The voltage stability problem has become a major concern in operating and planning today's power system as a result of heavier loading conditions, without sufficient transmission and/or generation enhancements Voltage stability problems normally occur in heavily stressed systems. While the disturbance leading to Voltage collapse may be initiated by a variety of causes, the underlying problem is an inherent weakness in the power system. The factors contributing to Voltage collapse are the generator reactive power voltage control limits, load characteristics, characteristics of reactive compensation devices and the action of the voltage control devices such as (ULTCs) (Thukaram et al., 2004).
The under load tap changer (ULTC) an important voltage regulation device automatically
adjusts its turns ratio in order to keep the load side voltage within an acceptable
range. The time constants of ULTCs are usually between 20s and 100s and therefore
can be considered to be a slow dynamic device. In transient stability analysis,
the dynamics of the ULTC can be neglected and its turns ratio assumed
to be constant. However, a mid-term voltage instability incident leading toward
voltage collapse is often a slow, gradual process (Dong et al., 2004).
ULTCs have been shown to play an important role in long term voltage collapse,
since they aim to keep load voltages and therefore the load power constant even
though transmission system voltages may be reduced. Considerable effort has
been given to voltage collapse are closely linked to dynamic interaction between
dynamics of voltage collapse are closely linked to dynamic interaction between
the ULTCs and loads. It led to significant progress in the area of dynamic load
modeling (Larsson, 2000). The increase of power demand at a higher rate than
the expansion of generation and transmission facilities has resulted in power
systems functioning closer to their operational and physical limits. In heavily
loaded systems, voltage profile in the transmission systems is often maintained
by generator reactive power injection or by adjusting the tap ratio of load
tap changing transformers (Roman, 2006). Many papers analyze tap changer behavior
at distribution points, where the interaction between the load and the system
characteristics can lead to voltage collapse (Vu, 1992; Venkatasubramanian,
2000; Popovic, 1996). Zhu (2000) and Vournas (2002) use static analysis to show
the ULTCs effect on the increase of maximum power transfer and its impact
on system stability.
It is well known that the operation of the ULTC has a significant influence on mid-term voltage instability (Dong et al., 2004). In (Van Cutsem, 1998), a stability region around the equilibrium point is constructed. In (Vu, 1992), the mechanisms of voltage collapse have been studied by considering a dynamic load characteristic as well as ULTC dynamics. In both references (Vu, 1992; Van Cutsem, 199), the voltage V at the load bus is considered as a state variable in the load model. Voltage stability of a power system is related to load characteristics and voltage control devices such as ULTC transformers.
The main focus of these studies which mainly relate to voltage stability questions has been of understanding of the complex dynamic nature of voltage collapse to which ULTC dynamics significantly contributes This study investigates the ULTCs effect on dynamic loads and attempts to underscore some of the discrepancies with static analysis results. Voltage instability in a system leads to a loss of post-disturbance equilibrium. Extending the power transfer capability helps to make the system re-enter a reasonable post-disturbance operating condition. We use static analysis to show the ULTCs effect on the increase of maximum power transfer and its impact on system stability. However, voltage stability is a dynamic phenomenon and analysis based on static modeling is not sufficient and usually leads to erroneous results. In the third section of this paper, the effect of the ULTC on dynamic analysis is theoretically studied for a simple test system. This paper further explores the oscillatory behavior of power supply systems with emphasis on illustrating interactions between ULTC and load dynamics. These oscillatory behavior curves serve their purpose well on the conservative side in predicting several system limits such as voltage stability limit or loadability limit. Load models are known to have profound impacts on power system behaviors.
Static voltage stability analysis:
Static load model:
Voltage-sensitive loads can be modeled as:
Where P0 is the real power at V0, Q0 is the reactive power at V0, V is the bus voltage magnitude, α is the voltage sensitivity exponent of the real power, β is the voltage sensitivity exponent of the reactive power. If α = 0, β = 0, then the load model essentially represents a constant power type load; if α = 1, β = 1, the load model becomes a constant current type and if α = 2, β = 2, then it becomes a constant impedance type load. An ULTC can automatically adjust and keep the load-side bus voltage constant given that enough buck or boost taps exist (Van Cutsem, 1998).
Two bus sample system: A simple two bus example system considering static analysis is shown in Fig. 1. Where X = 0.5 p.u and B1= 0.01 p.u are reactance and shunt capacity of transmission line, respectively. The load power demand is Pd + jQd . For the sake of simplicity but without loss of generality, the resistance of the transmission line is neglected; i.e.V1 and V2 are voltage values of generator and load bus, respectively.
Active and reactive power values fed from buses using load flow equations are
given Eq. 3-5
Where a and Xt= 0.1 p.u are tap rate and leakage reactance of ULTC, respectively. Figure 2 shows the extended maximum power transfer (knee points on the PV curves) by the increased tap ratio settings. All three curves, showing voltage at bus 2 versus power transfer, are with unity power factor load at bus 2. Based on the above analysis, one can conclude that if the improvement of the power transfer capability due to ULTC operation is more than the increase in the load, then there exists an equilibrium point and the system can maintain stability. This phenomenon has potentially beneficial effects on static voltage stability of the system.
|| A simple two bus example system
|| PV curves with ULTC of bus 2
The effect of shunt capacity B1: Here, the effects of voltage
stability of various B1 values stated in Eq. (4)
are examined. In order to this analysis, the best suitable method is to obtain
PV curves. In this analysis, tap rate and initial condition of ULTC are taken
as 1.0 and 0.01 p.u., respectively. In PV curves pictured in Fig.
3, with discrete and points curves are to belong to values of 0.009 p.u.
and 0,011 p.u. of B1, respectively. Increasing of B1 has
affected to improve of voltage stability as positive direction.
Dynamic analysis: The ULTC is beneficial in extending the power transfer capability and thus increasing the static stability margin. However, the margin obtained by the static method is rather optimistic. In general, the voltage stability problem involves complex dynamic phenomena. A common situation that is often encountered is that the system can collapse after a disturbance even if a post disturbance equilibrium point exists. In such cases, detailed dynamic models need to be used to analyze systemstability.
Power system model: The simple two bus system of show in Fig. 1. The p.u dynamic equations that represent this system, using a basic dynamic generator model, a frequency and voltage dependent dynamic model for the load, are given by
|| Curves with ULTC in different B1 values
where δ is the generator rotor angle, ω the generator angular speed, M the generator inertia constant, PM the mechanical power of prime mover, DG the generator damping, τ the voltage time constant of the dynamic-load and V2 the bus voltage of the dynamic-load. The load power demand is Pd + jQd. PM = Pd (Canizares, 1995). Furthermore, it is assumed that the load, in steady-state conditions, has a constant power factor, i.e., Qd =kPd, where k is a given constant. The active power demand Pd of the dynamic load is the parameter that can be varied, All other system parameters are as follows; M= 0.9 s, DG= 0.1, k = 0.3 and τ = 2 s.
Generic dynamic load model: The typical load-voltage response characteristics can be modeled by a generic dynamic load model proposed in Fig. 4.
In this model, x is the state variable. Pt(V) and Ps(V) are the transient and steady-state load characteristics respectively and can be expressed as,
Where V is the per-unit magnitude of the voltage imposed on the load. It can
be seen that, at steady-state, state variable x of the model is constant. The
input to the integration block, e = Ps-Pd must be zero
and, as a result, the model output is determined by the steady-state characteristics
Ps = Pd. For any sudden voltage change, x maintains its
predisturbance value initially. Because the integration block cannot change
its output instantaneously.
|| A generic dynamic load model
The transient output is then determined by the transient characteristics Pd
= xPt. The mismatch between the model output and the steady-state
load demand is the error signal e. This signal is fed back to the integration
block that gradually changes the state variable x. This process continues until
a new steady-state (e = 0) is reached. Analytical expressions of the load model
including real (Pd) and reactive (Qd) power dynamics are,
Typical load parameters are αt = 0.72~1.30, βt = 2.96~4.38 for residential load; αt = 0.99~1.51, βt = 3.15~3.95 for commercial load and αt = 0.18, βt = 6.0 for industrial load. The steady-state load parameters and load time constant depend strongly on the voltage level at which the load is aggregated. Since the downstream ULTCs play a very important role. To authors knowledge, no data has been fully documented and αs = 0, βs = 0 are commonly used (Xu and Marisour, 1994).
Continuous ULTC model: The continuous ULTC model is based on the assumption of a continuously changing tap a (t), which can take all real values between amin and amax. Usually the effect of the dead band is neglected in a continuous ULTC model, so that the following differential equation results,
Where V02 is the reference voltage and Tc
is the time constant and taken 120 s as simulation time. The ULTC is modeled
as an integral controller using Eq. (13).
Simulations: In the Fig. 1, that load bus has dynamic
load is admitted. Generic dynamic load is used as load model. It is analyzed
though being three type loads as residential, commercial and industrial load.
Typical parameters commercial and industrial load.
|| PV curves for different load types
Typical parameters in order to each load are chosen as following; αt =
0.75, βt = 3.0 for residential load (Type I), αt = 1.25, βt =
3.75 for commercial load (Type II) and αt = 0.18, βt = 6.0 for industriaload
(Type III). αs and βs are taken as zero for
PV curves attained each load type are shown in Fig. 5. It
can be observed that the system has led toward unstable at Pd = 0.6
p.u. As from this point, the curves obviously have bifurcated. That parameter
βt is bigger with respect to other load types are delayed collapse
time. This result with respect to being stable state again at heavy loading
conditions of post disturbance of system is very important. The simulations
obtained the relation to this results are shown in Fig. 6 (a,
b). In this stable system having to Po = 0.6 p.u
and Qo = 0.2 p.u values, the load at t = 500 s is supposed to increase
Po = 0.7 p.u as disturbance effect. The same ratio increasing is
also validity the reactive power. The disturbance effect has been applied to
each three load types are shown to return to initial operation point at t =
750 s. The post disturbance, the voltage of load bus has begun to step down.
Thus, according to Eq. (12), Qs (V)-yQβt
is bigger than the zero. This difference is bigger at Type I having the smallest
βt parameters. Thus, deficit of reactive power of load bus is
getting more increasing. In case of this increasing has trigger to collapse,
so the collapse time of Type I have accelerated. After the disturbance is cleaned,
in order to being a stable state of the system that the difference Qs(V)
and yQβt has converged to zero has required. It is
said that Type III having more big βt value is being to stable
state more fast with respect to the other load types. In order to each three
load types, the state space trajectories for Type II and Type III are stable,
for also Type I is unstable (Fig. 6a). The state space trajectory
for state t= 0s are shown in Fg. 6b. Since the load voltage is below from V02,
the transformers begin to step down its tap ratio.
The effect of Tc: The stability analysis in order to two
different values of Tc parameter in ULTC model given Eq.
(13) is arranged and plotted to phase portraits as shown in Fig.
7a-c, also the curves attained from changing voltage and
angle as shown in Fig. 7b and d.
||In during disturbance and post disturbance of each load types
(a) The change of between voltage and time, (b) The state space trajectory
||Phase portraits for Type I (a),(b) Time series and V-δ
phase plane for Tc = 20s, respectively, (c),(d) Time series and
V-δ phase plane for Tc = 120s, respectively, (V──── , δ
|| Stable limit cycle behavior of ULTC after the transient chaos
The system loaded at a value close to the collapse point is observed to being
stable, when the parameter Tc is chosen as at interval 0-20s. If
parameter Tc is chosen upper than this interval, the system has been
unstable as shown in Fig. 7d.
The system has converged to stable operating point with damped transient oscillation as shown in Fig. 8. When δ0 is equal to 0.6. After the oscillation damped, while the system is getting to stable state, δ and ω values try to return their initial points. The system has been stable limit cycle, due to tap changer of ULTC.
An electrical power system consists of many loads that have different characteristics.
Load characteristics are known to have a significant effect voltage dynamics.
Voltage stability depends on the details particularly the load characteristics.
It is very important have to know characteristics of loads and behaviors in
the voltage stability. The energy quality is usually broken down by nonlinear
loads. In this study investigated effect different loads those voltage stability
behaviors of against voltage changes. In this paper, the results of a static
analysis show that ULTCs can increase the power transfer capability and improve
the voltage stability. The effects of ULTCs are investigated by both static
and dynamic analysis. In respect to power transfer and voltage stability, Increasing
shunt capacity of line have given good results (Fig. 3). Thus,
in the state being outage one of the parallel lines, the benefit effects of
shunt capacitance of the line must also be thought. In dynamic analysis, the
load is modeled as a generic dynamic Load model. The ULTC is also modeled as
continuous dynamic model. The steady-state load parameters and time constant
of load depend strongly on the voltage level at which the load having different
parametric values is applying. Since the downstream ULTCs play a very
important role. When the time constants and τ increase, Pd has
also increased. In the mean while voltage stability margin is also increased.
Thus, in the voltage stability studies must be given an importance to time constants
of loads and ULTCs. Voltage stability of a power system is related to load characteristics
and voltage control devices such as (ULTC) transformers. Voltage collapse period
of the post disturbance have shown the difference as depend on parametric values
of loads with different characteristic as shown in Fig. 6a.
Because the power system with ULTC having loads as Type II and Type III have
collapsed immediately. The interferences such as supporting reactive power,
tap changing, cleaning the disturbance effect have to been more soon.