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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. 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

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Ece 476 power system analysis


Lecture 18

Optimal Power Flow, LMPs

Professor Tom Overbye

Department of Electrical andComputer Engineering


  • 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

    • 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

Awards scholarships articles
Awards, Scholarships, Articles

  • Grainger Power EngineeringAward forms are due


  • 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.

Area supply curve
Area Supply Curve

The area supply curve shows the cost to produce the

next MW of electricity, assuming area is economically



curve for

thirty bus


Economic dispatch summary
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
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
Electricity Markets

  • 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).

Ideal power market
“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

Market marginal incremental cost
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
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
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 cont d
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

Opf solution methods
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

Lp opf solution method
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
Two Bus with Unconstrained Line

With no overloads the

OPF matches

the economic


Transmission line is not overloaded

Marginal cost of supplying

power to each bus (locational marginal costs)

Two bus with constrained line
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
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
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
B3 with Line Limits Enforced

LP OPF redispatches

to remove violation.

Bus marginal

costs are now


Verify bus 3 marginal cost
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
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 cont d
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.

Both lines into bus 3 congested
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


Typical electricity markets
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.


  • 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 11 06 07
MISO LMP Contours – 11/06/07

LMP at one location was actually negative ($-302/MWh)!

Why not pay as bid
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