REACTIVE POWER AND VOLTAGE REACTIVE POWER AND VOLTAGE CONTROL

Salvador Acha Daza, Ph. D.

IEEE Distinguished Power Lecturer

June 2014

Guide

1 MODELINGAssumptions and V, I, P, QTransformers, Transmission Lines, GeneratorsSVC’s (static VAR compensators)

2 BASICS ABOUT VOLTAGE CONTROLQ-V relationReactive power flow and incremental modelDecupled Load-FlowTap’s control and generalized controlTap’s control and generalized control

3 SENSITIVITY Q-V AND COORDINATED CONTROLSensitivity coefficientsApplication and coordinated controlQ-V CongestionRadial networks

4 VOLTAGE COLLAPSEAngle stabilityVoltage collapse

2

Modeling

3

Single phase, steady state: Voltage, Current

• AC voltage and current, f = 60 Hz• rms values: 120 V, 50 A• If current lags voltage by 30o

+

V

-

I

Loadconvention

150

v(t)

4

V

I

f

Phasor plane0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016

-150

-100

-50

0

50

100

time, sec

volta

ge V

, cu

rren

t A

i(t)

v(t)

Complex power S, P and Q

VAjS

jQPIVS

oo 000,31.196,5)3050()0120( *

*

Qf

S

IP

Q

5

P

Qf

Power plane

• cos f is the power factor• The case shows a “lagging power factor”

+

V

-

Loadconvention

R

jwL

What is P and Q?

Instantaneous power p(t):

)sin(

)cos(

)2sin()sin())2cos(1()cos()(

)cos()cos()()()(

IVQ

IVP

tIVtIVtp

tItVtitvtp mm

2,

2mm I

IV

V

rms values

12000

10000

6

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016-2000

0

2000

4000

6000

8000

10000

time, sec

insta

nta

neous p

ow

er

p(t

)

P

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016

-2000

0

2000

4000

6000

8000

10000

time, sec

pow

er,

com

ponents

I a

nd I

I

Q

Voltage and current, steady state

)()( DD

s

XxjRr

VI

Positive sequencexjr

DD XjR sVI

Assuming load and voltage as a fixed value (no use of load flow), ZLine= 0.05 + j0.5 W

Power factor effect on voltage, angle and losses

7

Power factor effect on voltage, angle and lossesNominal load voltage and current (rms values)

120.00 50.00

Voltage source angle PS QS PL QL pf(load) Losses %136.20 8.61 5321.15 4250.00 5196.15 3000.00 -0.87 2.35131.01 10.33 5920.55 2802.91 5795.55 1552.91 -0.97 2.11125.02 11.53 6125.00 1250.00 6000.00 0.00 +1.00 2.04118.57 12.07 5920.55 -302.91 5795.55 -1552.91 +0.97 2.11112.03 11.80 5321.15 -1750.00 5196.15 -3000.00 +0.87 2.35

Transformer Model

For an ideal transformer with complex relation

8

tV

V

k

k 1

'~

~

** ''~~

kmkkmk iViV

t

V

V

i

i

k

k

km

km~'

~

' *

*

Complex, real and reactive power, from k to m:

**** ~'

~~'

~~kmmkkkmkkmkkm yVVtVitViVs

mkkmmkmkkmmkkmkkm btVVgtVVgtVp sincos22

mkkmmkmkkmmkkmkkm btVVgtVVbtVq cossin22

Transmission line model

A symmetric arrangement for a balanced transmission line, positive sequence values, using p equivalent

.

9

kjkk eVV ~

mjmm eVV ~

kmkmkm jbgy

2/2/

2/1

shC

sh jBjX

y

Complex power flow from k to m

mkkm

*, )(

~shkkmkkm iiVs

kmkmmkkmkmmkkmkkm bVVgVVgVp sincos2

kmkmmkkmkmmkshkmkkm bVVgVVBbVq cossin)( 2/2

Simple synchronous machine model

Single phase model, synchronous machine no saliency:

.

asIXjVE

I

VEjX

Equivalent circuit for synchronous machine

Synchronous machine

+E-

Ia +V-

Zs = Ra + jXs

10

IajXs Synchronous machine connected to an infinite bus

Ia

V

E

Zs Iad

IaV

E

Zs Iad

Ia

V

E

Zs Iadq

Load with lagging power factor

Load with unity pf Load with leading pf

SVC characteristics.

Q-V relation

-V

Vm QB = Bmin

B = Bmax

V

1.0

11

+Vo

TCR

capacitiveinductive

B = Bmax

Q-0.2 0 +0.2

TSC (Thyristor Switched Capacitor)TCR (Thyristor Controlled Reactors)

Basics about Voltage Control

12

Reactive power flow, Incremental Model

For a complex transformer in a balanced three phase power system, reactive power:

13

mkkmmkmkkmmkkmkkm btVVgtVVbtVq cossin22

Incremental reactive power Dqkm:

t

tt

t

q

V

VV

V

q

V

VV

V

qqqqq km

m

mm

m

km

k

kk

k

kmkmm

m

kmk

k

kmkm

Approximations

Reasonable assumptions will simplify the incremental reactive power flow, like:

mkkmkk

km bVq

mkkmkm

km bVq

mkkmkkm bV

q0.1)cos(

mkmk )sin(

kmkm bg

14

mkkmk bV

kmkk

km bVV

q

kmkm

km bVV

q

kmkkm bVt

q

Substitution into the incremental form:

tVVV

qx mk

k

kmkm

0.1)cos( mk

Ohm’s law and equivalent circuit

Ohm’s circuit to study the incremental reactive power flow in a complex tap transformer:

tVVV

qx mk

k

kmkm

15

B. Stott, O. Alsac, Fast Decoupled Load Flow, IEEE Trans. On Power Apparatus and Systems, Vol. PAS-93, May-June, 1974.

Equivalent circuit for Dq in a transmission line

A similar procedure will give us an incremental circuit equivalent for a transmission line. The right hand side comes from the line charging susceptance.

kkmmkk

kmkm VBshxVV

V

qx

2/2)(

+

DVk

-

+

DVm

-

+ - + -

xkmDqkm/Vk

16

Apply an incremental model for each transformer and transmission line.• Ohm’s law and Kirchhoff's current law are used to set up a system of

equations• Solve for control actions on nodal variables and transmission elements.

Voltage control by tap transformers

Q3 Q4 Q5 Q6

54(4) (5)

17

7 RefQ1Q2

t : 17

1

654

3

2

q71q21q23

q34 q45 q56 q76q25

(1)(2)

(3)

(4) (5)

(6)

(7)

(8)

Q7

7 RefQ1Q2

Q3 Q4 Q5 Q6

t : 17

1

654

3

2

q71q21q23

q34 q45 q56 q76q25

(1)(2)

(3)

(4) (5)

(6)

(7)

(8)

Q7

Ohm´s law for every element(1) 0177171 VVqx

(2) tVVqx 122121

(3) 0322323 VVqx

(4) 0433434 VVqx

(5) 0544545 VVqx

(6) 0522525 VVqx

(7) 0655656 VVqx

(8) 0677676 VVqx

Node equations, Kirchoff´s law

node 1 12171 Qqq node 2 2252321 Qqqq node 3 33423 Qqq node 4 44534 Qqq node 5 5564525 Qqqq node 6 67656 Qqq

18

Sensitivity factors

00110000008

1000

0001100000010

100

00001100000015

10

000001000000015

1

31128.0)4(

31128.0)3(

10117.2)2(

10117.2)1(

Second column of the inverse

00000011000000

00000001110000

00000000011000

00000000001100

00000000100110

00000000000011

10000010

10000000

11000005

1000000

0100100020

100000

01100000016

10000

8

21012.06

63035.05

64981.04

68872.03

71984.02

14008.01

10117.2)8(

10117.2)7(

78988.1)6(

31128.0)5(

31128.0)4(

19

SENSITIVITY Q-V AND COORDINATED CONTROLCOORDINATED CONTROL

20

Network model and voltage-reactive power controls

1

- b2323

- b12

2

2VQ

- b13+

DV2

-1

- b2323

slack

- b12

2

2VQ

- b13

+

DV3

-

3

3V

Q

0

0

322

2323

311

1313

211

1212

VVV

qx

tVVV

qx

VVV

qx

Series Elements

(1)

(2)

(3)

slack

Nodal Balance (including shunt currents)

Node 1

Node 2

Node 3

3

3

3

3)3(2

2

23

1

13

2

2

2

2)3(2

)1(2

2

23

1

12

1

1

1

1)1(2

1

13

1

12

2

)(2

2

V

Q

V

VB

V

q

V

q

V

Q

V

VBB

V

q

V

q

V

Q

V

VB

V

q

V

q

sht

shtsht

sht

21

- slack -

Matrix equation and Data

33

22

11

33

22

11

223

113

112

)3(2

)3(2

)1(2

)1(2

23

13

12

/

/

/

0

0

/

/

/

/

200110

0)(20101

002011

11000

10100

01100

VQ

VQ

VQ

t

VV

VV

VV

Vq

Vq

Vq

B

BB

B

x

x

x

sht

shtsht

sht

Nnods = 3 Elements = 3

ConnectivityElements Nsal Nlleg R X B/2 Code

1 1 2 0.0230 0.1380 0.2710 02 1 3 0.0012 0.0150 0.0000 13 2 3 0.0230 0.1380 0.2710 0

Node type: 0 comp, -1 load, +1 generatorNode type

1 02 -13 1

22

If it is required to change the reactive power flow in a given line. Controlsavailable are: a tap changer and power injection into nodes 2 and 3. Sensitivityvalues are used to solve for an optimal change to attain the reactive flow change.

223

113

112

0

0

0498.04935.003089.33224.35764.3

9540.04956.003224.30638.35910.3

0539.05474.005764.35910.32797.3

/

/

/

t

Vq

Vq

Vq

A change in power flow

A cost function and constraints are required. To minimize control effort tochange the flow in line from node 2 to node 3. From sensitivity matrix:

33

22

11

33

22

11

223

/

/

/

0

0143.00074.000489.09540.00539.0

0074.00755.004935.04956.05474.0

000000

0498.04935.003089.33224.35764.3

/

/

/

/

VQ

VQ

VQ

VV

VV

VV

Vq

33362235322

23 // VQSVQStSV

q

23

Lagrangian and its gradient:

0///.. 3336223532223 VQSVQStSVqas

)()()()(),,(3

336

2

23532

2

232

3

33

2

2

22

21

3

3

2

2V

QS

V

QStS

V

q

V

Qk

V

Qktk

V

Q

V

Qt

2333

2222

21 )/()/()(min VQkVQktk

Cost function and constraints

Gradient must be zero for an optimal solution. We find a set of equationsto be solved for the control actions required.

0

0

0

0

///

/2

/2

2

.

/

/

3336223532223

36333

35222

321

33

22

VQSVQStSVq

SVQk

SVQk

Stk

VQ

VQ

t

24

This case requires to solve a system of linear equations

The change in controls to reduce the reactive power flow q23 in 0.2:

223

33

22

363532

363

352

321

/

0

0

0

/

/

0

200

020

002

Vq

VQ

VQ

t

SSS

Sk

Sk

Sk

The change in controls to reduce the reactive power flow q23 in 0.2:

Smin =2.0000 0 0 -3.3224

0 2.0000 0 -0.49350 0 2.0000 0.0498

-3.3224 -0.4935 0.0498 0

Dy =-0.0589-0.00870.0009-0.0354

ans = -0.2000

0354.0

0009.0

0087.0

0589.0

/

/

33

22

VQ

VQ

t The coordinated changes work to modify the required value, but control limits must be included (i. e. maximum/minimum setting must be observed).

25

Previous solution used k’s as unit values.

Different weight or importance in a control action can be handled through the k values. A change to reduce (or make larger) the control action of injected reactive power is shown by values k2 = 0.1 and k3 = 0.1

Dy =-0.0492-0.07310.0074-0.0296

ans = -0.2000

0296.0

0074.0

0731.0

0492.0

/

/

33

22

VQ

VQ

t

26

A real network might have a structure as shown, in which nodes andtransmission elements show a radial structure; considered “longitudinal”; it isnot a very robust network.

This type of network is very interesting to study control actions and magnitudeof their influence.

Weak and Radial type networks

10 114 10 114

27

104

2 3 9

15 12 15

6

7

8

13

14

104

2 3 9

15 12 15

6

7

8

13

14

15 nodes, 15 elements longitudinal network

(1)(2)

Figure shows the matrix structure for circuit study; elements andnodal information are in the tables.

Nodo PG (MW) PD (MW)

1 145

2

3

4 52

5 71

6 81.5

7 71.9

8 79.4

9 1.5

10

11 88.7

12 13.5

13

14 26.6

15 52.528

15 node system and elements, connectivity and reactance values.

Elemento Nodo Salida - Nodo Llegada x (pu)

(1) 2 - 3 0.1189

(2) 3 - 4 0.0568

(3) 5 - 7 0.0807

(4) 7 - 8 0.0451(4) 7 - 8 0.0451

(5) 4 - 10 0.1316

(6) 3 - 9 0.1458

(7) 10 - 9 0.0449

(8) 12 - 15 0.1699

(9) 12 - 13 0.0849

(10) 1 - 2 0.09303

(11) 3 – 5 0.1885

(12) 6 – 5 0.1330

(13) 11 – 10 0.0930

(14) 9 – 12 0.1885

(15) 14 - 13 0.1331

transformers

lines

29

S =Columns 1 through 8 -3.1237 -0.9899 0 0 -0.9899 -1.0824 0.6391 0-0.9899 -3.4485 0 0 -3.4485 1.8060 -1.9376 -0.0000

0 0 0 0 0 0 0 00 0 0 0 0 0 0 0

-0.9899 -3.4485 0 0 -3.4485 1.8060 -1.9376 -0.0000-1.0824 1.8060 0 0 1.8060 -3.6018 2.9980 00.6391 -1.9376 0 0 -1.9376 2.9980 -4.4065 0

0 0 0 0 0 0 0 0-0.4433 -0.1316 0 0 -0.1316 -0.6039 -1.4085 0-3.1237 -0.9899 0 0 -0.9899 -1.0824 0.6391 0-1.0513 0.6526 0 0 0.6526 0.7135 -0.4213 0.00001.0513 -0.6526 0 0 -0.6526 -0.7135 0.4213 -0.00001.0513 -0.6526 0 0 -0.6526 -0.7135 0.4213 -0.00001.6290 1.5109 0 0 1.5109 1.1920 -2.4688 0-0.4433 -0.1316 0 0 -0.1316 -0.6039 -1.4085 00.4433 0.1316 0 0 0.1316 0.6039 1.4085 0

0.0000 0.0000 0 0 0.0000 0.0000 -0.0000 -0.00000.2906 0.0921 0 0 0.0921 0.1007 -0.0595 0-0.3380 0.2098 0 0 0.2098 0.2294 -0.1354 0-0.2818 -0.5943 0 0 0.4057 0.1268 -0.0254 -0.0000-0.1398 0.0868 0 0 0.0868 0.0949 -0.0560 0.0000-0.0000 0.0000 0 0 0.0000 0.0000 -0.0000 0-0.1398 0.0868 -1.0000 0 0.0868 0.0949 -0.0560 0.0000-0.1398 0.0868 -1.0000 -1.0000 0.0868 0.0949 -0.0560 0.0000-0.1802 -0.0535 0 0 -0.0535 -0.2455 -0.5725 0-0.1515 -0.1405 0 0 -0.1405 -0.1109 0.2296 0-0.0000 -0.0000 0 0 -0.0000 -0.0000 0.0000 0-0.0966 -0.0287 0 0 -0.0287 -0.1316 -0.3070 0-0.0590 -0.0175 0 0 -0.0175 -0.0804 -0.1875 0-0.0000 -0.0000 0 0 -0.0000 -0.0000 -0.0000 -0.0000-0.0966 -0.0287 0 0 -0.0287 -0.1316 -0.3070 -1.0000

Q-V RELATION AND VOLTAGE SUPPORTSUPPORT

31

Voltage and current, steady state

)()( DD

s

XxjRr

VI

Positive sequencexjr

DD XjR sVI

Assuming load and voltage as a fixed value (no use of load flow), ZLine= 0.05 + j0.5 W

Power factor effect on voltage, angle and losses

32

Power factor effect on voltage, angle and lossesLoad voltage and current, rms values

120.00 50.00

Voltage source angle PS QS PL QL pf(load) Efficiency136.20 8.61 5321.15 4250.00 5196.15 3000.00 -0.87 2.35131.01 10.33 5920.55 2802.91 5795.55 1552.91 -0.97 2.11125.02 11.53 6125.00 1250.00 6000.00 0.00 +1.00 2.04118.57 12.07 5920.55 -302.91 5795.55 -1552.91 +0.97 2.11112.03 11.80 5321.15 -1750.00 5196.15 -3000.00 +0.87 2.35

PV curve and load characteristics

0.4

0.6

0.8

1

volta

ge V

m, p

u

P-V curve, DC circuit

33

0 0.5 1 1.5 2 2.50

0.2

power PL, pu

System’s PV curve and for a resistive load, RL = 1.5, 1.1 pu.

.

By

Salvador Acha Daza, Ph. D.dr.acha@niat.com.mx(512) 623 9953http://www.niat.com.mx/

34