Figure 4.11

active power ¬‚ow control region when phase shifters PS1 and PS2 are connected in series

with the transmission lines connecting North“Lake and South“Lake, respectively. The range

of phase angle variation is speci¬ed to be Æ10 . The following combinations of phase angles

give the boundaries:

A: PS1 ¼ 10 , and PS2 ¼ À10 ;

Point

B: PS1 ¼ 10 , and PS2 ¼ 10 ;

Point

C: PS1 ¼ À 10 , and PS2 ¼ À10 ;

Point

D: PS1 ¼ À 10 , and PS2 ¼ 10 .

Point

Extensive power ¬‚ow simulations verify the feasibility region shown in Figure 4.11 for

this example. For instance, simulations are presented in Table 4.6, where the phase shifter

parameters, initial conditions, and control targets are given.

As expected, all the power ¬‚ows speci¬ed inside the feasible region are successfully

upheld (Cases 1“3). Power ¬‚ows speci¬ed outside the feasibility region lead to phase-shifter

limits violations; these are indicated by an asterisk. The size of the feasible active power

control region is a function of the phase angle controller range; as the range increases so too

does the sizes of the regions.

144 CONVENTIONAL POWER FLOW

Table 4.6 Feasible active power ¬‚ow control by PS1 and PS2

Final phase angle Active power ¬‚ow Active power ¬‚ow

value (deg) North“Lake (MW) South“Lake (MW)

PS1 PS2

Case Iteration Speci¬ed Final Speci¬ed Final

À 5.64 À 3.62

1 4 50 50 30 30

À5 À5

2 4 8.7 8.78 70 70

À 7.6 À 15 À 15

3 4 6.62 70 70

À5

10*

4 7 0.96 2.2 30 30

À 10* À 4.87

5 7 70 66.7 30 30

À 20 À 13.5

10*

6 6 5.2 30 30

À 7.87 À 10*

7 7 50 50 70 54

À 10 À 30 À 7.6

10* 10*

8 9 13.8

* Phase-shifter limit violation.

4.5 FURTHER CONCEPTS IN POWER FLOWS

4.5.1 Sparsity-oriented Solutions

When dealing with large-scale electrical power systems, the formation of actual matrices is

not desirable because of the exorbitant processing times associated with their numerical

solution. Instead, the Jacobian and nodal admittance matrices of the power system are stored

Next ptr 0

1 First ptr

Column

Column

2

3 Value

Value

:

: 0

Next ptr

Next ptr

2 — n b ’1

Column

Column

Column

2 — nb First ptr

Value

Value

Value

Figure 4.12 Linked lists for storing a sparse Jacobian matrix. Note: ptr, pointer. Redrawn, with

permission, from C.R. Fuerte-Esquivel, E. Acha, S.G. Tan, and J.J. Rico, ˜Ef¬cient Object Oriented

Power System Software for the Analysis of Large-scale Networks Containing FACTS Controlled

Branches™, IEEE Trans. Power Systems 3(2) 464“472, # 1998 IEEE

145

FURTHER CONCEPTS IN POWER FLOWS

and processed in vector form, where only nonzero elements are explicitly handled. In

computer languages with no linked list facilities several one-dimensional arrays and skilful

programming schemes are required in order to obtain ef¬cient power ¬‚ow analysis

solutions. In modern programming languages such as Cþþ , programming efforts are greatly

reduced owing to the existence of pointers and structures.

In theory, Cþþ allows sparsity techniques to be implemented following a rather purist

object-oriented programming (OOP) approach. However, this programming philosophy

incurs excessive cpu overheads. Alternatively, a more ef¬cient OOP approach may be

adopted where sparsity is implemented using an array of pointers pointing to structures.

Structures allow the encapsulation, in a single variable, of all the information associated

with a sparse coef¬cient (e.g. value, column, and pointer to next element). Pointers are used

to move from one structure to another. This is illustrated in Figure 4.12 for the case of a

system containing nb buses.

An array of pointers is created, the size of which equals the number of rows in the matrix.

Each element points to the address of the start of a list. Moreover, one list is created for each

row. In the case of conventional power ¬‚ows, where storage locations are kept for the slack

bus, an array of pointers of size equal to 2 ‚ (nb À 1) is created, where nb is the number of

buses in the network. Each list consists of an array of structures used to store information

associated with off-diagonal Jacobian elements. The information associated with diagonal

elements is stored in a separate array of structures.

4.5.2 Truncated Adjustments

The Newton“Raphson algorithm may perform poorly when solving large-scale power

systems that are either heavily loaded or contain a substantial number of power system

controllers in close proximity, such as LTCs and phase shifters. In such circumstances, large

increments in the state variables may take place during the iterative solution, with this in

turn inducing large ÁP and ÁQ residual terms. The result may be poor convergence, or

more seriously, divergent solutions.

Such unwanted problems can be avoided quite effectively by limiting the size of

correction, with the actual computed adjustments being replaced by truncated adjustments.

This is a straightforward software solution to a common problem when dealing with utility-

size power systems.

4.5.2.1 Test case of truncated adjustments involving three

load tap-changing transformers

The AEP30 test network (Freris and Sasson, 1968) modi¬ed to assess the impact of

truncated LTC solutions. The network contains two generators and four synchronous

condensers. Transformers connected between buses 4“12, 6“10, and 27“28 are taken to be

LTC transformers. The nodal voltage magnitudes at buses 4, 6, and 12 are controlled at 1, 1,

and 1.04 p.u., respectively, using the primary taps of the three LTCs. The transformer

connected between buses 6“9 is taken to be a phase shifter with a ¬x tap on the primary

winding of 1¬ À3.75 .

146 CONVENTIONAL POWER FLOW

To show the effectiveness of truncated solutions, two types of adjustments are carried out:

Truncation of the size of correction (TA);

Use of full correction (NTA).

Adjusted solutions are achieved in 6 iterations to a power mismatch tolerance of 1e À 12.

However, the algorithm fails to reach convergence if the state variable increments are not

truncated. This is illustrated in Figure 4.13, where maximum active and reactive power

mismatches for both kinds of adjustments are shown.

10 3

10 2

10 1

10 0

10 ’1

10 ’2

10 ’3

Mismatch (p.u.)

10 ’4

10 ’5 P(TA)

10 ’6 Q(TA)

10 ’7 T (TA)

10 ’8 P(NTA)

10 ’9 Q(NTA)

10 ’1 0 T (NTA)

10 ’1 1

10 ’1 2

10 ’1 3

10 ’1 4 0 1 2 3 4 5 6 7 8 9 10 11

Iteration

Convergence pro¬le as a function of power mismatch

Figure 4.13

The ¬nal LTC parameters are shown in Table 4.7. It is assumed in the study that none of

the LTCs violates tap limits. The active and reactive powers generated by the two

synchronous generators (GE) and four synchronous condensers (CO) are shown in Table

4.8, where it is shown that one generator and two condensers hit their upper reactive power

limits.

Final settings of power system controllers

Table 4.7

Controller Magnitude (p.u.) Angle (deg)

LTC 4“12 0.9013 0.00

LTC 6“10 0.8821 0.00

LTC 27“28 1.0667 0.00

À 3.75

PS 6“9 1.0000

Note: LTC, load tap changer; PS, phase shifter.

147

FURTHER CONCEPTS IN POWER FLOWS

Power generation

Table 4.8

Source Active power (MW) Reactive power (MVAR)

À 3.1

GE-1 261.29

50.0*

GE-2 40.0

40.0*

CO-5 0.0

40.0*

CO-8 0.0

CO-11 0.0 13.17

À 2.27

CO-13 0.0

* Violation reactive power limit.

Note: GE, generator; CO, condenser.

4.5.3 Special Load Tap Changer Con¬gurations

Groups of LTCs may be operated in a coordinated fashion enabling more general control

strategies than those afforded by a single LTC. Series and parallel LTC con¬gurations are

the most obvious possibilities. The series condition occurs when one or more LTCs regulate

the nonregulated terminal of another LTC. This situation is shown in Figure 4.14(a), in

which LTC 1 regulates bus k, and LTC 2 regulates bus m. The parallel condition occurs

when bus k is regulated by two or more LTCs, as shown in Figure 4.14(b). It must be noted

that buses m and n may not necessarily be electrically connected.

OPEN

m n

n m k

LTC 2 LTC 1

T1 T2

T1

T2

PVT

PQ PVT

PVT

k

(a) (b)

Figure 4.14 Control con¬gurations: (a) series and, (b) parallel. Reproduced, with permission, from

C.R. Fuerte-Esquivel, E. Acha, S.G. Tan, and J.J. Rico, ˜Ef¬cient Object Oriented Power System

Software for the Analysis of Large-scale Networks Containing FACTS Controlled Branches™, IEEE

Trans. Power Systems 3(2) 464“472, # 1998 IEEE

The parallel condition does not belong to the category of single criterion control, where

only one control variable is adjusted in order to maintain another dependent variable at a

speci¬ed value. When two or more LTCs are controlling one nodal voltage magnitude

multiple solutions become a possibility because the number of unknown variables is greater

than the number of equations. An entire group of parallel LTCs may be treated as a single

control criterion if they are started from the same tapping initial condition. One equation and

148 CONVENTIONAL POWER FLOW

one variable corresponding to the common tap position may be suf¬cient to describe the

group performance. This Equation is linearised with respect to the common tap and

incorporated in the overall Jacobian Equation (4.57).

From the LTC set, the LTC that draws less reactive power is selected to be the master, and

its tapping position becomes the master tapping position. Since the various LTCs in the

group may have the same tapping position but different tap limits, it may be appropriate to

consider the following options:

If an LTC different from the master hits one of its limits, the tapping position is ¬xed at

the offending limit and the LTC is removed from the linearised system of equations.

If the master LTC hits a limit, it follows the same treatment as a slave LTC. Moreover, a

new master is selected from the remaining active LTCs. If no active LTC remains

following limit violation by the master then the bus becomes PQ.

A sensitivity factor, , may be used when the various LTCs in the parallel set have different

tapping positions; refers the slave tap position to the master tap position.

Assuming a group of np LTCs operating in parallel, and taking Tk to be the master

position, the sensitivity factor is calculated as:

Tk

p ¼ ; p ¼ 1; . . . ; np : °4:97Þ

Tp

The expression used for computing the Jacobian entry for the master tap position is also

used for the other LTCs in the group. The tap is adjusted by using Equation (4.97), where

each LTC in the group has its own adjusting pattern and where the sensitivity factor is taken

into account:

ÁTk °iÞ °iÀ1Þ

°iÞ °iÀ1Þ °4:98Þ

¼ þ Tp ; p ¼ 1; . . . ; np :

Tp Tp

Tk

An alternative adjusting strategy is given by Equation (4.98), where equal corrections are

given to all the LTCs in the group:

°iÞ

ÁTk

°iÞ °iÀ1Þ

°iÞ Tp ;

°iÀ1Þ °4:99Þ

Tp ¼ Tp þ p ¼ 1; . . . ; np :

p

Tk

4.5.3.1 Test case of sensitivity factors in parallel

load tap-changing operation

The AEP30 test system (Freris and Sasson, 1968) is modi¬ed to include four LTCs. The

nodal voltage magnitude at bus 6 is kept at 1.01 p.u. with LTCs 6“9 and 6“10 exerting

parallel control in bus 6. The voltage magnitude at buses 4 and 27 are controlled at 1.01 p.u.

and 1 p.u. by LTCs 4“12 and 27“28, respectively. The transformers reactance and off-

nominal tap values given in (Freris and Sasson, 1968) are taken to be on the secondary and

primary windings, respectively. The primary windings of the four transformers are assumed

connected to buses 6, 4, and 27, respectively.

149

SUMMARY

Initial position of load tap changer (LTC) taps

Table 4.9

Case LTC 6“9 LTC 6“10 LTC 4“12 LTC 27“28

1 0.978 0.969 0.932 0.968

2 1.1 1.1 1.1 1.1

3 1.0 1.0 1.0 1.0

4 0.9 0.9 0.9 0.9

5 1.0 0.9 1.0 1.0

Following on the discussion started in Section 4.5.3, the adjustment of the two LTCs

operating in parallel is carried out by using: (1) sensitivity factors and (2) equal updating of

taps. A comparison is made for the various cases given in Table 4.9. The number of

iterations taken to obtain the solution as well as the ¬nal tapping values required to maintain

the nodal voltage magnitudes at the speci¬ed values are given in Table 4.10.

As expected, both adjusting methods give the same solution for a speci¬ed LTC initial

condition. However, the use of sensitivity factors guarantees better results in terms of the

number of iterations required to get to the solution, compared with the case in which

identical tapping updates is carried out.

Table 4.10 Final position of load tap changer (LTC) taps: (a) updating using

sensitivity factors and (b) equal updating

Tap position

Case Iteration LTC 6“9 LTC 6“10 LTC 4“12 LTC 27“28

(a)

1 5 0.976 0.967 0.915 0.998

2 5 0.974 0.974 0.915 0.998

3 5 0.974 0.974 0.915 0.998

4 5 0.974 0.974 0.915 0.998

5 5 1.008 0.907 0.913 0.995

(b)

1 6 0.976 0.967 0.915 0.998

2 5 0.974 0.974 0.915 0.998

3 5 0.974 0.974 0.915 0.998

4 5 0.974 0.974 0.915 0.998

5 10 1.008 0.908 0.913 0.995

4.6 SUMMARY

In this chapter we have addressed the basic theory of power ¬‚ows. Building upon

elementary concepts afforded by circuit theory and complex algebra, we have derived

150 CONVENTIONAL POWER FLOW

equations for active and reactive powers injections at a bus. Owing to the idiosyncrasies of

the electrical power network, the mathematical model that describes its operation during

steady-state is nonlinear. Furthermore, for most practical situations, the power network is a

very large-scale system. Hence, solution of the nonlinear set of equations, which must be

reached by iteration, requires a robust and ef¬cient numerical technique. For several decades

the Newton“Raphson method, with its quadratic convergence characteristic, has proved

invaluable in solving the power ¬‚ow problem. The additional burden imposed on the

numerical solution by the many constraint actions resulting from the various power system

controllers in the network does not impair the ability of the Newton“Raphson method to

converge in a quadratic fashion. Derived Newton“Raphson formulations, such as the fast

decoupled method, also possess strong convergence characteristics. Both methods have been

explained in full detail in this chapter. The calculated power equations, mismatch powers,

and Jacobian terms all have been derived from ¬rst principles. The relevant equations

making up the Newton“Raphson and fast decoupled methods have been coded in Matlab1

and the programs used to solve a classical test case. The test system is small and yet it

provides suf¬cient realism and ¬‚exibility for the reader to explore different loading

scenarios, active power generator schedules, and transmission-line parameters. This is

something we certainly encourage the user to do.

The material presented in this chapter progressed to tackle the most specialised issue of

constrained power ¬‚ow solutions. To this end, ¬‚exible models of tap-changing and phase-

shifting transformers were developed from ¬rst principles. Together with the generator,

these two power controllers are capable of providing automatic regulation at speci¬c points

of the network provided their design limits are not exceeded. The generator and the tap-

changing transformer provide voltage magnitude regulation whereas the phase-shifting

transformer provides active power regulation. Inclusion of such regulating characteristics

within the power ¬‚ow solution is a matter of great engineering importance. However, they

introduce additional complexity in power ¬‚ow theory and may impose an extra burden on

the numerical solution. We believe that suf¬cient breadth and depth was provided in the

second part of this chapter to make accessible the concepts associated with constrained

power ¬‚ow solutions. This is in preparation for the widespread constrained solutions

associated with the various FACTS controllers presented in the next chapter.

REFERENCES

Arrillaga, J., Arnold, C.P., 1990, Computer Analysis of Power Systems, John Wiley & Sons,

Chichester.

Brown, H.E., 1975, Solution of Large Networks by Matrix Methods, John Wiley & Sons,

Chichester.

Elgerd, O., 1982, Electric Energy System Theory: An Introduction, McGraw-Hill, New York.

Freris, L.L., Sasson, A.M., 1968, ˜Investigation of the Load-¬‚ow Problem™, Proceedings of the IEE,

Part C 115(10), 1459“1470.

Fuerte-Esquivel, C.R., Acha, E., Tan, S.G., Rico, J.J., 1998, ˜Ef¬cient Object Oriented Power

System Software for the Analysis of Large-scale Networks Containing FACTS Controlled

Branches™, IEEE Trans. Power System 3(2) 464“472.

Grainger, J.J., Stevenson, W.D., 1984, Power System Analysis, McGraw-Hill, New York.

Kundur, P.P., 1994, Power System Stability and Control, McGraw-Hill, New York.

151

REFERENCES

Peterson, N.M., Scott Meyer, W., 1974, ˜Automatic Adjustment of Transformer and Phase Shifter Taps

in the Newton Power Flow™, IEEE Trans. Power Apparatus and Systems PAS-90(1) 103“108.

Stagg, G.W., El-Abiad, A.H., 1968, Computer Methods in Power System Analysis, McGraw-Hill, New

York.

Stott, B., 1974, ˜Review of Load-¬‚ow Calculation Methods™, IEEE Proceedings 62(July) 916“929.

Stott, B., Alsac, O., 1978, ˜Fast Decoupled Load Flow™, IEEE Trans. Power Apparatus and Systems

PAS-93, 859“862.

Tinney, W.F., Hart, C.E., 1967, ˜Power Flow Solution by Newton™s Method™, IEEE Trans. Power

Apparatus and Systems PAS-86(11) 1449“1460.

Weedy, B.M., 1987, Electric Power Systems, John Wiley & Sons, Chichester.

Wood, A.J., Wollenberg, B.F., 1984, Power Generation, Operation and Control, John Wiley & Sons,

Chichester.

Zollenkoff, K., 1970, ˜Bifactorization: Basic Computational Algorithm and Programming Tech-

niques™, in J.K. Reid (ed.), Large Sparse Sets of Linear Equations, Academic Press, Oxford,

pp. 75“96.

5

Power Flow Including

FACTS Controllers

5.1 INTRODUCTION

FACTS controllers narrow the gap between the noncontrolled and the controlled power

system mode of operation, by providing additional degrees of freedom to control power

¬‚ows and voltages at key locations of the network (Hingorani and Gyugyi, 2000). Key

objectives of the technology are: to increase transmission capacity allowing secure loading

of the transmission lines up to their thermal capacities; to enable better utilisation of

available generation; and to contain outages from spreading to wider areas (Song and Johns,

1999).

In order to determine the effectiveness of this new generation of power systems

controllers on a network-wide basis, it has become necessary to upgrade most of the

analysis tools on which power engineers rely to plan and to operate their systems (IEEE/

´

CIGRE, 1995). For the purpose of steady-state network assessment, power ¬‚ow solutions

are probably the most popular kind of computer-based calculations carried out by planning

and operation engineers. The reliable solution of power ¬‚ows in real-life transmission and

distribution networks is not a trivial matter and, over the years, owing to its very practical

nature, many calculation methods have been put forward to solve this problem. Among

them, Newton“Raphson type methods, with their strong convergence characteristics, have

proved the most successful and have been embraced by industry (Tinney and Hart, 1967).

In preparation for the material covered in this chapter, in Chapter 4 we provided a

thorough grounding on conventional power ¬‚ow theory with particular reference to the

Newton“Raphson method. Similar material can also be found in many of the excellent

power system analysis books that address the subject (Arrillaga and Arnold, 1990; Grainger

and Stevenson, 1994; Kundur, 1994). The aim of this chapter is to introduce a systematic

and coherent way to study models and methods for the representation of FACTS controllers

in power ¬‚ow studies. This aspect of power ¬‚ow theory has not been covered in existing

textbooks in the breadth and depth that the importance and complexity of the subject

demands (Fuerte-Esquivel et al., 1998). It should be emphasised that the material presented

in this chapter is a distillation of the wealth of research contributions on the subject that have

FACTS: Modelling and Simulation in Power Networks.

´ ´

Enrique Acha, Claudio R. Fuerte-Esquivel, Hugo Ambriz-Perez and Cesar Angeles-Camacho

# 2004 John Wiley & Sons, Ltd ISBN: 0-470-85271-2

154 POWER FLOW INCLUDING FACTS CONTROLLERS

´

been published over recent years (Ambriz-Perez, Acha, and Fuerte-Esquivel, 2000; Fuerte-

´

Esquivel and Acha, 1996, 1997; Fuerte-Esquivel, Acha, and Ambriz-Perez, 2000a, 2000b,

2000c; Nabavi-Niaki and Iravani, 1996; Noroozian and Andersson, 1993). It is in this

respect that the chapter gives an up-to-date and authoritative account of the power ¬‚ow

models and methods of power electronics-based controllers currently available in the power

transmission industry.

5.2 POWER FLOW SOLUTIONS INCLUDING FACTS CONTROLLERS

The technical literature is populated with clever and elegant solutions for accommodating

models of controllable plant in Newton“Raphson power ¬‚ow algorithms; load tap-changing

(ltc) and phase-shifting transformers are early examples of such work. The model-

ling approach used to represent controllable equipment can be broadly classi¬ed into two

main categories, namely, sequential and simultaneous solution methods. The former

approach is amenable to easier implementations in Newton“Raphson algorithms. However,

its major drawback is that the bus voltage magnitudes and angles are the only state variables

that are calculated in true Newton fashion, and a subproblem is formulated for updating the

state variables of the controllable devices at the end of each iteration. Such an approach

yields no quadratic convergence (Acha, 1993; Chang and Brandwajn, 1988).

Alternatively, the uni¬ed approach combines the state variables describing controllable

equipment with those describing the network in a single frame of reference for uni¬ed,

´

iterative solutions using the Newton“Raphson algorithm (Ambriz-Perez, Acha, and Fuerte-

Esquivel, 2000; Fuerte-Esquivel and Acha, 1996, 1997; Fuerte-Esquivel, Acha, and Ambriz-

´

Perez, 2000a, 2000b, 2000c; Fuerte-Esquivel et al., 1998). The method retains Newton™s

quadratic convergence characteristics.

The uni¬ed approach blends the alternating-current (AC) network and power system

controller state variables in a single system of simultaneous equations:

f°XnAC ; RnF Þ ¼ 0;

°5:1Þ

g°XnAC ; RnF Þ ¼ 0;

where XnAC stands for the AC network state variables, namely, nodal voltage magnitudes and

phase angles, and RnF stands for the power system controller state variables.

The increase in the dimensions of the Jacobian, compared with the case when there are no

power system controllers, is proportional to the number and kind of such controllers. In very

general terms, the structure of the modi¬ed Jacobian is shown in Figure 5.1.

Building upon the basic principles of steady-state operation and modelling of FACTS

controllers described in Chapter 2 and the power ¬‚ow theory detailed in Chapter 4, key

aspects of modelling implementation of FACTS controllers are presented in this chapter,

within the context of the Newton“Raphson power ¬‚ow algorithm. The FACTS controllers

that receive attention are:

Static VAR compensator (SVC);

Thyristor-controlled series compensator (TCSC);

Static compensator (STATCOM);

Uni¬ed power ¬‚ow controller (UPFC);

High-voltage direct-current-based voltage source converter (HVDC-VSC).

155

STATIC VAR COMPENSATOR

x n AC rn F

r1

x1

f1

network

AC

fn AC

F1 FACTS

controllers

FnF

Augmented Jacobian

Figure 5.1

5.3 STATIC VAR COMPENSATOR

Conventional and advanced power ¬‚ow models of SVCs are presented in this section. The

advanced models depart from the conventional generator-type representation (Erinmez,

1986; IEEE SSCWG, 1995) of the SVC and are based instead on the variable shunt

susceptance concept. In the latter case, the SVC state variables are combined with the nodal

voltage magnitudes and angles of the network in a single frame of reference for uni¬ed,

iterative solutions using the Newton“Raphson method. Two models are presented in this

´

category (Ambriz-Perez, Acha, and Fuerte-Esquivel, 2000), namely, the variable shunt

susceptance model and the ¬ring-angle model. Moreover, a compound transformer and SVC

model based on the SVC ¬ring-angle representation is also given.

5.3.1 Conventional Power Flow Models

Early SVC models for power ¬‚ow analysis treat the SVC as a generator behind an inductive

reactance (Erinmez, 1986; IEEE SSCWG, 1995). The reactance accounts for the SVC

voltage-regulation characteristic.

A simpler representation assumes that the SVC slope is zero; an assumption that may be

acceptable as long as the SVC operates within its design limits, but one which may lead to

gross errors if the SVC is operating close to its limits (Kundur, 1994). This point is

illustrated in Figure 5.2 with reference to the upper characteristic when the system is

operating under low loading conditions. If the slope is taken to be zero then the generator

will violate its minimum limit, point AXSL¼0 . However, the generator will operate well within

limits if the SVC voltage“current slope is taken into account at, point A.

156 POWER FLOW INCLUDING FACTS CONTROLLERS

V

V1

A

Vmax

D A xSL = 0

D System reactive

Vmin

load characteristics

V2

Capacitive rating

Inductive rating

IS

Imin 0 Imax

Figure 5.2 Static VAR compensator and power system voltage“current characteristics. From P.P.

Kundur, Power System Stability and Control, # 1994 McGraw-Hill. Reproduced by permission of

The McGraw-Hill Companies

The reasons for including the SVC voltage“current slope in power ¬‚ow studies are

compelling. The slope can be represented by connecting the SVC model to an auxiliary bus

coupled to the high-voltage bus by an inductive reactance consisting of the transformer

reactance and the SVC slope, in per unit (p.u.) on the SVC base. The auxiliary bus is

represented as a PV bus and the high-voltage bus is taken to be PQ. This model is shown

schematically in Figure 5.3(a). Alternatively, the SVC coupling transformer may be

represented explicitly as shown in Figure 5.3(b).

High-voltage bus (PQ)

k

XSL

Auxiliary bus (PV)

High voltage bus (PQ) Vref

k

XT’SL XT

Low voltage bus

Auxiliary bus (PV)

Vref

(PV with remote control)

(b)

(a)

Figure 5.3 Conventional static VAR compensator power ¬‚ow models: (a) slope representation and

(b) slope and coupling transformer representation

157

STATIC VAR COMPENSATOR

These SVC representations are quite straightforward but are invalid for operation outside

the limits (IEEE SSCWG, 1995). In such cases, it becomes necessary to change the SVC

representation to a ¬xed reactive susceptance, given by

Qlim

BSVC ¼ À ; °5:2Þ

2

VSVC

where VSVC is the newly freed voltage due to the reactive power limit Qlim being exceeded.

The combined generator“susceptance representation yields accurate results. However, a

drawback of such a representation is that both models use a different number of buses. The

generator uses two or three buses, as shown in Figure 5.3, whereas the ¬xed susceptance

uses only one bus. In Newton“Raphson power ¬‚ow solutions such a difference in the number

of buses required to represent the same plant component may lead to Jacobian reordering

and redimensioning during the iterative solution. Also, extensive checking becomes

necessary in order to verify whether or not the SVC has returned to operation within limits

at any stage of the iterative solution.

It should be remarked that for operation outside limits the SVC must be modelled as a

susceptance and not as a generator set at its violated limit, Qlim. Ignoring this point will lead

to inaccurate results. The reason is that the amount of reactive power drawn by the SVC is

given by the product of the ¬xed susceptance, B¬x, and the nodal voltage magnitude, Vk.

Since Vk is a function of network operating conditions, the amount of reactive power drawn

by the ¬xed susceptance model differs from the reactive power drawn by the generator

model; that is,

Qlim 6¼ ÀBfix Vk : °5:3Þ

2

This point is exempli¬ed in Figure 5.4, where the reactive power output of the generator is

set at 100 MVAR. This value is constant as it is voltage-independent. The result given by the

112

110

108

Generator model

Reactive power (MVAR)

106

Susceptance model

104

102

100

98

96

94

92

90

0.95 0.96 0.97 0.98 0.99 1.00 1.01 1.02 1.03 1.04 1.05

Voltage magnitude (p.u.)

Figure 5.4 Comparison of reactive power drawn by the generator and susceptance models.

´

Reproduced, with permission, from H. Ambriz-Perez, E. Acha, and C.R. Fuerte-Esquivel, ˜Advanced

SVC Models for Newton“Raphson Load Flow and Newton Optimal Power Flow Studies™, IEEE Trans.

Power Systems 15(1) 129“136, # 2000 IEEE

158 POWER FLOW INCLUDING FACTS CONTROLLERS

constant susceptance model varies with nodal voltage magnitude. The voltage range

considered is 0.95“1.05 p.u. The susceptance value, on a 100 MVA base, is of 1 p.u.

5.3.2 Shunt Variable Susceptance Model

In practice the SVC can be seen as an adjustable reactance with either ¬ring-angle limits or

´

reactance limits (Ambriz-Perez, Acha, and Fuerte-Esquivel, 2000). The equivalent circuit

shown in Figure 5.5 is used to derive the SVC nonlinear power equations and the linearised

equations required by Newton™s method.

Vk

ISVC

BSVC

´

Figure 5.5 Variable shunt susceptance. Reproduced, with permission, from H. Ambriz-Perez, E.

Acha, and C.R. Fuerte-Esquivel, ˜Advanced SVC Models for Newton“Raphson Load Flow and

Newton Optimal Power Flow Studies™, IEEE Trans. Power Systems 15(1) 129“136, # 2000 IEEE

With reference to Figure 5.5, the current drawn by the SVC is

ISVC ¼ jBSVC Vk ; °5:4Þ

and the reactive power drawn by the SVC, which is also the reactive power injected at bus k,

is

QSVC ¼ Qk ¼ ÀVk BSVC : °5:5Þ

2

The linearised equation is given by Equation (5.6), where the equivalent susceptance BSVC is

taken to be the state variable:

!°iÞ !°iÞ !°iÞ

Ák

ÁPk 00

¼ : °5:6Þ

ÁBSVC BSVC

ÁQk 0 Qk

At the end of iteration (i), the variable shunt susceptance BSVC is updated according to

ÁBSVC °iÞ °i-1Þ

°iÞ °i-1Þ

BSVC ¼ BSVC þ BSVC : °5:7Þ

BSVC

The changing susceptance represents the total SVC susceptance necessary to maintain the

nodal voltage magnitude at the speci¬ed value.

159

STATIC VAR COMPENSATOR

Once the level of compensation has been computed then the thyristor ¬ring angle can be

calculated. However, the additional calculation requires an iterative solution because the

SVC susceptance and thyristor ¬ring angle are nonlinearly related.

5.3.3 Static VAR Compensator Computer Program in Matlab1 Code

Program 5.1 incorporates the SVC representation, modelled as a variable shunt susceptance

model, within the Newton“Raphson power ¬‚ow program given in Section 4.3.6. The

functions PowerFlowsData, YBus, and PQ¬‚ows are also used here. In the main SVC

Newton“Raphson program, the function SVCBData is added to read the SVC data,

SVCNewtonRaphson replaces NewtonRaphson, and SVCPQ¬‚ows is used to calculate

power ¬‚ows and losses in the SVC.

Function SVCNewtonRaphson borrows the following functions from NewtonRaphson:

NetPowers; CalculatedPowers; GeneratorsLimits; PowerMismatches; Newton-

RaphsonJacobian; and StateVariablesUpdates. Furthermore, four new functions are

added to cater for the SVC representation: SVCCalculatedPowers; SVCUpdates;

SVCLimits; and SVCNewtonRaphsonJacobian.

Program 5.1 Program written in Matlab1 to incorporate static VAR compensator (SVC)

variable shunt susceptance model within the Newton“Raphson power ¬‚ow algorithm

% - - - Main SVC Program

PowerFlowsData; %Function to read network data

SVCBData; %Function to read Static VAR Compensator data

[YR,YI] = YBus(tlsend,tlrec,tlresis,tlreac,tlsuscep,tlcond,ntl,nbb);

[VM,VA,it,B] = SVCNewtonRaphson(tol,itmax,ngn,nld,nbb,bustype,...

genbus, loadbus,PGEN,QGEN,QMAX,QMIN,PLOAD,QLOAD,YR,YI,VM,VA,NSVC,...

SVCsend,B,BLo, BHi,TarVol,VSta);

[PQsend,PQrec,PQloss,PQbus] = PQ¬‚ows(nbb,ngn,ntl,nld,genbus,loadbus,

tlsend,tlrec,tlresis,tlreac,tlcond,tlsuscep,PLOAD,QLOAD,VM,VA);

[QSVC] = SVCQpower(VM,NSVC,SVCsend,B);

%Print results

it %Number of iterations

VM %Nodal voltage magnitude (p.u)

VA=VA*180/pi %Nodal voltage phase angles (Deg)

QSVC %Final reactive power (p.u.)

B %Final susceptance (p.u)

%End of MAIN FOR SVC SHUT VARIABLE SUSCEPTANCE

160 POWER FLOW INCLUDING FACTS CONTROLLERS

function [VM,VA,it,B] = SVCNewtonRaphson(tol,itmax,ngn,nld,nbb,...

bustype, genbus,loadbus,PGEN,QGEN,QMAX,QMIN,PLOAD,QLOAD,YR,YI,...

VM,VA,NSVC,SVCsend,B,BLo,BHi,TarVol,VSta);

% GENERAL SETTINGS

¬‚ag = 0;

it = 1;

% CALCULATE NET POWERS

[PNET,QNET] = NetPowers(nbb,ngn,nld,genbus,loadbus,PGEN,QGEN,...

PLOAD,QLOAD);

while (it < itmax & ¬‚ag==0)

% CALCULATED POWERS

[PCAL,QCAL] = CalculatedPowers(nbb,VM,VA,YR,YI);

%SVC CALCULATED POWER

[QCAL] = SVCCalculatePower(QCAL,VM,NSVC,SVCsend,B)

% POWER MISMATCHES

[DPQ,DP,DQ,¬‚ag] = PowerMismatches(nbb,tol,bustype,¬‚ag,PNET,...

QNET, PCAL,QCAL);

if ¬‚ag == 1

break

end

% JACOBIAN FORMATION

[JAC] = NewtonRaphsonJacobian(nbb,bustype,PCAL,QCAL,DPQ,VM,VA,...

YR,YI);

% MODIFICATION THE JACOBIAN FOR SVC

[JAC] = SVCJacobian(JAC,VM,NSVC,SVCsend,B,VSta);

% SOLVE JOCOBIAN

D = JAC\DPQ™;

% UPDATE THE STATE VARIABLES VALUES, WITH TRUNCATED CORRECTIONS

% IF NECESSARY (VM increments < +-0.1 p.u. and VA increments < +- 5 deg)

[VA,VM] = StateVariablesUpdating(nbb,D,VA,VM,it);

% UPDATE THE SVC VARIABLES

[VM,B] = SVCUpdating(VM,D,NSVC,SVCsend,B,BLo,BHi,TarVol,VSta);

161

STATIC VAR COMPENSATOR

%CHECK SUSCEPTANCE FOR LIMITS

[B] = SVCLimits(NSVC,B,BLo,BHi);

it = it + 1;

end

%Function to calculate injected bus powers by the SVC function

[QCAL]= SVCCalculatePower(QCAL,VM,NSVC,SVCsend,B);

for ii = 1 : NSVC

QCAL(SVCsend(ii))=QCAL(SVCsend(ii))-VM(SVCsend(ii))^2*B(ii);

end

%Function to upgrade the Jacobian matrix with SVC elements

function [JAC] = SVCJacobian(JAC,VM,NSVC,SVCsend,B,VSta);

for ii = 1 : NSVC

if (VSta(ii) == 1)

%Delete the voltage magnitud for the SVC bus

JAC( : , 2*SVCsend(ii) ) = 0;

JAC(2*SVCsend(ii)-1,2*SVCsend(ii)-1) = ...

JAC(2*SVCsend(ii)- 1,2*SVCsend(ii)-1)- ...

VM(SVCsend(ii))^2*B(ii);

JAC(2*SVCsend(ii),2*SVCsend(ii))= - VM(SVCsend(ii))^2*B(ii);

end

end

%Function to update SVC state variable

function [VM,B] = SVCUpdating(VM,D,NSVC,SVCsend,B,BLo,BHi,TarVol,...

VSta);

for ii = 1 : NSVC

if (VSta(ii) == 1)

% Adjust the Voltage Magnitud target

VM(SVCsend(ii)) = TarVol(ii);

% Truncation

value = B(ii)*D(2*SVCsend(ii));

value2 = D(2*SVCsend(ii));

if (value > 0.1)

value2 = 0.1/B(ii);

elseif (value < -0.1)

value2 = -0.1/B(ii);

end

B(ii) = B(ii) + B(ii)*value2;

end

end

162 POWER FLOW INCLUDING FACTS CONTROLLERS

%Function to check the susceptance limits

function [B] = SVCLimits(NSVC,B,BLo,BHi);

% Check susceptance limits in SVC

for ii = 1 : NSVC

if (B(ii) > BHi(ii))

B(ii) = BHi(ii);

elseif (B(ii) < BLo(ii))

B(ii) = BLo(ii);

end

end

%Function to calculate the reactive power in SVC

function [QSVC] = SVCQpower(VM,NSVC,SVCsend,B);

for ii = 1 : NSVC

QSVC(ii) = -VM(SVCsend(ii))^2*B(ii);

end

5.3.4 Firing-angle Model

An alternative SVC model, which circumvents the additional iterative process, consists in

handling the thyristor-controlled reactor (TCR) ¬ring angle as a state variable in the

power ¬‚ow formulation (Ambriz-Perez, Acha, and Fuerte-Esquivel, 2000). The variable

´

will be designated here as SVC, to distinguish it from the TCR ¬ring angle used in the

TCSC model.

The positive sequence susceptance of the SVC, given by Equation (2.20), is used in

Equation (5.5):

& '

ÀVk 2

XC

Qk ¼ XL À ½2°p À SVC Þ þ sin°2SVC Þ : °5:8Þ

p

XC XL

From Equation (5.8), the linearised SVC equation is given as

2 3°iÞ

!°iÞ !°iÞ

0 0

ÁPk Ák

¼4 5 : °5:9Þ

2

2Vk

ÁQk ÁSVC

½cos°2SVC Þ À 1

0

XL

At the end of iteration (i), the variable ¬ring angle SVC is updated according to

°iÞ °iÀ1Þ °iÞ

SVC ¼ SVC þ ÁSVC : °5:10Þ

5.3.5 Static VAR Compensator Firing-angle Computer

Program in Matlab1 Code

Program 5.2 incorporates the SVC ¬ring-angle (SVC-FA) model within the Newton“

Raphson power ¬‚ow program given in Section 4.3.6. The functions PowerFlowsData,

YBus, and PQ¬‚ows are also used here. In the main SVC-FA Newton“Raphson program, the

163

STATIC VAR COMPENSATOR

function SVCFAData is added to read the SVC-FA data, SVCFANewtonRaphson replaces

NewtonRaphson, and SVCFAPQ¬‚ows is used to calculate power ¬‚ows and losses in the

SVC-FA model.

Function SVCFANewtonRaphson borrows the following functions from Newton-

Raphson: NetPowers; CalculatedPowers; PowerMismatches; NewtonRaphson-

Jacobian; and StateVariablesUpdates. Furthermore, four new functions are added

to cater for the SVC-FA representation, namely: SVCFACalculatedPowers; SVCFA-

Updates; SVCFALimits; and SVCFANewtonRaphsonJacobian.

PROGRAM 5.2 Program written in Matlab1 to incorporate the static VAR compensator

¬ring-angle (SVC-FA) model within the Newton“Raphson power ¬‚ow algorithm

% - - - Main SVC-FA Program

PowerFlowsData; %Function to read network data

SVCFAData; %Function to read SVC-FA data

[YR,YI] = YBus(tlsend,tlrec,tlresis,tlreac,tlsuscep,tlcond,ntl,nbb);

[VM,VA,it,FA] = SVCFANewtonRaphson(tol,itmax,ngn,nld,nbb,bustype,...

genbus, loadbus,PGEN,QGEN,QMAX,QMIN,PLOAD,QLOAD,YR,YI,VM,VA,NSVC,...

SVCsend,FA,Xc,Xl,FALo,FAHi,TarVol,VSta);

[PQsend,PQrec,PQloss,PQbus] = PQ¬‚ows(nbb,ngn,ntl,nld,genbus,...

loadbus,tlsend,tlrec,tlresis,tlreac,tlcond,tlsuscep,PLOAD,QLOAD,...

VM,VA);

[QSVC,B] = SVCFAQpower(VM,NSVC,SVCsend,FA,Xc,Xl);

%Print online results

it %Number of iterations

VM %Nodal voltage magnitude (p.u)

VA=VA*180/pi %Nodal voltage phase angles (Deg)

QSVC %Final reactive power value(p.u.)

B %Final susceptance value (p.u.)

FA=FA*180/pi %Final ¬ring angle value (Deg)

%End of MAIN SVC-FA PROGRAM

function [VM,VA,it,FA] = SVCFANewtonRaphson(tol,itmax,ngn,nld,nbb,...

bustype, genbus,loadbus,PGEN,QGEN,QMAX,QMIN,PLOAD,QLOAD,YR,YI,VM,...

VA,NSVC,SVCsend, FA,Xc,Xl,FALo,FAHi,TarVol,VSta);

% GENERAL SETTINGS

¬‚ag = 0;

it = 1;

164 POWER FLOW INCLUDING FACTS CONTROLLERS

% CALCULATE NET POWERS

[PNET,QNET] = NetPowers(nbb,ngn,nld,genbus,loadbus,PGEN,QGEN,...

PLOAD,QLOAD);

while ( it < itmax & ¬‚ag==0 )

% CALCULATED POWERS

[PCAL,QCAL] = CalculatedPowers(nbb,VM,VA,YR,YI);

%SVC CALCULATED POWER

[QCAL,B] = SVCFACalculatePower(QCAL,VM,NSVC,SVCsend,FA,Xc,Xl);

% POWER MISMATCHES

[DPQ,DP,DQ,¬‚ag] = PowerMismatches(nbb,tol,bustype,¬‚ag,PNET,QNET,...

PCAL,QCAL);

% JACOBIAN FORMATION

[JAC]=NewtonRaphsonJacobian(nbb,bustype,PCAL,QCAL,DPQ,VM,...

VA,YR,YI);

% SVC-FA JACOBIAN UPDATING

[JAC]=SVCFAJacobian(JAC,VM,NSVC,SVCsend,FA,Xl,B,VSta);

% SOLVE FOR THE STATE VARIAVLES VECTOR

D = JAC\DPQ™;

% UPDATE THE STATE VARIABLES

[VA,VM] = StateVariablesUpdating(nbb,D,VA,VM,it);

% UPDATE THE SVC-FA VARIABLES

[VM,FA] = SVCFAUpdating(VM,D,NSVC,SVCsend,FA,FALo,FAHi,TarVol,...

VSta);

%CHECK SVC-FA FIRING ANGLE FOR LIMITS VIOLATIONS

[FA] = SVCFALimits(NSVC,FA,FALo,FAHi);

it = it + 1;

end

%Function to calculate injected bus powers by the SVC-FA

function [QCAL,B] = SVCFACalculatePower(QCAL,VM,NSVC,SVCsend,FA,...

Xc,Xl);

for ii = 1 : NSVC

FA(ii) = FA(ii)*pi/180;

165

STATIC VAR COMPENSATOR

B(ii) = (2*(pi-FA(ii)) + sin(2*FA(ii)))*Xc(ii)/pi;

B(ii) = (Xl(ii) - B(ii))/(Xc(ii)*Xl(ii));

QCAL(SVCsend(ii))=QCAL(SVCsend(ii))-VM(SVCsend(ii))^2*B(ii);

end

%Function to add up the SVC-FA elements to Jacobian matrix function

[JAC] = SVCFAJacobian(JAC,VM,NSVC,SVCsend,FA,Xl,B,VSta);

for ii = 1 : NSVC

if VSta(ii) == 1

%Delete the voltage magnitud for the SVC bus

JAC(:,2*SVCsend(ii))=0;

% Element add by the SVC to the Jacobian

FA(ii)=FA(ii)*pi/180;

JAC(2*SVCsend(ii)-1,2*SVCsend(ii)-1) = JAC(2*SVCsend(ii)-1,2*...

SVCsend(ii)-1) - VM(SVCsend(ii))^2*B(ii);

JAC(2*SVCsend(ii),2*SVCsend(ii))= 2*VM(SVCsend(ii))^2*...

(cos(2*FA(ii))-1)/(Xl(ii)*pi);

end

end

%Function to update SVC-FA state variable

function[VM,FA]=SVCFAUpdating(VM,D,NSVC,SVCsend,FA,FALo,FAHi,...

TarVol,VSta);

for ii = 1 : NSVC

if (VSta(ii) == 1)

% Adjust the Volatge Magnitud target

VM(SVCsend(ii)) = TarVol(ii);

% Truncation

value = D(2*SVCsend(ii));

if (value > 0.5236)

value = 0.5236;

elseif (value < -0.5236)

value = -0.5236;

end

FA(ii) = FA(ii) + value*180/pi;

if (FA(ii)<0.0)

FA(ii) = FA(ii)*(-1);

end

end

end

%Function to check the ¬ring angle limits

function [FA] = SVCFALimits(NSVC,FA,FALo,FAHi);

%Check SVC-FA Limits

for ii = 1 : NSVC

if (FA(ii) > FAHi(ii))

166 POWER FLOW INCLUDING FACTS CONTROLLERS

FA(ii) = FAHi(ii);

elseif (FA(ii) < FALo(ii))

FA(ii) = FALo(ii);

end

end

%Function to calculate the reactive power in SVC

function [QSVC,B] = SVCFAQpower(VM,NSVC,SVCsend,FA,Xc,Xl);

for ii = 1 : NSVC

FA(ii) = FA(ii)*pi/180;

B(ii) = (2*(pi-FA(ii)) + sin(2*FA(ii)))*Xc(ii)/pi;

B(ii) = (Xl(ii) - B(ii))/(Xc(ii)*Xl(ii));

QSVC(ii)=-VM(SVCsend(ii))^2*B(ii);

end

5.3.6 Integrated Transformer Firing-angle Model

The SVC ¬ring angle model is extended in this section to include the explicit representation

´

of the step-down transformer (Fuerte-Esquivel, Acha, and Ambriz-Perez, 2000a). Both

components are combined to form a single model, which allows for direct voltage

magnitude control at the high-voltage side of the transformer without compromising the

quadratic convergence characteristics of the Newton“Raphson method.

The total admittance of the combined SVC“transformer set, YT“SVC, as seen from the

high-voltage side of the transformer, consists of the series combination of admittances YT

and YSVC, as shown schematically in Figure 5.6.

k High-voltage bus (PVB)

k

ZT = RT + jXT

+

I SVC Y T-SVC

’

X SVC

XC

XL

Figure 5.6 Combined static VAR compensator“transformer representation. Reproduced, with

´

permission, from C.R. Fuerte-Esquivel, E. Acha, and H. Ambriz-Perez, ˜Integrated SVC and Step-

down Transformer Model for Newton“Raphson Load Flow Studies™, IEEE Power Engineering Review

20(2) 45“46, # 2000 IEEE

167

STATIC VAR COMPENSATOR

It should be noted that the equivalent admittance, YT“SVC, is a function of the SVC ¬ring

angle:

YT YSVC

YT--SVC °SVC Þ ¼ : °5:11Þ

YT þ YSVC

The admittance of the combined variable shunt compensator is given by

YT--SVC ¼ GT--SVC þ jBT--SVC ; °5:12Þ

where

RT

GT--SVC ¼ ; °5:13Þ

R2 þ XEq

2

T

XEq

BT--SVC ¼ À 2 ; °5:14Þ

RT þ XEq

2

XEq ¼ XT þ XSVC ; °5:15Þ

XC XTCR

XSVC ¼ ; °5:16Þ

XC À XTCR

pXL

XTCR ¼ : °5:17Þ

2°p À SVC Þ þ sin°2SVC Þ

The linearised power ¬‚ow equations are given as

2 3°iÞ

2 qGT--SVC

!°iÞ !°iÞ

6 0 Vk qSVC 7

ÁPk Ák

¼6 7 ; °5:18Þ

4 qBT--SVC 5 ÁSVC

ÁQk

0 ÀVk 2

qSVC

where the Jacobian terms in explicit form are:

qGT--SVC RT qD

¼À 2 ; °5:19Þ

qSVC D qSVC

qBT--SVC qXSVC qD

1

¼ 2 ÀD þ XEq ; °5:20Þ

qSVC qSVC qSVC

DT

qD qXSVC

¼ 2XEq ; °5:21Þ

qSVC qSVC

qXSVC 2XSVC

2

¼ °1 À cos 2SVC Þ; °5:22Þ

qSVC XL

D ¼ R2 þ XEq : °5:23Þ

2

T

At the end of iteration (i), the ¬ring angle SVC is updated according to

°iÞ °i-1Þ °iÞ

SVC ¼ SVC þ ÁSVC : °5:24Þ

5.3.7 Nodal Voltage Magnitude Control using Static

VAR Compensators

The SVC connecting bus is a voltage-controlled bus where the voltage magnitude and active

and reactive powers are speci¬ed and either the SVC ¬ring angle, SVC, or the SVC

168 POWER FLOW INCLUDING FACTS CONTROLLERS

equivalent susceptance, BSVC, are handled as state variables. This bus is de¬ned to be PVB-

type. If SVC or BSVC are within limits, the speci¬ed voltage magnitude is attained and

the controlled bus remains PVB. However, if SVC or BSVC go outside the limits then these

variables are ¬xed at the violated limit and the bus becomes PQ. This is, of course, in the

absence of any other controller capable of providing reactive power control at the bus.

The reactive power mismatch values at the controlled buses are used to check whether or

not the SVC is operating within limits, a process that starts just after the reactive power

mismatch at the controlled bus is less than a speci¬ed tolerance; a value of 1e À 3 p.u. is

normally used.

5.3.8 Control Coordination between Reactive Sources

The use of different kinds of reactive power sources to control voltage magnitude at a given

bus calls for a prioritisation of reactive power sources in order to have a single control

criterion. Synchronous generators are normally selected to be the regulating plant

components with the highest priority, holding all the other reactive power sources ¬xed

at their initial values as long as the generators operate within limits. If all the generators

connected to the bus violate their reactive power limits then other kinds of reactive power

sources become activated (e.g. SVC). In such a case, the generators™ reactive powers are set

at their violated limits and the bus is transformed from PV to PVB. The control sequence is

shown schematically in Figure 5.7.

Generator active

PV bus

Reactive

no

Remains limits

PV bus violation

yes

yes

SVC a ctive Looking for

PVB bus active SVC

no

Reactive yes

no yes LTC a ctive

Remains Looking for

limits

PVT bus

active LTC

PVB bus violation

no

no

PQ yes Violation of Remains

tap changer

bus PVT bus

limits

Figure 5.7 Coordination between nodal voltage magnitude controllers. Note: LTC, load tap charger;

SVC, static VAR compensator

169

STATIC VAR COMPENSATOR

5.3.9 Numerical Example of Voltage Magnitude Control using

One Static VAR Compensator

The ¬ve-bus network (see Figure 5.8) is modi¬ed to examine the voltage-control capabilities

of the SVC models. The generators are set to control voltage magnitudes at the Slack bus

(North) and the PV bus (South) at 1.06 p.u. and 1 p.u., respectively. One SVC is placed at

Lake to keep voltage magnitude at that bus at 1 p.u.

40+j5

45+j15

131.06 85.34

North Main

Lake

19.65 19.59

41.95 40.55

89.11 74.06

6.78

11.28 12.41 11.19 13.02 3.25

27.18

24.09

20.47

6.69

4.77

72.99

86.66

27.66

9.51

24.49

6.71 7.91

7.32

South Elm

53.29

54.48

20 + j10 60 + j10

2.09

2.75

77.07

40

Figure 5.8 Power ¬‚ow results in the ¬ve-bus network with one static VAR compensator

In order to compare the various SVC models, three different power ¬‚ow simulations are

carried out. First, the SVC susceptance model is used to attain the speci¬ed voltage

magnitude. The other two simulations are for the ¬ring-angle model and for the integrated

transformer“¬ring-angle model, respectively. The aim in all cases is to achieve 1 p.u.

voltage magnitude at Lake.

The SVC inductive and capacitive reactances are taken to be 0.288 p.u. and 1.07 p.u.,

respectively. The SVC ¬ring angle is set initially at 140 , a value that lies on the capacitive

region of the SVC characteristic. The SVC transformer impedance is ZT ¼ j0.11 p.u.

In all three cases, the SVC upholds its target value and, as expected, identical power ¬‚ows

and bus voltages are obtained. Power ¬‚ows are shown in Figure 5.8, and nodal voltages are

given in Table 5.1. Moreover, the three SVC models contribute the same amount of reactive

power to the system.

170 POWER FLOW INCLUDING FACTS CONTROLLERS

Nodal voltages of modi¬ed network

Table 5.1

Network bus

Nodal voltage North South Lake Main Elm

Magnitude (p.u.) 1.06 1 1 0.994 0.975

À 2.05 À 4.83 À 5.11 À 5.80

Phase angle (deg) 0

Convergence is achieved in 5 iterations, satisfying a prespeci¬ed tolerance of 1e À 12 for

all the variables involved. The SVC susceptance values and ¬ring-angle values are shown in

Table 5.2 for each step of the iterative process. It should be noted that the ¬nal ¬ring-angle

solutions for the ¬ring-angle model and the combined transformer“¬ring-angle model differ

slightly because of the inclusion of the reactance of the transformer in the latter model.

Table 5.2 Static VAR compensator state variables

Firing-angle model Transformer“¬ring angle model

Susceptance model

SVC (deg) T“SVC (deg)

Iteration BSVC (p.u.) BSVC (p.u.) BSVC (p.u.)

1 0.1 0.4798 140 0.5066 140

2 0.1679 0.1038 130.23 0.1166 130.48

3 0.2047 0.2013 132.47 0.2029 132.40

4 0.2047 0.2047 132.55 0.2047 132.44

5 0.2047 0.2047 132.55 0.2047 132.44

The SVC data for both variable shunt susceptance and ¬ring angle are given in function

SVCBData and SVCFAData, respectively; function PowerFlowsData remains as the

original:

Function SVCBData is as follows:

%This function is used exclusively to enter data for:

% STATIC VAR COMPENSATION

% VARIABLE SHUNT SUSCEPTANCE MODEL

% NSVC : Number of SVC™s

% SVCsend : Compensated bus

% B : Initial SVC™s susceptance value (p.u.)

% BLo : Lower limit of variable susceptance (p.u.)

% BHi : Higher limit of variable susceptance (p.u)

% TarVol : Target nodal voltage magnitude to be controlled by SVC (p.u.)

% VSta : Indicate control status for nodal voltage magnitude:1 is on and 0

% is off

NSVC=1;

SVCsend(1)=3; B(1)=0.02; BLo(1)= -0.25; BHi=0.25;

TarVol(1)=1.0; VSta(1)=1;

171

THYRISTOR-CONTROLLED SERIES COMPENSATOR

Function SVCFAData is as follows:

%This function is used exclusively to enter data for:

% STATIC VAR COMPENSATION

% FIRING ANGLE MODEL

% NSVC : Number of SVC™s

% SVCsend : Compensated bus

% Xc : Capacitive reactance (p.u.)

% Xl : Inductive reactance (p.u.)

% FA : Initial SVC™s ¬ring angle value (Deg)

% FALo : Lower limit of ¬ring angle (Deg)

% BHi : Higher limit of ¬ring angle (Deg)

% TarVol : Target nodal voltage magnitude to be controlled by SVC (p.u.)

% VSta : Indicate the status to get control over voltage magnitude nodal : 1

% is on; 0 is off

NSVC=1;

SVCsend(1)=3; Xc(1)=1.07; Xl(1)=0.288; FA(1)=140; FALo(1)=90;

FAHi(1)=180; TarVol(1)=1.0; VSta(1)=1;

The SVC injects 20.5 MVAR into Lake and keeps the nodal voltage magnitude at 1 p.u.

The action of the SVC results in an overall improved voltage pro¬le. The SVC generates

reactive power in excess of the local demand, which stands at 15 MVAR and, compared with

the base case, there is an almost fourfold export increase of reactive power to Main. Also,

there is an export of reactive power to South via transmission line Lake“South, with the

larger amount of reactive power available at the bus being absorbed by the synchronous

generator. It draws 77.1 MVAR as opposed to 61.59 MVAR in the base case.

5.4 THYRISTOR-CONTROLLED SERIES COMPENSATOR

Two alternative power ¬‚ow models to assess the impact of TCSC equipment in network-

´

wide applications are presented in this section (Ambriz-Perez, Acha, and Fuerte-Esquivel,

2000; Fuerte-Esquivel and Acha, 1996). The simpler TCSC model exploits the concept of a

variable series reactance. The series reactance is adjusted automatically, within limits, to satisfy a

speci¬ed amount of active power ¬‚ows through it. The more advanced model uses directly the

TCSC reactance“¬ring-angle characteristic, given in the form of a nonlinear relation. The TCSC

¬ring angle is chosen to be the state variable in the Newton“Raphson power ¬‚ow solution.

5.4.1 Variable Series Impedance Power Flow Model

The TCSC power ¬‚ow model presented in this section is based on the simple concept of a

variable series reactance, the value of which is adjusted automatically to constrain the power

¬‚ow across the branch to a speci¬ed value. The amount of reactance is determined

ef¬ciently using Newton™s method. The changing reactance XTCSC, shown in Figures 5.9(a)

and 5.9(b), represents the equivalent reactance of all the series-connected modules making

up the TCSC, when operating in either the inductive or the capacitive regions.

172 POWER FLOW INCLUDING FACTS CONTROLLERS

reg reg

P P

k km k km

m m

(a) (b)

Figure 5.9 Thyristor-controlled series compensator equivalent circuit: (a) inductive and (b) capa-

citive operative regions

The transfer admittance matrix of the variable series compensator shown in Figure 5.9 is

! ! !

given by

Ik jBkk jBkm Vk

¼ : °5:25Þ

Im jBmk jBmm Vm

For inductive operation, we have

)

Bkk ¼ Bmm ¼ À XTCSC ;