Modelling of Power System and Stability Analysis in Load Flow

I INTRODUCTIONoperating point (xo, zo, ?o, ?o) to obtain the system
            Power system stability analysisstate matrix A:
tools and techniques, and the test cases used-----1.6
throughout the thesis is presented. A discussion of the 
critical points one needs to take into consideration inFor slowly varying parameters ?, the power system
different system studies, such as continuation powermodel has been shown to present local bifurcations, on
flow or small-disturbance stability analysis, is alsowhich most stability indices in the current literature are
presented here.based.
1.1 Modeling3.1 Voltage Stability
Models for power system components have to be 
selected according to the purpose of the systemIn a power system, voltage stability is directly related to
study, and hence, one must be aware of what modelsthe voltage on the system buses, and is defined as the
in terms of accuracy and complexity should be usedpower system ability to maintain steady acceptable
for a certain type of system studies, while keeping thevoltages at all buses under normal operating conditions
computational burden as low as possible. Selectingand after a contingency [6]. Thus, if the bus voltage
improper models for power system components maymagnitude decreases as the reactive power injection
lead to erroneous conclusions. For example, the authorat the same bus increases, the power system is
in [2] studied the effect of using various load modelsvoltage unstable. This may lead to voltage collapse, if
on the system stability margin, showing that for somegenerators or other reactive power sources do not
case studies, when only load models are changed,provide enough reactive power support. Voltage
different stability margins in terms of MWs arecollapse can be explained within the context of
obtained. In the following sections, the main elements ofbifurcation theories applied to DAEs in nonlinear
power systems, for the purpose of this thesis, aresystems, namely, SNB and LIB [3]. Saddle-node
briefly discussed, and the corresponding models areBifurcations (SNB) When the system state matrix A
reviewed.has a simple and unique zero eigenvalue with nonzero
1.2 Generatorsleft and right eigenvectors, the equilibrium point (xo, zo,
Generators are important in system stability studies,?o, ?o) is typically referred to as SNB point (other
and are modeled in dissimilar ways depending on thetransversality conditions must also be met). In power
objective of the study. For instance, in a power flowsystems, this bifurcation point is associated with
study, a generator is modeled as a PV bus (defined asvoltage stability problems due to the local merger and
a bus with fixed voltage and power). For otherdisappearance of equilibrium (operating points) as ?
complex analyses, such as small-disturbance stability, itchanges.
may be required to use either generator subtransient 
or transient stability models that are represented by3.2 Continuation Power Flow
means of DAEs. The per unit stator voltage equations 
for generator detailed model in dq reference frameFor given dispatch scenarios, the continuation power
are typically written as [6]:ed = p?d ? ?q?r ? Raideq =flow [43] technique is used to obtain P-V curves similar
p?q + ?d?r   ? Raiq -------------1.1to the one depicted in Figure 2.6, and thus determine
 the static loading margin (SLM) of the system (nose
Where ed and eq are the instantaneous stator phasepoint) associated with a voltage collapse point, which
voltages; p is the differential operator d/dt; id and iq arecould be the result of an SNB or an LIB. Figure 2.6 also
the instantaneous stator phase currents; ?d and ?q aredemonstrates the dynamic loading margin (DLM) of a
the flux linkages; ?r is the rotor electrical speed; and Rasystem, which is associated with an angle instability
is the armature resistance per phase. The two mosthappening before the nose point. All the P-V curves in
common simplifications in obtaining generator stabilitythis work have been
models are: First, neglect the stator transients, which 
are represented by the p?d and p?d terms in 2.1; these 
terms are associated with network transients, which 
decay rapidly. Second, neglect the effect of speedFigure 1.6: A typical PV curve and corresponding SLM
variations on stator voltages, i.e. ?r = 1 in 2.1. In additionand DLM.
to the abovementioned simplifications, other as it has been developed in C and C++, and hence
assumptions, such as balanced voltages with slowlyappropriate to study large systems.
varying phase and angle, yield generator stability3.3 Small-Disturbance Stability Analysis
models represented by differential equations withAs explained before, matrix A and its eigenvalues can
orders ranging from II (classical model) to VIprovide valuable information
(subtransient model) . For instance, a generator About the system stability for small perturbations
subtransient model is obtained assuming two q-axisthat may occur in the system.
and one d-axis damper windings on the rotor, and X?dThis is also referred to as small-disturbance stability
= X?? q , where X?? d and X??q are subtransientanalysis or eigenvalue analysis. In this work, matrix A
reactances. On the other hand, a generator classicaland its eigenvalues for the test cases have been
model is obtained by modeling the generator as aobtained by means of the linearized transient stability
constant voltage source behind a reactance, andmodels in the Power System Toolbox (PST) , which is
hence, only two differential equations are used toa MATLAB based program. PST, when compared to
represent the electromechanical swing equations.other programs, is user-friendly but slow, and hence
A generator is normally equipped with an exciter forinappropriate for large systems (more than 50 buses).
primary voltage control and a governor for frequencyTherefore, for large systems, the Small Signal Analysis
control. Fast exciters are known to enhance generatorTool (SSAT) is used; as it is able to deal with systems
synchronizing torque, but may deteriorate the dampingmade up of several thousand buses.
[7], and hence, for some generators, a Power SystemIt offers powerful features, such as complete
Stabilizer (PSS) is installed to improve the damping.eigenvalue analysis; Single-Machine Infinite-Bus (SMIB)
Several types of exciters, governors and PSSs areanalysis; eigenvalue analysis within specified frequency
readily available (for more details, please refer to [6]),and damping ranges; computation of modes closest to
and are incorporated in most small-disturbance stabilitya specified frequency and damping; computation of
and transient stability analysis programs, such as themodes related to a generator; sensitivity analysis;
Power System Toolbox (PST) .mode trace;etc.
These models are not typically modeled in a powerIV Time-Domain Simulation
flow study; however, they have to be adequatelyTime-domain simulation is mainly used for transient
represented in an eigenvalue analysisstability analysis of power systems following large
(small-disturbance analysis) or a transient stabilityperturbations, as it accounts for all the nonlinear
analysis.effects by solving the complete set of DAEs by
1.3 Loadsmeans of step-by-step trapezoidal or
Load models are categorized as static and dynamic.predictor-corrector integration [6]. However, in this
Dynamic load models are more complicated, and arethesis, this time-domain response of the power system
used mainly for transient stability analysis. On the otheris also used to obtain important small-disturbance
hand, static models are better suited for power flowstability information. Time-domain simulations of test
and small-disturbance stability analysis. The three maincases were carried out by means of both the PST
static load models are known as constant PQ (orand the Transient Stability Analysis Tool (TSAT) [10];
MVA), constant current and constant impedance; all ofhowever, the simulation of large systems was only
them can be mathematically expressed byfeasible with the later. TSAT has two simulation
 engines: A conventional time-domain simulation engine
-------------1.2that uses full numerical integration techniques and a
 fast time-domain simulation engine based on a quasi
Where P0 and Q0 are the active and reactive powerSteady-state system model. It has several useful
consumed at voltage V0, respectively. The type of thefeatures for transient stability analysis, such as the
load model depends on exponents a and b, i.e.possibility of running multi-contingency cases or
constant PQ for a = b = 0, constant current for a = bmulti-dispatch scenarios, obtaining a security index
= 1, and constant impedance for a = b = 2.based on critical clearing time, etc. A wide range of
II Synchronous static compensator (STATCOM)dynamic models of power system components is
Shunt compensators are primarily used to regulate theavailable, and well-known formats, such as PTI PSS/E,
voltage in a bus by providing or absorbing reactiveGE PSLF, and BPA can be used as input data.
power. They are also known to be effective in4.1 Test Systems
damping electromechanical oscillations [4, 5]. DifferentA variety of test cases, ranging from a
kinds of shunt compensators are currently being usedSingle-Machine-Infinite-Bus (SMIB) to a real power
in power systems, of which the most popular ones aresystem with 14,000 buses, were used to test the
 STATIC COMPENSATOR (STATCOM) [37];feasibility of the proposed stability indices and system
however, in this research, only the STATCOM, whichidentification techniques. In some cases, several
has a more complicated topology, is explained anddispatch scenarios were considered in order to
studied. SVCs and STATCOMs are thyristor basedemulate the operation of a real power system. The
and GTO based FACTS controllers, respectively. Ageneral characteristics of these test cases are briefly
thyristor has only turn-on capability thus cannot bereviewed in this section.
used in switch mode applications. Advanced devices4.2 Single-Machine-Infinite-Bus (SMIB)
such as Gate Turn-Off Thyristors (GTO) andThis is the simplest but the most widely used test
Integrated Gate Bipolar Transistors (IGBT) have bothcase, as it consists of only a generator, a transmission
turn-on and turn-off capabilities; hence, it is possible toline and a load as depicted in Figure 2.7. The load bus
use them in switched mode applications such asis modeled as an infinite bus, which is normally used to
Voltage-Source Converters (VSC) in power systemsreplace a stiff large system with a constant voltage
he function of the converter output voltage denotedmagnitude and angle. This system can be used to
as Vout in Figure 2.1, i.e.investigate the behavior of a generator or group of
 generators, labeled as G1 in
.Figure 2.7, with respect to the infinite bus.
------1.34.3 IEEE 3-bus System
  
Where ?conv is the angle between the ac systemThis corresponds to a case where two areas are
voltage V and Vout. Two control strategies may beconnected through a long transmission line (weak
used for a STATCOM; namely, Phase Control andconnection); hence, power oscillations are observed in
PWM Control. In phase control, the DC bus voltagethe tie-line.
Vdc is regulated by changing ?conv, i.e. charging and 
discharging the DC capacitor, which ultimately controls. 
Vout, as this voltage is proportional to Vdc; the blockFigure 1.7: IEEE 3-bus test system.
diagram of a phase control is shown in Figure 2.2. On 
the other hand, in the PWM control, both angle andA single-line diagram of the test system is shown in
magnitude of the converter output voltage areFigure 2.8 [2]. The base load used at Bus 3 is a 900
regulated as shown in Figure 2.3.MW and 300 MVar load, and is modeled as a
 constant PQ. Each machine has a simple exciter, and
Although less low frequency harmonics are produceda simple governor is used for the machine at Bus 1.
by a STATCOM with a PWM control, the highThe generators are modeled in detail by means of
switching losses due to the high switching frequencysubtransient models.
are the main constraints for its application inThe corresponding static and dynamic data is
transmission systems. The maximum and minimumpresented in Appendix A.1.
operating points of a STATCOM are independent 
Figure 1.8: IEEE 14-bus test system.
 Figure 1.9: Two-area benchmark system.
Figure 1.1: Basic structure of STATCOM. 
 Tie-lines, hence resulting in an inter-area mode with a
From the system voltage as opposed to an SVC. Thefrequency of about 0.7 Hz. However, the individual
V-I characteristic of a STATCOM is limited only by themachines in each area also contribute to a local mode
maximum voltage and current rating as depicted inin the same area with a frequency of about 1.3 Hz.
Figure 2.4. This controller can be operated over its fullTherefore, an inter-area rotor angle mode and two
output current range even at very low voltageslocal modes are observed for this test case. The
(typically 0.2 p.u.).generators were modeled using subtransient models
 and their exciters are simple exciters equipped with
STATCOM Transient Stability (TS) Model For thePSSs. The corresponding static and dynamic data is
case that the output voltage of the STATCOM isgiven in Appendix A.3. The total base loading level is
balanced and harmonic free, a TS model has been2734 MW and 200 MVar.
proposed, which does not include converter switchingV.CONCLUSION
phenomena [1]. The STATCOM TS model replacesA brief explanation of some of the key power system
the detailed model with a variable voltage source ascomponents used in this thesis, such as loads and
shown in Figure 2.5, in which the magnitude ofgenerators, is presented in this chapter. Also discussed
capacitor voltage is determined by a differentialin this chapter is the importance of selecting the right
equation derived based on the power exchangemodels for different kinds of analyses. Power system
 stability concepts and the analysis techniques and tools
used throughout this thesis, such as voltage and angle
Figure 1.2: STATCOM control block diagram withstability, continuation power
phase control.Flow and system identification are briefly explained.
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