Lecture 10Transformers, Generators, Load, Ybus

Professor Tom OverbyeDepartment of Electrical and

Computer Engineering

ECE 476

POWER SYSTEM ANALYSIS

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Announcements

Be reading Chapter 6. HW 3 is due now. HW 4 is 3.4, 3.10, 3.14, 3.19, 3.23, 3.60; due September 29

in class. First exam is October 11 during class. Closed book, closed

notes, one note sheet and calculators allowed

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Load Tap Changing Transformers

LTC transformers have tap ratios that can be varied to regulate bus voltages

The typical range of variation is 10% from the nominal values, usually in 33 discrete steps (0.0625% per step).

Because tap changing is a mechanical process, LTC transformers usually have a 30 second deadband to avoid repeated changes.

Unbalanced tap positions can cause "circulating vars"

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LTCs and Circulating Vars

slack

1 1.00 pu

2 3

40.2 MW

40.0 MW

1.7 Mvar

-0.0 Mvar

1.000 tap 1.056 tap

24.1 MW 12.8 Mvar

24.0 MW-12.0 Mvar

A

MVA

1.05 pu 0.98 pu

24 MW

12 Mvar

64 MW

14 Mvar

40 MW 0 Mvar

0.0 Mvar

80%A

MVA

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Phase Shifting Transformers

Phase shifting transformers are used to control the phase angle across the transformer

Since power flow through the transformer depends upon phase angle, this allows the transformer to regulate the power flow through the transformer

Phase shifters can be used to prevent inadvertent "loop flow" and to prevent line overloads.

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Phase Shifter Example 3.13

slack

Phase Shifting Transformer

345.00 kV 341.87 kV

0.0 deg 216.3 MW 216.3 MW

283.9 MW 283.9 MW

1.05000 tap

39.0 Mvar 6.2 Mvar

93.8 Mvar 125.0 Mvar

500 MW

164 Mvar 500 MW 100 Mvar

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ComED Control Center

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ComED Phase Shifter Display

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Phase Shifting Transformer Picture

230 kV 800 MVA Phase Shifting Transformer During factory

testingSource: Tom Ernst, Minnesota Power

Costs about $7 million,weighs about 1.2million pounds

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Autotransformers

Autotransformers are transformers in which the primary and secondary windings are coupled magnetically and electrically.

This results in lower cost, and smaller size and weight.

The key disadvantage is loss of electrical isolation between the voltage levels. Hence auto-transformers are not used when a is large. For example in stepping down 7160/240 V we do not ever want 7160 on the low side!

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Could it Happen Tomorrow?

Geomagnetic disturbances (GMDs) impact the power grid by causing geomagenetic induced dc currents (GICs) that can push the transformers into saturation.

Saturated transformershave high harmonics whichleads to highreactive losses andheating

Image from Ed Schweitzer June 2011 JASON Presentation

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Could It Happen Tomorrow?

• A 1989 storm caused a major blackout in Quebec. Much larger storms have occurred in the past, such as in 1859, which knocked out much of the telegraph system in the Eastern US

• A 2010 Metatech Report indicated an 1859 typeevent could destroyhundreds of EHVtransformers, cripplingour grid for months!

Metatech R-319, Figure 4.11

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Load Models

Ultimate goal is to supply loads with electricity at constant frequency and voltage

Electrical characteristics of individual loads matter, but usually they can only be estimated– actual loads are constantly changing, consisting of a large number of individual

devices– only limited network observability of load characteristics

Aggregate models are typically used for analysis Two common models

– constant power: Si = Pi + jQi

– constant impedance: Si = |V|2 / Zi

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Generator Models

Engineering models depend upon application Generators are usually synchronous machines For generators we will use two different models:

– a steady-state model, treating the generator as a constant power source operating at a fixed voltage; this model will be used for power flow and economic analysis

– a short term model treating the generator as a constant voltage source behind a possibly time-varying reactance

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Power Flow Analysis

We now have the necessary models to start to develop the power system analysis tools

The most common power system analysis tool is the power flow (also known sometimes as the load flow)– power flow determines how the power flows in a network– also used to determine all bus voltages and all currents– because of constant power models, power flow is a

nonlinear analysis technique– power flow is a steady-state analysis tool

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Linear versus Nonlinear Systems

A function H is linear if

H(11 + 22) = 1H(1) + 2H(2)

That is

1) the output is proportional to the input

2) the principle of superposition holds

Linear Example: y = H(x) = c x

y = c(x1+x2) = cx1 + c x2

Nonlinear Example: y = H(x) = c x2

y = c(x1+x2)2 ≠ (cx1)2 + (c x2)2

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Linear Power System Elements

Resistors, inductors, capacitors, independent

voltage sources and current sources are linear

circuit elements

1V = R I V = V =

Such systems may be analyzed by superposition

j L I Ij C

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Nonlinear Power System Elements

Constant power loads and generator injections are nonlinear and hence systems with these elements can not be analyzed by superposition

Nonlinear problems can be very difficult to solve,and usually require an iterative approach

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Nonlinear Systems May Have Multiple Solutions or No Solution

Example 1: x2 - 2 = 0 has solutions x = 1.414…

Example 2: x2 + 2 = 0 has no real solution

f(x) = x2 - 2 f(x) = x2 + 2

two solutions where f(x) = 0 no solution f(x) = 0

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Multiple Solution Example 3

The dc system shown below has two solutions:

where the 18 wattload is a resistiveload

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Load

Load

Load

The equation we're solving is

9 voltsI 18 watts

1 +R

One solution is R 2

Other solution is R 0.5

Load LoadR R

What is themaximumPLoad?