ECEN 615 Methods of Electric Power Systems Analysis ...overbye.engr.tamu.edu/.../sites/146/2018/11/ECEN615_Fall2018_Lect… · Lecture 22: Voltage Stability, PV and QV Curves, Geomagnetic

Lecture 22: Voltage Stability, PV and QV

Curves, Geomagnetic Disturbances

ECEN 615Methods of Electric Power Systems Analysis

Prof. Tom Overbye

Dept. of Electrical and Computer Engineering

Texas A&M University

[email protected]

mailto:[email protected]

Announcements

• Homework 5 is due today

• Homework 6 is due on Tuesday Nov 27

• Read Chapters 3 and 8 (Economic Dispatch and

Optimal Power Flow)

2

Small Disturbance Voltage Collapse

• At constant frequency (e.g., 60 Hz) the complex power

transferred down a transmission line is S=VI*

– V is phasor voltage, I is phasor current

– This is the reason for using a high voltage grid

• Line real power losses are given by RI2 and reactive

power losses by XI2

– R is the line’s resistance, and X its reactance; for a high

voltage line X >> R

• Increased reactive power tends to drive down the

voltage, which increases the current, which further

increases the reactive power losses

3

PowerWorld Two Bus Example

slack

Bus 1 Bus 2

x=0.2

x=0.20.933 pu

MW 150

Mvar 50

Commercial power flow

software usually auto

converts constant power

loads at low voltages;

set these fields to zero

to disable this

conversion

4

Power Flow Region of Convergence

Convergence

regions with

P=100 MW,

Q=0 Mvar

5

Load Parameter Space Representation

• With a constant power model there is a maximum

loadability surface, S

– Defined as point in which the power flow Jacobian is

singular

– For the lossless two bus system it can be determined as

2

LL

P 1Q B 0

B 4

6

Load Model Impact

• With a static load model regardless of the voltage

dependency the same PV curve is traced

– But whether a point of maximum loadability exists

depends on the assumed load model

• If voltage exponent is > 1 then multiple solutions do not exist

(see B.C. Lesieutre, P.W. Sauer and M.A. Pai “Sufficient

conditions on static load models for network solvability,”NAPS

1992, pp. 262-271)

7

slack

Bus 1 Bus 2

x=0.2

x=0.2

0.943 pu

MW 133

Mvar 44

Change load to

constant impedance;

hence it becomes a

linear model

ZIP Model Coefficients

• One popular static load model is the ZIP; lots of

papers on the “correct” amount of each type

Table 7 from A, Bokhari, et. al., “Experimental Determination of the ZIP Coefficients for Modern Residential, Commercial, and Industrial

Loads,” IEEE Trans. Power Delivery, June. 2014

Table 1 from M. Diaz-Aguilo, et. al., “Field-Validated Load Model for the Analysis of CVR in Distribution Secondary Networks: Energy

Conservation,” IEEE Trans. Power Delivery, Oct. 2013

8

Application: Conservation Voltage Reduction (CVR)

• If the “steady-state” load has a true dependence on

voltage, then a change (usually a reduction) in the

voltage should result in a total decrease in energy

consumption

• If an “optimal” voltage could be determined, then this

could result in a net energy savings

• Some challenges are 1) the voltage profile across a

feeder is not constant, 2) the load composition is

constantly changing, 3) a decrease in power

consumption might result in a decrease in useable

output from the load, and 4) loads are dynamic and an

initial increase might be balanced by a later increase 9

Determining a Metric to Voltage Collapse

• The goal of much of the voltage stability work was to

determine an easy to calculate metric (or metrics) of

the current operating point to voltage collapse

– PV and QV curves (or some combination) can determine such

a metric along a particular path

– Goal was to have a path independent metric. The closest

boundary point was considered,

but this could be quite misleading

if the system was not going to

move in that direction

– Any linearization about the current operating point (i.e., the

Jacobian) does not consider important nonlinearities like

generators hitting their reactive power limits 10

Determining a Metric to Voltage Collapse

• A paper by Dobson in 1992 (see below) noted that at a

saddle node bifurcation, in which the power flow

Jacobian is singular, that

– The right eigenvector associated with the Jacobian zero

eigenvalue tells the direction in state space of the voltage

collapse

– The left eigenvector associated with the Jacobian zero

eigenvalue gives the normal in parameter space to the

boundary S. This can then be used to estimate the minimum

distance in parameter space to bifurcation.

I. Dobson, “Observations on the Geometry of Saddle Node Bifurcation and Voltage Collapse in Electrical Power

Systems,” IEEE Trans. Circuits and Systems, March 1992 11

Determining a Metric to Voltage Collapse Example

• For the previous two bus example we had

L L

sin

cos

cos sin( , )

sin cos

Singular when cos

So consider , . , . , then P = 3.317, Q =1.400

.

. .

L

2

L

P BV 0

Q BV BV 0

BV BV

BV B 2BV

2V 1 0

B 10 V 0 6 33 56

5 5 528

3 317 3 667

J

J

12

Determining a Metric to Voltage Collapse Example

• Calculating the right and left eigenvalues

associated with the zero eigenvalue we get

.

. .

. .,

. .

5 5 528

3 317 3 667

0 742 0 553

0 671 0 833

J

v w

13

Quantifying Power Flow Unsolvability

• Since lack of power flow convergence can be a major

problem, it would be nice to have a measure to

quantify the degree of unsolvability of a power flow

– And then figure out the best way to restore solvabiblity

• T.J. Overbye, “A Power Flow Measure for Unsolvable

Cases,” IEEE Trans. Power Systems, August 1994

14


15


• To setup the problem, first consider the power flow

iteration without and with the optimal multiplier

1

1

1

( ) ( )

With the optimal multiplier we are minimizing

1F( ) ( ) ( )

2

When there is a solution 1 and the cost function

goes to zero

k k k

k k k

Tk k k k k

x x Δx

Δx J x f x S

x f x Δx S f x Δx S

16


• However, when there is not solution the standard

power flow would diverge. But the approach with the

optimal multiplier tends to point in the direction of

minimizing F(xk+1). That is,

17

k k k

k k 1 k

F( ) = ) - ( )

Also

= - ( ) ) -

where how far to move in this direction is

limited by .

T

x f(x S J x

x J x f(x S


• The only way we cannot reduce the cost function

some would be if the two directions were

perpendicular, hence with a zero dot product. So

18

k k k 1 k

k k

= ) - ( ) ( ) ) -

= ) - ) -

(provided the Jacobian is not singular). As we approach singularity

this goes to zero. Hence we converge to a poi

Tk k

k k

T

k

F

x x f(x S J x J x f(x S

x x

f(x S f(x S

x

nt on the boundary

, but not necessarilty at the closest boundary point. S


• The left eigenvector associated with the zero

eigenvalue of the Jacobian (defined as wi*) is

perpendicular to S (as noted in the early 1992 Dobson

paper)

• We can get the closest point on the S just by iterating,

updating the S Vector as

(here S is the initial power injection, xi* a boundary

solution)

• Converges when

1 i* i* i* = + [( ( ) - ) ] i S S f x S w w

19

i*( ( ) - )i f x S


20

If S were flat then w is

parallel to wm

Challenges

• The key issues is actual power systems are quite

complex, with many nonlinearities. For example,

generators hitting reactive power limits, switched

shunts, LTCs, phase shifters, etc.

• Practically people would like to know how far some

system parameters can be changed before running into

some sort of limit violation, or maximum loadability.

– The system is changing in a particular direction, such as a

power transfer; his often includes contingency analysis

• Line limits and voltage magnitudes are considered

– Lower voltage lines tend to be thermally constrained

• Solution is to just to trace out the PV or QV curves 21

PV and QV Analysis in PowerWorld

• Requires setting up what is known in PowerWorld as

an injection group

– An injection group specifies a set of objects, such as

generators and loads, that can inject or absorb power

– Injection groups can be defined by selecting Case

Information, Aggregation, Injection Groups

• The PV and/or QV analysis then varies the injections

in the injection group, tracing out the PV curve

• This allows optional consideration of contingencies

• The PV tool can be displayed by selecting Add-Ons,

PV

22

PV and QV Analysis in PowerWorld: Two Bus Example

• Setup page defines the source and sink and step size

23


• The PV Results Page does the actual solution

– Plots can be defined to show the results

– Other Actions, Restore initial state restores the pre-study state

24

Click the Run button

to run the PV analysis;

Check the Restore

Initial State on

Completion of Run to

restore the pre-PV

state (by default it is

not restored)


25

PV and QV Analysis in PowerWorld: 37 Bus Example

slack

SLACK138

PINE345

PINE138

PINE69

PALM69

LOCUST69

PEAR69

PEAR138

CEDAR138

CEDAR69

WILLOW69

OLIVE69

BIRCH69

PECAN69

ORANGE69

MAPLE69

OAK69

OAK138

OAK345

BUCKEYE69

APPLE69

WALNUT69

MAPLE69

POPLAR69

PEACH69

ASH138

PEACH138

SPRUCE69

CHERRY69

REDBUD69

TULIP138

LEMON69

LEMON138

ELM138 ELM345

20%A

MVA

57%A

MVA

73%A

MVA

38%A

MVA

58%A

MVA

46%A

MVA

46%A

MVA

A

MVA

67%A

MVA

15%A

MVA

27%A

MVA

80%A

MVA

34%A

MVA

69%A

MVA

21%A

MVA

79%A

MVA

12%A

MVA

49%A

MVA

15%A

MVA

18%A

MVA

A

MVA

14%A

MVA

56%A

MVA

79%A

MVA

73%A

MVA

71%A

MVA

85%A

MVA

1.01 pu

0.98 pu

1.01 pu

1.03 pu

0.96 pu

0.96 pu

0.99 pu

0.97 pu

0.97 pu

0.95 pu

0.97 pu

0.96 pu

0.97 pu

0.96 pu

0.99 pu

0.90 pu

0.90 pu

0.95 pu

0.787 pu

0.78 pu

0.82 pu

0.90 pu

0.90 pu

0.93 pu

0.91 pu

0.93 pu 0.92 pu

0.94 pu

0.93 pu0.93 pu

1.02 pu

69%A

MVA

PLUM138

16%A

MVA

0.99 pu

A

MVA

0.95 pu

70%A

MVA

877 MW 373 Mvar

24 MW

0 Mvar

42 MW

12 MvarMW 180

130 Mvar

60 MW

16 Mvar

14 MW 6 Mvar

MW 150

60 Mvar

64 MW

14 Mvar

44 MW 12 Mvar

42 MW

4 Mvar

49 MW

0 Mvar

67 MW

42 Mvar

42 MW

12 Mvar

MW 10

5 Mvar

26 MW

17 Mvar

69 MW

14 Mvar

MW 20

40 Mvar

26 MW

10 Mvar

38 MW

15 Mvar 14.3 Mvar 21 MW

6 Mvar

66 MW

46 Mvar 297 MW

19 Mvar

Mvar 0.0

126 MW

8 Mvar

89 MW 7 Mvar

27 MW

7 Mvar 16 MW

0 Mvar

7.8 Mvar

6.0 Mvar

11.5 Mvar

26.0 Mvar

6.8 Mvar

16.3 Mvar

MW 106

60 Mvar

22 MW

9 Mvar

MW 150

60 Mvar

19 MW

3 Mvar

MW 16

26 Mvar

35 MW

12 Mvar

Total Losses: 99.11 MW

80%A

MVApu 1.000

tap0.9812

Load Scalar:1.00

90%A

MVA

86%A

MVA

90%A

MVA

89%A

MVA

88%A

MVA

94%A

MVA

94%A

MVA

96%A

MVA

97%A

MVA

104%A

MVA

115%A

MVA217%A

MVA

130%A

MVA

136%A

MVA

103%A

MVA

103%A

MVA

325%A

MVA

132%A

MVA

113%A

MVA

114%A

MVA

122%A

MVA

111%A

MVA

112%A

MVA

102%A

MVA

26

Usually other limits also need to be considered in

doing a realistic PV analysis

High-Impact, Low-Frequency Events

• Growing concern to consider what the NERC calls

calls High-Impact,

Low-Frequency Events

(HILFs); others call them

black sky days

– Large-scale, potentially long duration blackouts

– HILFs identified by NERC

were 1) a coordinated cyber,

physical or blended attacks, 2) pandemics, 3)

geomagnetic disturbances (GMDs), and 4) HEMPs

• The next several slides will consider GMDs and

HEMPs

Image Source: NERC, 2012

27

Geomagnetic Disturbances (GMDs)

• GMDs are caused by solar corona mass ejections

(CMEs) impacting the earth’s magnetic field

• A GMD caused a blackout in 1989 of Quebec

• They have the potential to severely disrupt the

electric grid by causing quasi-dc geomagnetically

induced currents (GICs) in the high voltage grid

• Until recently power engineers had few tools to

help them assess the impact of GMDs

• GMD assessment tools are now moving into the

realm of power system planning and operations

engineers; required by NERC Standards (TPL

007-1, 007-2) 28

Earth’s Magnetic Field

29Image Source: Wikepedia

The earth’s

magnetic

field is

usually

between

25,000 and

65,000 nT

Earth’s Magnetic Field Variations

• The earth’s magnetic field is constantly changing,

though usually the variations are not significant

– Larger changes tend to occur closer to the earth’s magnetic

poles

• The magnitude of the variation at any particular location

is quantified with a value known as the K-index

– Ranges from 1 to 9, with the value dependent on nT variation

in horizontal direction over a three hour period

– This is station specific; higher variations are required to get a

k=9 closer to the poles

• The Kp-index is a weighted average of the individual

station K-indices; G scale approximately is Kp - 430

Space Weather Prediction Center has an Electric Power Dashboard

www.swpc.noaa.gov/communities/electric-power-community-dashboard31

GMD and the Grid

• Large solar corona mass ejections (CMEs) can

cause large changes in the earth’s magnetic field

(i.e., dB/dt). These changes in turn produce a

non-uniform electric field at the surface

– Changes in the magnetic flux are usually expressed in

nT/minute; from a 60 Hz perspective they are almost dc

– 1989 North America storm produced

a change of 500 nT/minute, while a

stronger storm, such as the ones in

1859 or 1921, could produce

2500 nT/minute variation

– Storm “footprint” can be continental in scale

32

Solar Cycles

• Sunspots follow an 11 year cycle, and have

been observed for hundreds of years

• We're in solar cycle 24 (first numbered cycle

was in 1755); minimum was in 2009,

maximum in 2014/2015

33Images from NASA, NOAA

But Large CMEs Are Not Well Correlated with Sunspot Maximums

The large

1921 storm

occurred

four years

after the

1917

maximum

34

July 2012 GMD Near Miss

• In July 2014 NASA said in July of 2012 there

was a solar CME that barely missed the earth

– It would likely have

caused the largest

GMD that we have

seen in the last 150

years

• There is still lots of

uncertainly about

how large a storm

is reasonable to

consider in electric utility planning Image Source: science.nasa.gov/science-news/science-at-nasa/2014/23jul_superstorm/ 35

Overview of GMD Assessments

Image Source: http://www.nerc.com/pa/Stand/WebinarLibrary/GMD_standards_update_june26_ec.pdf

The two key concerns from a big storm are 1) large-scale blackout

due to voltage collapse, 2) permanent transformer damage due to

overheating

In is a quite interdisciplinary problem

Starting Here

36

Geomagnetically Induced Currents (GICs

• GMDs cause slowly varying electric fields

• Along length of a high voltage transmission line,

electric fields can be modeled as a dc voltage

source superimposed on the lines

• These voltage sources

produce quasi-dc

geomagnetically induced

currents (GICs) that are

superimposed on the ac

(60 Hz) flows

37

37

GIC Calculations for Large Systems

• With knowledge of the pertinent transmission

system parameters and the GMD-induced line

voltages, the dc bus voltages and flows are found

by solving a linear equation I = G V (or J = G U)

– J and U may be used to emphasize these are dc values,

not the power flow ac values

– The G matrix is similar to the Ybus except 1) it is

augmented to include substation neutrals, and 2) it is just

resistive values (conductances)

• Only depends on resistance, which varies with temperature

– Being a linear equation, superposition holds

– The current vector contains the Norton injections

associated with the GMD-induced line voltages 38

GIC Calculations for Large Systems

• Factoring the sparse G matrix and doing the

forward/backward substitution takes about 1

second for the 60,000 bus Eastern Interconnect

Model

• The current vector (I) depends upon the assumed

electric field along each transmission line

– This requires that substations have correct geo-

coordinates

• With nonuniform fields an exact calculation would

be path dependent, but just a assuming a straight

line path is probably sufficient (given all the other

uncertainties!)39

Four Bus Example (East-West Field)

,3

150 volts93.75 amps or 31.25 amps/phase

1 0.1 0.1 0.2 0.2GIC PhaseI

The line and transformer resistance and current values are per phase

so the total current is three times this value. Substation grounding

values are total resistance. Brown arrows show GIC flow.

40

slack

Substation A with R=0.2 ohm Substation B with R=0.2 ohm

765 kV Line

3 ohms Per Phase

High Side of 0.3 ohms/ PhaseHigh Side = 0.3 ohms/ Phase

DC = 28.1 VoltsDC = 18.7 Volts

Bus 1 Bus 4Bus 2Bus 3

Neutral = 18.7 Volts Neutral = -18.7 Volts

DC =-28.1 Volts DC =-18.7 Volts

GIC Losses = 25.5 Mvar GIC Losses = 25.4 Mvar

1.001 pu 0.999 pu 0.997 pu 1.000 pu

GIC/Phase = 31.2 Amps

GIC Input = -150.0 Volts

Case name is GIC_FourBus

Four Bus Example GIC G Matrix

41

1

118.75 15 0 10 0 0

18.75 0 15 0 10 0

28.12 10 0 11 1 150

28.12 0 10 1 11 150

U G J

GICs, Generic EI, 5 V/km East-West

42

GICs, Generic EI, 5 V/km North-South

43

Determining GMD Storm Scenarios

• The starting point for the GIC analysis is an

assumed storm scenario; sets the line dc voltages

• Matching an actual storm can be complicated, and

requires detailed knowledge of the geology

• GICs vary linearly with the assumed electric field

magnitudes and reactive power impacts on the

transformers is also mostly linear

• Working with space weather community to

determine highest possible storms

• NERC proposed a non-uniform field magnitude

model that FERC has partially accepted44

Electric Field Linearity

• If an electric field is assumed to have a uniform

direction everywhere (like with the current NERC

model), then the calculation of the GICs is linear

– The magnitude can be spatially varying

• This allows for very fast computation of the impact of

time-varying functions (like with the NERC event)

• PowerWorld now provides support for loading a

specified time-varying sequence, and quickly

calculating all of the GIC values

45

Overview of GMD Assessments

Image Source: http://www.nerc.com/pa/Stand/WebinarLibrary/GMD_standards_update_june26_ec.pdf

The two key concerns from a big storm are 1) large-scale blackout

due to voltage collapse, 2) permanent transformer damage due to

overheating

In is a quite interdisciplinary problem

Next we go here

46

Documents

ECEN 615 Methods of Electric Power Systems Analysis ...overbye.engr.tamu.edu/.../sites/146/2018/11/ECEN615_Fall2018_Lect… · Lecture 22: Voltage Stability, PV and QV Curves, Geomagnetic