543© Springer-Verlag London 2015S. Corsi, Voltage Control and Protection in Electrical Power Systems, Advances in Industrial Control, DOI 10.1007/978-1-4471-6636-8

Appendix

Appendix A

Synchronous Machine Ideal Model

The three-phase electrical machine can be schematised [A.1] with three windings distributed along the stator circumference at 120° apart from each other. The fourth winding determining the magnetic field is solid with the rotor. The three stator windings and the field winding are respectively named “a”, “b”, “c” and “f ”.

Two reference axes, solid with the rotor, are called the “direct” (“d”) and “quadra-ture” (“q”) axes. The angle θ is between the direct axis and the phase “a” axis.

• The direct (or polar) d-axis is coincident with the f-axis, that is, with the mag-netic field produced by the rotor.

• The quadrature (or interpolar) q-axis is 90° ahead of d.

Under the hypotheses to overlook the damping windings, magnetic saturations and iron losses, the instantaneous voltage at each winding can be computed as follows:

where R, I and Ψ are the winding resistance, current and flux linkage. Solving this equation requires knowledge of the auto and mutual inductances and their depen-dence on angle θ( t).

With Ω( t) being the rotor angular speed (nominal value ΩN),

,d

V RIdt

Ψ= +

00

( ) ( ) .t

t dϑ γ γ ϑ= Ω +∫

544 Appendix

The Park transform allows us to rewrite the equations referring to the a-, b-, and c-axes by taking as new references the d- and q-axes solid with the rotor.

From here, the Park transform result is shown as:

(A.1)

where Id and Iq are positive if they determine a demagnetising effect (the synchro-nous maching operates as a generator with exiting currents).

Overlooking armature winding resistence R and representing the operator d/dt by “p”, the “per unit” equations become

The terms p/ΩN consider only flux variations; therefore, they are null at steady state and numerically negligible during transients because they are divided by ΩN.

As a consequence, the generator terminal voltage amplitude Vm substantially de-pends on the magnetic flux and the rotor speed:

The exciter fixes the flux while the term Ω/ΩN significantly contributes to voltage value. Under normal operating conditions, Ω/ΩN ≈ 1; therefore, in these cases the exciter acquires more roles in imposing the V value:

Returning to Eq. (A.1) and considering the Ψd, Ψq, Ψf dependence from the auto and mutual inductances, eliminating Ψf by substitution results in the following:

,

,

,

dd q d

qq d q

ff f f

dV RI

dtd

V RIdt

dV R I

dt

Ψ= −ΩΨ + +

Ψ= ΩΨ + +

Ψ= +

,

.

d q dN N

q d qN N

pV

pV

Ω= − Ψ + ΨΩ ΩΩ= Ψ + ΨΩ Ω

2 2 .m d qV V V= +

,

.

d q

q d

V

V

≈ −Ψ

≈ Ψ

545Appendix

where transfer functions A( s), B( s), Ld( s), Lq( s) depend on machine-building char-acteristics.

The per-unit model becomes

where AN is alternator apparent nominal power; *fP is exciting power value deter-

mining nominal stator voltage at no-load, and operating point on the air-gap linear characteristic. Moreover:

• a(0) = b(0) = 1;• Ld(0) = Xd ( direct axis synchronous reactance);• Lq(0) = Xq ( quadrature axis synchronous reactance).

Starting from this introductory and simplified model, the reader can now intuitively understand its possible extension to less simplified representations.

General, Linearised Model

A general alternator model, also valid when including the damping circuits on the d- and q-axes, is here after shown:

(A.2)

These equations also show the alternator subtransient constants on both the direct and quadrature axes. They take into account the damping windings and massive

( ) ( ) ,

( ) ,

( ) ( ) ,

d f d d

q q q

f f d

A s V L s I

L s I

I B s V sA s I

Ψ = −

Ψ = −

= +

( )*

( ) ( ) ,

( ) ,

/( ) ,

( )

d f d d

q q q

N ff f

N d

a s V L s I

L s I

A P sI b s V

a s I

Ψ = −

Ψ = −

= +Ω

( )( )( )( )( )( )0 0 0 0

0

0 0 0 0

1 11

1 1 1 1

1,

1

, , , .

,d dad f d d q

d d d d

qq q q d

q

q qd dd d q q d d q q

d q d q

sT sTsTV X I V

sT sT sT sT

sTX I V

sT

T TT TX X X X X X X X

T T T T

′ ′′+ ++Ψ = − =

′ ′′ ′ ′′+ + + +′+

Ψ = − = −′+

′ ′′′ ′′′ ′ ′′ ′′= = = =

′ ′ ′′ ′′

546 Appendix

rotor effects. On the direct axis the impact is negligeble due to the dominant effect of the field winding, whereas on the quadrature axis the damping winding repre-sents the sole dynamic effect.

Neglecting the dT ′′ and d0T ′′ subtransient time constants, the (A.2) become

(A.3)

Because Vd and Vq are the components on the d- and q-axes of the Vm vector (hav-ing constant amplitude during a sinusoidal regime) solid with the rotor, then Va, Vb, Vc represent the Vm vector projections with respect to the a, b, c phase axes, respectively.

Analogously, for im and Ψ. This means:

Typical values of the synchronous machine are given in Table A.1:

Generator Operating on a Large Power System

We refer to the case represented in Fig. A.1:The scheme shows a generator connected through a transformer and an equivalent

line to an “infinite bus” characterised by infinite ( SSC = ∞) apparent power and con-stant voltage VR. This schematisation is corrently used to analyse the synchronous machine dynamics. In evidence is the generator voltage control loop that operates in parallel to the turbine speed regulator.

The overal reactance connecting the generator to the grid is obtained combining the transformer reactance XT with the equivalent line reactance XL: Xe = XT + XL.

The synchronous machine Eq. (A.3) represent the generator model, Park trans-formed. The electrical connection between the alternator and the infinite bus require new equations, here after introduced.

( )( )( )

( )( )

( )( )

11,

'1 1 10 0 0

.1 0

V X X IsT f d d ddV V X I X Iq f d d d dsT sT sTd d d

X X Iq q qV X Iq qd

sTq

− − ′+ ′= − = − ′

+ +′ ′ +

− ′= + ′

+ ′

( )( )( )( )

*

,

,

,

activepower,

reactivepower.

m d q

m d q

m m d q d q

d d q q

d d q q

v V jV

i I jI

A v i V jV I jI P jQ

P V I V I

Q V I V I

= +

= +

= = + − = +

= + =

= − =

547Appendix

The angle difference between the Rv phasor (solid to the infinite bus rotor: ( δR = 0.0) and the q-axis of the generator rotor is called δ (Fig. A.2).

The electrical link between the two generators is described by the following equations:

Table A.1 Synchronous machine typical parameter valuesAxis Parameter Turbo alternator Hydraulic generatorDirect ( d) Xd 1.9 p-u. 1.1 p-u.

dX ′ 0.3 p.u. 0.35 p.u.

dX ′′ 0.25 p.u. 0.33 p.u.

0dT ′ 7–10 s 6 s

dT ′ 1.1 s 1.9 s

0dT ′′ 0.01 s 0.08 s

dT ′′ 0.008 s 0.07 s

Quadrature ( q) Xq 1.7 p.u. 0.7 p.u.

qX ′ 0.25 p.u. 0.33 p.u.

0qT ′ 0.25 s 0.15 s

qT ′ 0.04 s 0.07 s

Fig. A.1 Generator connected to a prevailing grid (infinite bus) with AVR in closed loop

δ=

Fig. A.2 Phasor diagram of the infinite bus rotor with respect to the d- and q- axes of the alternator rotor

δ

q

d

R

ΩΩN

ῡ

548 Appendix

Therefore,

The generator voltage amplitude is 2 2m d qV V V= + , whereas Vf is provided by the

voltage regulator AVR having μ( p) as the control function.The generator electrical power is given by the real part of the product:

while the generator reactive power is given by the imaginary part of the product:

All the above equations related to the link with the equivalent grid provide a non-linear model.

Mechanical Equations

Moving to the mechanical part of the process, where Pm represents the motor power from the turbine, the link between the rotor speed and the accelerating power is

(A.4)

( ),

cos sin.

m R e m

q R R dm R R mm

e e e

v v jX i

V V j V Vv v v vi j

jX X X

δ δ= +

− + −− −= = =

cos sin, .q R R d

d qe e

V V V VI I

X X

δ δ− −= =

* ,e e m m d d q qP v i V I V I= ℜ = +

* .e m m m q d d qQ v i V I V I= ℑ = −

( )

( )

2

1,

,

1,

m eN m

MNm

N

N

P PpT

JT

A

pδ

Ω= −

Ω

Ω=

= Ω −Ω

549Appendix

where:Tm = at no-load, generator starting time with the nominal mechanical torque;J = moment of inertia of rotating masses (generator rotor + turbine shaft);ΩΜΝ = mechanical nominal angular speed while ΩN is electrical nominal

angular speed;δ = angle given by integral of speed difference between generator and pre-

vailing grid rotating at ΩN.

The above electrical and mechanical equations together describe the block diagram in Fig. A.3. Examining the scheme, the system alternator-grid puts in evidence two control loops:

• The voltage loop operating through the voltage regulator;• The electromechanical nature loop, with dynamics determined by the above me-

chanical equations characterised by two series integrators.

As can be seen, these two control loops interact with one another.

Fig. A.3 Block representation of generator model in Fig. A.1

= +

_=

_=

550 Appendix

The Linearised Model

The considered nonlinear system, linearised by imposing small variations around an operating point, and therefore representable with transfer functions by substituting “p” for the complex variable “s”, is provided by Fig. A.4:

In fact, from § A.1, the alternator-grid linearised model is given by Eq. (A.3):

(A.5)

In the field of interest, high frequency phenomena are analysed; therefore, Eq. (A.5) can be further approximated as follows:

(A.6)

Moreover, because X'd ≈ X'q ≈ Xi, (A.6) becomes

0 0

0

1,

1 1

1.

1

f dq d d

d d

qd q q

q

V sTV X I

sT sT

sTV X I

sT

′+= −

′ ′+ +′+

=′+

''0

'

fq d d

d

d q q

VV X I

sT

V X I

≅ −

≅

Fig. A.4 Block diagram of the linear model of the generator in Fig. A.1

551Appendix

(A.7)

Therefore,

This equation corresponds to Fig. A.5, representing the equivalent scheme of the alternator within the frequency field of interest.

As said, equations that model the electrical link between the two system buses indicate the nonlinear relationships:

Moreover, at the VR bus:

''0

;

fq i d q i d

d

d i q

VV X I e X I

sT

V X I

≅ − = −

≅

( ) .m d q q i d q i i mv V jV je jX I jI e jX i′= + = − + = −

( )( )

( )sin cos

,

cos cos sinsin.

q R Rm R i Rm

e i e i e

q R q R R dRd q

i e i e e e

je V jVv v e vi

jX j X X j X X

e V V V V VVI jI j j

X X X X X X

δ δ

δ δ δδ

− +′− −= = =

+ +

− −′ −+ = + = +

+ +

0

2

* ,

, ,

,

sin cos, .

j j

i R R R R

j

R

R

t e

R R R

R m

q q

q

q q

P jQ v

e je e e v jV V e V

e e VP jQ V j

X X

e V e V VP Q

X X

iδ

δ

δ δ

−

+ =

′ ′= = = = =

′ −+ =

+

′ ′ −= =

Fig. A.5 Equivalent scheme of the alternator ( left) grid interconnected ( right)

552 Appendix

Without line losses, the active power equation is the same for both the extreme buses, while the expression of reactive power only refers to the Q entering into the infinite bus. In fact, the Q delivered by the generator internal bus differs by the amount Im2X due to the reactive power absorbed by the line.

In general, referring to Fig. A.5, the delivered power and generator voltage are given by nonlinear equations:

These nonlinear equations link the electromechanical and voltage loops.Figure A.4 shows the linearised links obtained by differentiating the above Pe

and Vm equations. They are:

with

From these results it is evident that linearisation based on sensitivity gives propor-tional coefficients and not transfer functions, as was preliminarily introduced in the Fig. A.4 links.

Therefore, based on the approximations used, the linear model of the system is represented in Fig. A.6, which includes the speed regulator and additional stabilis-ing feedback, discussed in Chap. 3.

The scheme clearly evidences the two loops’ interaction due to the h and h1 blocks:

• A variation in δ determines a ∆Vm and, through the voltage regulator, a ∆Vf;• A variation in qe′ determines a ∆P and, through the electromechanical loop, a ∆δ.

The electromechanical and voltage loops are coupled, unless parameters h and h1 are zero. This happens if δ0 = 0 (that is, Pe

0 = 0) because under this condition,

( ) ( ), , , , , .q R m q RP f e V V g e Vδ δ′ ′= =

1 2, .q m qP K h e V h h eδ δ∆ = ∆ + ∆ ∆ = ∆ + ∆′ ′

0 0

0 01 2

, ,

, .

e e e e

m m m m

qP P P P

m m

qV V V V

P PK h

e

V Vh h

e

δ

δ

= =

= =

∂ ∂ = = ∆ ∆ ′

∂ ∂ = = ∆ ∆ ′

0 0 0 0, .d q mi i V V= =

553Appendix

Fig. A.6 Block diagram of linear model of power station in Fig. A.1

Therefore,

Moreover,

0 00, 0,d qV i= =

( )0

0sin0.

e e

R

q P P

VPh

e X

δ

=

∂= = =

∆ ′

( )0

( ) ( ) ,

( ) ( ) ,

sin.

m q f d d

m q f d d

q Rd

e

V V a s V X s I

V V a s V X s I

V VI

X

δ δ

= = −

∆ = ∆ = ∆ − ∆

∆ − ∆∆ =

554 Appendix

∆Vm being independent from ∆δ determines that h1 = 0.To sum up, with δ0 = 0:

• Active power is zero;• ∆Pe is independent from ∆Vf , and h = 0;• ∆Vm is independent from ∆δ and h1 = 0.

Conversely, with 0 0eP∆ ≠ , the two control loops interact unless the voltage control loop is open. In fact, under manual voltage control, Vf is a constant value, therefore it is independent from ∆δ.

Analogously, with constant Vf , no contribution comes to Pe from the voltage loop.

Reference

A.1. Kimbark EW (1956) Power system stability, vol 3. Wiley, New York

555

Index

© Springer-Verlag London 2015S. Corsi, Voltage Control and Protection in Electrical Power Systems, Advances in Industrial Control, DOI 10.1007/978-1-4471-6636-8

AAncillary services 248, 297, 298, 468Angle stability 282, 290, 302, 319, 320, 322,

330, 331, 339, 341Automatic real-time control 439Automatic Regulation of Reactive

Resources 471Automatic voltage regulator (AVR) 13, 20,

22, 26, 28–30, 42, 86, 87, 90–95, 234, 243, 249, 299, 306, 330

of generator stator edges 90, 91, 92, 93, 94, 95

BBenefits

voltage-VAR control, 302–304Bifurcation Analysis 389Block diagrams 24, 101, 125, 141, 217, 350,

477–479of PCVR control functions 477, 478, 479

Brazilian voltage control 254

CCapital costs 300, 301, 308China voltage control studies and

applications 260Closed loop control 60, 68, 76Continuation method 405Continuous Control Devices 20–23Control apparatuses 166Control effort 8, 82, 166, 169, 184, 186, 189,

195, 226, 228, 245, 257, 263, 265, 304, 442, 443, 470, 472, 474, 478, 537

Control Functions and Logics 481, 484, 518Control margin 43, 49, 83, 98, 106, 166, 171,

187, 219, 226, 241, 256, 270, 520Control parameters 91, 92, 148, 152, 180,

181, 212, 239, 320, 375, 520

Control schemes 46, 79, 148, 149, 468, 473, 503

PCVR basic 475UPFC 149, 150

Coordinated voltage regulation (CVR) 161Coordinated voltage regulation and

protection 470–472, 489Costs 44, 87, 228, 246, 299–301, 413

generation, 299transmission, 301

DDesign of SVR control parameters 180Distributed generators (DG) 465, 468, 479Distribution dispatching centres (DDC) 465,

493Distribution smart grids 82, 465Dynamic performance 23, 30, 38, 90, 157,

167, 465

EEconomic recognition 298, 311, 471Electrical power system 32, 41, 118, 133, 162,

321, 322, 397, 399stability, 321, 322

Electromechanical oscillations 270, 275, 330, 331, 339

Examples of economic benefits provided by SVR-TVR 299

Examples of pilot nodes and areas selections 196, 197

FFiscal meter 311, 315, 316French voltage control 233, 234, 240Frequency control 320, 321, 368, 431Functional performance of SVR-TVR based

indicators 443

556 Index

GGenerator tripping 84, 257, 259, 282, 290,

397, 466tests on 290

Grid losses minimisation 166, 187, 501

HHierarchical control 163, 166, 242, 377, 465High side voltage regulator (HSVR) 108, 112,

113, 115, 116, 118, 309High voltage grid control 145

IIdentification methodologies 462Identification of Saddle-Node

Bifurcation 319, 389, 394, 396, 397Indicator Based On Grid Area Reactive Power

Injection 450, 451Indicators Based On Thevenin Equivalent

Identification 452Islanded grid control 475Italian hierarchical voltage control 242–244

JJacobian singular values 406

LLine drop compensation 87, 90, 97, 101–106,

116, 156, 213, 224, 267simplified feedback 105, 106

Line losses 6, 10, 516, 517Line opening 282, 285, 331, 512, 534, 537Load increase and equilibrium points speed

along VP curve 350, 375, 528Load shedding 11, 82, 302, 498, 508, 509,

512, 518–520, 522, 528, 540criteria 519, 520

Load variation 69, 199, 252, 270, 531Long and short term phenomena 243, 321

MManual control 68, 71, 84, 85

voltage-reactive power 85Maximum line loadability 319, 341, 533Measuring of contributions to voltage

service 163Modal analysis 401, 407, 408

NNew identification algorithm theory 418, 420,

451, 462, 530

OOELs and OLTCs impact on stability 377Off-line indicators 401, 406, 450, 452, 462OLTC blocking 446, 497, 509OLTCs Inverse Operation, 320OLTC Tap Control 472, 489OLTCs Inverse Operation 320On-field tests 244Operation costs 246, 299, 300, 311Operation quality-security-efficiency 84, 85,

162, 440, 541Operator interfaces 212Oscillation damping 42, 60, 327, 331

electromechanical 333Over and under excitation limits 27, 28, 113,

238, 249, 310, 440, 518Over-excitation limit (OEL) 86, 90, 167, 169,

385, 534

PPMU based indicators 404, 411Power flows control 4, 10, 32, 82, 472Power plant meter 314, 315Power stabilising feedback (PSS) 90, 290,

331Power system security 405, 439, 462, 497,

499, 534, 541Primary cabin (PC) controls 465, 470Primary cabin hierarchical control

scheme 446Prony identification 405Protection by Jacobian computing and

sensitivity matrices 403, 498Protection by reactive power inflow

indicators 14, 52, 82Protection By SVR-TVR real-time

indicators 230Protection by Thevenin equivalent

identification 452PV curve under grid automatic voltage

regulation 377, 379

RReactive loads 7, 251Reactive power 4, 6, 8, 10, 11, 15, 20, 27, 32,

41, 86, 171, 211, 234, 244, 270, 310, 478, 506, 540

Reactive power control 20, 49, 113, 128, 129, 148, 505, 517

dynamic behaviour of SVC 129loops 113

Reactive power resources coordination 189Real-time indicators 401, 404, 530, 534

557Index

Real-time voltage instability indicators 399, 403, 439, 541

Regulating transformers 76, 77Romanian SVR studies 255Rotating machines 20

SSecondary voltage regulation (SVR) 131, 145,

150, 170, 171, 177, 182, 308, 327, 472principle of 170, 171, 177, 182

Sensitivity analysis 401, 407, 408, 421, 422, 426, 452, 538

of identification method 421, 422, 426Service indicators

voltage, 311Simplicity of Voltage Service recognition

under SVR-TVR 522Simulation results and smartness 230, 245,

498Simulation tests 421, 508Smart grid 82, 162Stability 22, 23, 35, 92, 106, 127, 320, 322,

323, 326, 327, 366, 517, 538index 516loop 22steady-state 326, 327transient 322, 323, 326

Static compensator (STATCOM) 20, 44–46, 78, 106, 133, 134, 136, 140–143, 472

dynamic behaviour of 141, 143, 144grid voltage regulation 134, 136

Static power electronic converters 13Static VAR compensator (SVC) 41–44, 503

regulation scope 43, 44voltage control requirements 42, 43

Steady state stability 323, 324Study and field test results 221SVR Areas 187, 193, 196, 202, 267, 512SVR contribution to angle stability 334–336SVR dynamic tests

with contingencies 263SVR pilot nodes 196, 387SVR stabilising effects 387SVR-TVR applications in the world 87, 230SVR-TVR dynamic models 157, 375SVR-TVR impact on voltage service 377Switching compensating equipment 82, 86System automatic regulation 233System dynamic model 245, 401, 405, 508,

538identification 401

System operation 84, 112, 163, 234, 270, 298, 498, 499

economic 234System static model 407

TTertiary voltage regulation (TVR) 161, 186,

187, 226, 306Tests on real large systems 373Tests on the considered real-time

indicators 163, 166, 213, 411Thevenin equivalent 137, 319, 349, 375, 376,

518, 528Transient stability 44, 60, 321, 322, 327

UUnder-excitation limit (UEL) 87, 90, 97, 169,

410Unified power flow controller (UPFC) 50, 55,

57, 58, 60, 62, 78, 106, 149–151, 156dynamic behaviour 152fundamentals of 55, 57, 58series converter control 151, 152shunt converter control 150, 151voltage control requirements 60, 61

VVoltage 10, 13, 60, 62, 78, 81, 84, 85, 87, 124,

163, 219, 298, 302, 319, 321, 341, 466, 497, 498, 512, 534

Voltage control tests in real power systems 186, 537

Voltage instability dynamics 403, 404Voltage instability indicators 399, 402, 403,

411, 412, 462real-time PMU-based 411, 412

Voltage protection 84, 381Voltage service 163, 248, 297, 299, 307, 309Voltage stability 28, 50, 81, 148, 245, 320,

322, 343, 356, 440limit increase 245

V-P curve nose 441, 528V-P Curve Nose Tip 530, 532, 534

WWide area control 404Wide-Area Protection 162