<<

. 7
( 17)



>>

Feasible region of active power ¬‚ow control for PS1 and PS2
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°2 SVC ފ : °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 Á SVC
½cos°2 SVC Þ À 1Š
0
XL
At the end of iteration (i), the variable ¬ring angle SVC is updated according to
°iÞ °iÀ1Þ °iÞ
SVC ¼ SVC þ Á SVC : °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°2 SVC Þ
The linearised power ¬‚ow equations are given as
2 3°iÞ
2 qGT--SVC
!°iÞ !°iÞ
6 0 Vk q SVC 7
ÁPk Ák
¼6 7 ; °5:18Þ
4 qBT--SVC 5 Á SVC
ÁQk
0 ÀVk 2
q SVC
where the Jacobian terms in explicit form are:
qGT--SVC RT qD
¼À 2 ; °5:19Þ
q SVC D q SVC
 
qBT--SVC qXSVC qD
1
¼ 2 ÀD þ XEq ; °5:20Þ
q SVC q SVC q SVC
DT
qD qXSVC
¼ 2XEq ; °5:21Þ
q SVC q SVC
qXSVC 2XSVC
2
¼ °1 À cos 2 SVC Þ; °5:22Þ
q SVC 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 Á SVC : °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 ;

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