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Modelling of Power System and Stability Analysis in Load Flow
I INTRODUCTION
Power system stability analysis tools and techniques, and the test cases used throughout the thesis is presented. A discussion of the critical points one needs to take into consideration in different system studies, such as continuation power flow or small-disturbance stability analysis, is also presented here.
1.1 Modeling
Models for power system components have to be selected according to the purpose of the system study, and hence, one must be aware of what models in terms of accuracy and complexity should be used for a certain type of system studies, while keeping the computational burden as low as possible. Selecting improper models for power system components may lead to erroneous conclusions. For example, the author in [2] studied the effect of using various load models on the system stability margin, showing that for some case studies, when only load models are changed, different stability margins in terms of MWs are obtained. In the following sections, the main elements of power systems, for the purpose of this thesis, are briefly discussed, and the corresponding models are reviewed.
1.2 Generators
Generators are important in system stability studies, and are modeled in dissimilar ways depending on the objective of the study. For instance, in a power flow study, a generator is modeled as a PV bus (defined as a bus with fixed voltage and power). For other complex analyses, such as small-disturbance stability, it may be required to use either generator subtransient or transient stability models that are represented by means of DAEs. The per unit stator voltage equations for generator detailed model in dq reference frame are typically written as [6]:
ed = p?d ? ?q?r ? Raid
eq = p?q + ?d?r ? Raiq ————-1.1
Where ed and eq are the instantaneous stator phase voltages; p is the differential operator d/dt; id and iq are the instantaneous stator phase currents; ?d and ?q are the flux linkages; ?r is the rotor electrical speed; and Ra is the armature resistance per phase. The two most common simplifications in obtaining generator stability 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 speed variations on stator voltages, i.e. ?r = 1 in 2.1. In addition to the abovementioned simplifications, other assumptions, such as balanced voltages with slowly varying phase and angle, yield generator stability models represented by differential equations with orders ranging from II (classical model) to VI (subtransient model) . For instance, a generator subtransient model is obtained assuming two q-axis and one d-axis damper windings on the rotor, and X?d = X?? q , where X?? d and X??q are subtransient reactances. On the other hand, a generator classical model is obtained by modeling the generator as a constant voltage source behind a reactance, and hence, only two differential equations are used to represent the electromechanical swing equations.
A generator is normally equipped with an exciter for primary voltage control and a governor for frequency control. Fast exciters are known to enhance generator synchronizing torque, but may deteriorate the damping [7], and hence, for some generators, a Power System Stabilizer (PSS) is installed to improve the damping. Several types of exciters, governors and PSSs are readily available (for more details, please refer to [6]), and are incorporated in most small-disturbance stability and transient stability analysis programs, such as the Power System Toolbox (PST) .
These models are not typically modeled in a power flow study; however, they have to be adequately represented in an eigenvalue analysis (small-disturbance analysis) or a transient stability analysis.
1.3 Loads
Load models are categorized as static and dynamic. Dynamic load models are more complicated, and are used mainly for transient stability analysis. On the other hand, static models are better suited for power flow and small-disturbance stability analysis. The three main static load models are known as constant PQ (or MVA), constant current and constant impedance; all of them can be mathematically expressed by
————-1.2
Where P0 and Q0 are the active and reactive power consumed at voltage V0, respectively. The type of the load model depends on exponents a and b, i.e. constant PQ for a = b = 0, constant current for a = b = 1, and constant impedance for a = b = 2.
II Synchronous static compensator (STATCOM)
Shunt compensators are primarily used to regulate the voltage in a bus by providing or absorbing reactive power. They are also known to be effective in damping electromechanical oscillations [4, 5]. Different kinds of shunt compensators are currently being used in power systems, of which the most popular ones are Static Var Compensator (SVC) and Synchronous STATIC COMPENSATOR (STATCOM) [37]; however, in this research, only the STATCOM, which has a more complicated topology, is explained and studied. SVCs and STATCOMs are thyristor based and GTO based FACTS controllers, respectively. A thyristor has only turn-on capability thus cannot be used in switch mode applications. Advanced devices such as Gate Turn-Off Thyristors (GTO) and Integrated Gate Bipolar Transistors (IGBT) have both turn-on and turn-off capabilities; hence, it is possible to use them in switched mode applications such as Voltage-Source Converters (VSC) in power systems he function of the converter output voltage denoted as Vout in Figure 2.1, i.e.
.
——1.3
Where ?conv is the angle between the ac system voltage V and Vout. Two control strategies may be used for a STATCOM; namely, Phase Control and PWM Control. In phase control, the DC bus voltage 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 block diagram of a phase control is shown in Figure 2.2. On the other hand, in the PWM control, both angle and magnitude of the converter output voltage are regulated as shown in Figure 2.3.
Although less low frequency harmonics are produced by a STATCOM with a PWM control, the high switching losses due to the high switching frequency are the main constraints for its application in transmission systems. The maximum and minimum operating points of a STATCOM are independent
Figure 1.1: Basic structure of STATCOM.
From the system voltage as opposed to an SVC. The V-I characteristic of a STATCOM is limited only by the maximum voltage and current rating as depicted in Figure 2.4. This controller can be operated over its full output current range even at very low voltages (typically 0.2 p.u.).
STATCOM Transient Stability (TS) Model For the case that the output voltage of the STATCOM is balanced and harmonic free, a TS model has been proposed, which does not include converter switching phenomena [1]. The STATCOM TS model replaces the detailed model with a variable voltage source as shown in Figure 2.5, in which the magnitude of capacitor voltage is determined by a differential equation derived based on the power exchange
Figure 1.2: STATCOM control block diagram with phase control.
Figure 1.3: STATCOM control block diagram with PWM control.
Figure 1.4: Voltage-Current characteristic of a STATCOM.
Between the STATCOM and the network [1, 40]:
—-1.4
Where a stands for the transformer ratio, and the resistance Rc represents the converter losses, which can be significant, depending on the number of switches and the switching frequency. All the blocks in Figure 2.5 are the same as the STATCOM detailed model in Figure 2.3, except that the converter and the blocks related to the switches are replaced by a voltage source kVdc\?; the coefficient k is proportional to the modulation index ma, which for a two-level inverter is ma2?2.
It has been shown by means of time-domain simulation results that the TS model response is reasonably close to that obtained with the detailed model when the transients are small [1].
Figure 1.5: STATCOM transient stability model and its control.
III Power System Stability
From the aforementioned models, a power system model can be represented using
a DAE model, such as :
x = f(x, z, ?, ?) ———————–1.5
0 = g(x, z, ?, ?)
y = h(x, z, ?, ?)
Where x ? ?nis a vector of state variables that represents the state variables of generators, loads and other system controllers; z ? ?m
is a vector of steady state algebraic variables that result from neglecting fast dynamics in some load phasor voltage magnitudes and angles; ? ? ?? is a set of uncontrollable parameters such as active and reactive power load variations; ? ? ?a is a set of controllable parameters such as tap or AVR set points; and y ? ?l is a vector of output variables such as power through the lines and generator output power. The nonlinear functions
f: ?n × ?m × ?? × ?a 7? ?n, g : ?n × ?m × ?? × ?a 7? ?m, and h : ?n × ?m ×?? ×?a 7? ?l
Stand for the differential equations, algebraic constraints and output variable measurements, respectively.
The DAE model in (2.8) can be linearized about an operating point (xo, zo, ?o, ?o) to obtain the system state matrix A:
—–1.6
For slowly varying parameters ?, the power system model has been shown to present local bifurcations, on which most stability indices in the current literature are based.
3.1 Voltage Stability
In a power system, voltage stability is directly related to the voltage on the system buses, and is defined as the power system ability to maintain steady acceptable voltages at all buses under normal operating conditions and after a contingency [6]. Thus, if the bus voltage magnitude decreases as the reactive power injection at the same bus increases, the power system is voltage unstable. This may lead to voltage collapse, if generators or other reactive power sources do not provide enough reactive power support. Voltage collapse can be explained within the context of bifurcation theories applied to DAEs in nonlinear systems, namely, SNB and LIB [3]. Saddle-node Bifurcations (SNB) When the system state matrix A has a simple and unique zero eigenvalue with nonzero left and right eigenvectors, the equilibrium point (xo, zo, ?o, ?o) is typically referred to as SNB point (other transversality conditions must also be met). In power systems, this bifurcation point is associated with voltage stability problems due to the local merger and disappearance of equilibrium (operating points) as ? changes.
3.2 Continuation Power Flow
For given dispatch scenarios, the continuation power flow [43] technique is used to obtain P-V curves similar to the one depicted in Figure 2.6, and thus determine the static loading margin (SLM) of the system (nose point) associated with a voltage collapse point, which could be the result of an SNB or an LIB. Figure 2.6 also demonstrates the dynamic loading margin (DLM) of a system, which is associated with an angle instability happening before the nose point. All the P-V curves in this work have been
Figure 1.6: A typical PV curve and corresponding SLM and DLM.
as it has been developed in C and C++, and hence appropriate to study large systems.
3.3 Small-Disturbance Stability Analysis
As explained before, matrix A and its eigenvalues can provide valuable information
About the system stability for small perturbations that may occur in the system.
This is also referred to as small-disturbance stability analysis or eigenvalue analysis. In this work, matrix A and its eigenvalues for the test cases have been obtained by means of the linearized transient stability models in the Power System Toolbox (PST) , which is a MATLAB based program. PST, when compared to other programs, is user-friendly but slow, and hence inappropriate for large systems (more than 50 buses). Therefore, for large systems, the Small Signal Analysis Tool (SSAT) is used; as it is able to deal with systems made up of several thousand buses.
It offers powerful features, such as complete eigenvalue analysis; Single-Machine Infinite-Bus (SMIB) analysis; eigenvalue analysis within specified frequency and damping ranges; computation of modes closest to a specified frequency and damping; computation of modes related to a generator; sensitivity analysis; mode trace;etc.
IV Time-Domain Simulation
Time-domain simulation is mainly used for transient stability analysis of power systems following large perturbations, as it accounts for all the nonlinear effects by solving the complete set of DAEs by means of step-by-step trapezoidal or predictor-corrector integration [6]. However, in this thesis, this time-domain response of the power system is also used to obtain important small-disturbance stability information. Time-domain simulations of test cases were carried out by means of both the PST and the Transient Stability Analysis Tool (TSAT) [10]; however, the simulation of large systems was only feasible with the later. TSAT has two simulation engines: A conventional time-domain simulation engine that uses full numerical integration techniques and a fast time-domain simulation engine based on a quasi
Steady-state system model. It has several useful features for transient stability analysis, such as the possibility of running multi-contingency cases or multi-dispatch scenarios, obtaining a security index based on critical clearing time, etc. A wide range of dynamic models of power system components is available, and well-known formats, such as PTI PSS/E, GE PSLF, and BPA can be used as input data.
4.1 Test Systems
A variety of test cases, ranging from a Single-Machine-Infinite-Bus (SMIB) to a real power system with 14,000 buses, were used to test the feasibility of the proposed stability indices and system identification techniques. In some cases, several dispatch scenarios were considered in order to emulate the operation of a real power system. The general characteristics of these test cases are briefly reviewed in this section.
4.2 Single-Machine-Infinite-Bus (SMIB)
This is the simplest but the most widely used test case, as it consists of only a generator, a transmission line and a load as depicted in Figure 2.7. The load bus is modeled as an infinite bus, which is normally used to replace a stiff large system with a constant voltage magnitude and angle. This system can be used to investigate the behavior of a generator or group of generators, labeled as G1 in
Figure 2.7, with respect to the infinite bus.
4.3 IEEE 3-bus System
This corresponds to a case where two areas are connected through a long transmission line (weak connection); hence, power oscillations are observed in the tie-line.
Figure 1.7: IEEE 3-bus test system.
A single-line diagram of the test system is shown in Figure 2.8 [2]. The base load used at Bus 3 is a 900 MW and 300 MVar load, and is modeled as a constant PQ. Each machine has a simple exciter, and a simple governor is used for the machine at Bus 1. The generators are modeled in detail by means of subtransient models.
The corresponding static and dynamic data is presented in Appendix A.1.
Figure 1.8: IEEE 14-bus test system.
Figure 1.9: Two-area benchmark system.
Tie-lines, hence resulting in an inter-area mode with a frequency of about 0.7 Hz. However, the individual machines in each area also contribute to a local mode in the same area with a frequency of about 1.3 Hz. Therefore, an inter-area rotor angle mode and two local modes are observed for this test case. The generators were modeled using subtransient models and their exciters are simple exciters equipped with PSSs. The corresponding static and dynamic data is given in Appendix A.3. The total base loading level is 2734 MW and 200 MVar.
V.CONCLUSION
A brief explanation of some of the key power system components used in this thesis, such as loads and generators, is presented in this chapter. Also discussed in this chapter is the importance of selecting the right models for different kinds of analyses. Power system stability concepts and the analysis techniques and tools used throughout this thesis, such as voltage and angle stability, continuation power
Flow and system identification are briefly explained.
REFERENCES
[1] E. Uzunovic, “Transient Stability and Power Flow Models of VSC FACTS controllers,” Ph.D. dissertation, University of Waterloo, Waterloo, ON, Canada, 2001.
[2] N. Mithulananthan, “Hopf bifurcation control and indices for power system with interacting generator and FACTS controllers,” Ph.D. dissertation, University of Waterloo, Waterloo, ON, Canada, 2002.
[3] C. A. Ca˜nizares, N. Mithulananthan, A. Berizzi, and J. Reeve, “On the linear
Profile of indices for the prediction of saddle-node and limit-induced bifurcation
Points in power systems,” IEEE Trans. Circuits and Systems, vol. 50, no. 2,
pp. 1588–1595, December 2003.
[4] N. Mithulananthan, C. A. Ca˜nizares, J. Reeve, and G. J. Rogers, “Comparison
of PSS, SVC and STATCOM controllers for damping power system oscillations,” IEEE Trans. Power Systems, vol. 18, no. 2, pp. 786–792, May 2003.
[5] N. Mithulananthan, C. A. Ca˜nizares, and J. Reeve, “Hopf bifurcation control
in power system using power system stabilizers and static var compensators,”
in Proc. of NAPS’99, San Luis Obispo, California, October 1999, pp. 155–163.
[6] P. Kundur, Power System Stability and Control. New York: McGraw-Hill,
1994.
[7] F.P.Demello and C.Concordia,“Concepts of synchronous machine stability as affected by excitation control” IEE E Trans.Power Apparatus and Systems, vol. PAS-88, no. 4, pp. 316 –329, April 1969.
[8] G. Hingrani, Understanding FACTS. New York: IEEE Press, 2000.
[9] C.A.Caizares and F.L.Alvarado, “Point of collapse and continuation methods for large ac/dc systems,” IEEE Trans.Power Systems, vol. 8, no. 1, pp. 1–8,
[10] Transient Security Assessment Tool (TSAT), User Manual, Power tech Labs
Inc., Surrey, BC, Canada, V3W 7R7, 2002.
About the Author
Assistant professor in lord venkateswara engineering college.I am doing phd in sathyabama university, Tamil Nadu,India.
