ECE 476 POWER SYSTEM ANALYSIS Lecture 4 Power System Operation, Transmission Line Modeling Professor Tom Overbye Department of Electrical and Computer Engineering

Reading and Homework For lectures 4 through 6 please be reading Chapter 4 we will not be covering sections 4.7, 4.11, and 4.12 in detail HW 1 is 2.7, 12, 21, 26 is due now HW 2 is 2.32, 43, 47 (You can download the latest educational version of PowerWorld (version 13) at http://www.powerworld.com/gloversarma.asp The Problem 2.32 case will also be on the website

Interesting Graph: Electricity Prices Real is inflation adjusted to 2000 prices; nominal is unad- justed Source: EIA 2007 Annual Energy Review, Fig 8.10

Operating (Balancing Authority) Areas Transmission lines that join two areas are known as tie-lines. The net power out of an area is the sum of the flow on its tie-lines. The flow out of an area is equal to total gen - total load - total losses = tie-flow

Area Control Error (ACE) The area control error is the difference between the actual flow out of an area, and the scheduled flow. Ideally the ACE should always be zero. Because the load is constantly changing, each utility must constantly change its generation to “chase” the ACE.

Automatic Generation Control Most utilities use automatic generation control (AGC) to automatically change their generation to keep their ACE close to zero. Usually the utility control center calculates ACE based upon tie-line flows; then the AGC module sends control signals out to the generators every couple seconds.

Three Bus Case on AGC Generation is automatically changed to match change in load Net tie flow is close to zero

Generator Costs There are many fixed and variable costs associated with power system operation. The major variable cost is associated with generation. Cost to generate a MWh can vary widely. For some types of units (such as hydro and nuclear) it is difficult to quantify. For thermal units it is much easier. These costs will be discussed later in the course.

Economic Dispatch Economic dispatch (ED) determines the least cost dispatch of generation for an area. For a lossless system, the ED occurs when all the generators have equal marginal costs. IC1(PG,1) = IC2(PG,2) = … = ICm(PG,m)

Power Transactions Power transactions are contracts between areas to do power transactions. Contracts can be for any amount of time at any price for any amount of power. Scheduled power transactions are implemented by modifying the area ACE: ACE = Pactual,tie-flow - Psched

100 MW Transaction Scheduled 100 MW Transaction from Left to Right Net tie-line flow is now 100 MW

Security Constrained ED Transmission constraints often limit system economics. Such limits required a constrained dispatch in order to maintain system security. In three bus case the generation at bus 3 must be constrained to avoid overloading the line from bus 2 to bus 3.

Security Constrained Dispatch Dispatch is no longer optimal due to need to keep line from bus 2 to bus 3 from overloading

Multi-Area Operation If Areas have direct interconnections, then they may directly transact up to the capacity of their tie-lines. Actual power flows through the entire network according to the impedance of the transmission lines. Flow through other areas is known as “parallel path” or “loop flows.”

Seven Bus Case: One-line System has three areas Area top has five buses Area left has one bus Area right has one bus

Seven Bus Case: Area View Actual flow between areas System has 40 MW of “Loop Flow” Scheduled flow Loop flow can result in higher losses

Seven Bus - Loop Flow? Note that Top’s Losses have increased from 7.09MW to 9.44 MW Transaction has actually decreased the loop flow 100 MW Transaction between Left and Right

Pricing Electricity Cost to supply electricity to bus is called the locational marginal price (LMP) Presently some electric makets post LMPs on the web In an ideal electricity market with no transmission limitations the LMPs are equal Transmission constraints can segment a market, resulting in differing LMP Determination of LMPs requires the solution on an Optimal Power Flow (OPF)

3 BUS LMPS - OVERLOAD IGNORED Gen 2’s cost is $12 per MWh Gen 1’s cost is $10 per MWh Line from Bus 1 to Bus 3 is over-loaded; all buses have same marginal cost

LINE OVERLOAD ENFORCED Line from 1 to 3 is no longer overloaded, but now the marginal cost of electricity at 3 is $14 / MWh

LMPs at 7:40 AM Today Source: www.midwestmarket.org

LMPs at 7:55 AM Today

Development of Line Models Goals of this section are develop a simple model for transmission lines gain an intuitive feel for how the geometry of the transmission line affects the model parameters

Primary Methods for Power Transfer The most common methods for transfer of electric power are Overhead ac Underground ac Overhead dc Underground dc other

Magnetics Review Ampere’s circuital law:

Line Integrals Line integrals are a generalization of traditional integration Integration along the x-axis Integration along a general path, which may be closed Ampere’s law is most useful in cases of symmetry, such as with an infinitely long line

Magnetic Flux Density Magnetic fields are usually measured in terms of flux density

Magnetic Flux

Magnetic Fields from Single Wire Assume we have an infinitely long wire with current of 1000A. How much magnetic flux passes through a 1 meter square, located between 4 and 5 meters from the wire? Direction of H is given by the “Right-hand” Rule Easiest way to solve the problem is to take advantage of symmetry. For an integration path we’ll choose a circle with a radius of x.

Single Line Example, cont’d For reference, the earth’s magnetic field is about 0.6 Gauss (Central US)

Flux linkages and Faraday’s law

Inductance For a linear magnetic system, that is one where B = m H we can define the inductance, L, to be the constant relating the current and the flux linkage l = L i where L has units of Henrys (H)

Inductance Example Calculate the inductance of an N turn coil wound tightly on a torodial iron core that has a radius of R and a cross-sectional area of A. Assume 1) all flux is within the coil 2) all flux links each turn

Inductance Example, cont’d

Inductance of a Single Wire To development models of transmission lines, we first need to determine the inductance of a single, infinitely long wire. To do this we need to determine the wire’s total flux linkage, including 1. flux linkages outside of the wire 2. flux linkages within the wire We’ll assume that the current density within the wire is uniform and that the wire has a radius of r.

Flux Linkages outside of the wire

Flux Linkages outside, cont’d

Flux linkages inside of wire

Flux linkages inside, cont’d Wire cross section x r

Line Total Flux & Inductance

Inductance Simplification

Two Conductor Line Inductance Key problem with the previous derivation is we assumed no return path for the current. Now consider the case of two wires, each carrying the same current I, but in opposite directions; assume the wires are separated by distance R. To determine the inductance of each conductor we integrate as before. However now we get some field cancellation R Creates counter- clockwise field Creates a clockwise field

Two Conductor Case, cont’d Rp Direction of integration Key Point: As we integrate for the left line, at distance 2R from the left line the net flux linked due to the Right line is zero! Use superposition to get total flux linkage. Left Current Right Current

Two Conductor Inductance