadline!)

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B3 with Line Limits EnforcedBus 2 Bus 1

Bus 3

Total Cost

60.0 MW

0 MW

180 MW

12.00 $/MWh

20 MW 20 MW

80 MW

80 MW100 MW

100 MW

10.00 $/MWh

14.00 $/MWh

120.0 MW

0 MW

1920 $/hr

100%

100%

LP OPF redispatchesto remove violation.Bus marginalcosts are nowdifferent. Prices will be different at each bus.

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Bus 2 Bus 1

Bus 3

Total Cost

62.0 MW

0 MW

181 MW

12.00 $/MWh

19 MW 19 MW

81 MW

81 MW100 MW

100 MW

10.00 $/MWh

14.00 $/MWh

119.0 MW

0 MW

1934 $/hr

81%

81%

100%

100%

Verify Bus 3 Marginal Cost

One additional MWof load at bus 3 raised total cost by14 $/hr, as G2 wentup by 2 MW and G1went down by 1MW.

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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.

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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.

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Both lines into Bus 3 CongestedBus 2 Bus 1

Bus 3

Total Cost

100.0 MW

4 MW

204 MW

12.00 $/MWh

0 MW 0 MW

100 MW

100 MW100 MW

100 MW

10.00 $/MWh

20.00 $/MWh

100.0 MW

0 MW

2280 $/hr

100% 100%

100% 100% For bus 3 loadsabove 200 MW,the load must besupplied locally.Then what if thebus 3 generator breaker opens?

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Typical Electricity MarketsElectricity 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.

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PaymentGenerators 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.”

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MISO LMP Contours – 10/30/08

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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.

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