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ECE 476POWER SYSTEM ANALYSIS

- Lecture 17
- Optimal Power Flow, LMPs
- Professor Tom Overbye
- Department of Electrical andComputer 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

Back of Envelope Values

- Often times incremental costs can be approximated

by a constant value - /MWhr fuelcost heatrate variable OM
- 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

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

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

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

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.

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).

Electricity Futures Example

Source Wall Street Journal Online, 10/30/08

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

Two Bus ED Example

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

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

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

OPF, contd

- 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

OPF Solution Methods

- Non-linear approach using Newtons 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

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

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)

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.

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

B3 with Line Limits NOT Enforced

Line from Bus 1 to Bus 3 is over- loaded all

buses have same marginal cost

B3 with Line Limits Enforced

LP OPF redispatches to remove violation. Bus

marginal costs are now different.

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

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.

Why is bus 3 LMP 14 / MWh, contd

- 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.

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?

Profit Maximization 30 Bus Example

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.

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.

MISO LMP Contours 10/30/08

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

Market Experiments

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.

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