ABSTRACT
The objective of the research is to provide the application of Generalized Average models of FACTS device, TCSC and custom power device, DVR for power system transient analysis within dynamic phasor approach. The dynamic phasor models of TCSC and DVR are evaluated through simulation of realistic network model of 3 machines 9 bus WECC system. The continuous-time large signal models of TCSC and DVR under a fault condition captured the phasor dynamics of key wave forms. It is found that the increase transient stability and improve power quality response of dynamic phasor model of TCSC and DVR, respectively are closely hugged with the detail time domain EMTP simulation. Numerical simulation of phasor dynamics with larger size steps through solving differential equation is also faster than that of EMTP. Both EMTP and theoretical framework of dynamic phasor models has been simulated using Simulink/Matlab dynamic system modelling software.
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DOI: 10.3923/jas.2006.1074.1081
URL: https://scialert.net/abstract/?doi=jas.2006.1074.1081
INTRODUCTION
The objective of the electric power system is to generate and to distribute power to its consumers in a stable, viable and optimal fashion. Its needs that the system can withstand certain major disturbances of the equipments without severe consequence. This motivates the notion of the transient stability (Perez et al., 1994), which is essentially the ability of the system operation to recover its stable operating point without major break ups after a special set of first contingencies. The instability problems is one of the severest fault in power system operation loss of synchronism may cause interruption of power supply in large areas, consequently introduces heavily damage to the distribution network and social life (Shuti et al., 1993).
Now days, voltage source power electronic converters are widely used often referred to a flexible AC transmission systems (FACTS) and custom power devices, promise to play an important role in emerging deregulated power system. The FACTS device likewise thyristor controlled series compensation (TCSC) is used in power systems for increasing power transfer capability and enhancement of transient stability or to dampen power oscillations and sub synchronous resonances. Dynamic voltage restorer (DVR) is a custom power device designed specially to improve dynamic response of power quality at distribution or transmission level.
Over the years, for the system equivalent and size of integration steps, the electromagnetic transient program, EMTP concept is universally accepted for simulation of complex power system containing non-linearties, power electronic components and their controllers (Adapa and Reeve, 1988). However, due to the limitation of practical computer storage and computation time, the implementation of a complete representation of a large power system in an electromagnetic transients program is difficult. It has practical restrictions on the extent of ac system that can be incorporated into a detailed simulation including valve by valve switching and discrete control timing (Bui et al., 1992).
Periodic steady-state approximations in terms of leading Fourier coefficients is a complex form, called phasor, which is classical technique applications such as power systems, electric machine and switching power electronics. Computationally phasor approximations reduce steady-state analysis to algebraic harmonic balancing equation. Traditional phasor representation is ill equipped to capture non periodic transients, now withstanding quasi steady-state extensions (Gilead, 2001). Moreover, the voltages and currents in the power electronic converters and electric drives are typically periodic, but often non-sinusoidal. This sinusoidal quasi steady-state approximation is widely used to study electromechanical dynamics and it is almost invariably included in software tools for power systems (Stankovic et al., 1999). For faster phenomena, time domain simulations are not only a significant computation burden, but also offer little insight into problem sensitivities to design quantities and no basis for design for protection schemes. This type of fundamental analytical problems is not mitigated by improvements in computational technology (Stankovic et al., 2002).
Overcoming the above problems, dynamic phasor has been used successfully to analyze the FACTS equipments-TCSC, UPFC and DVR systems, using its frequency selective approach (Shengli et al., 1991). It is well known that the time varying dynamic phasor offers a numbers of advantages over conventional methods. Firstly, it has wider bandwidth in the frequency domain than traditional slow quasi-stationary assumptions. Secondly, dynamic phasor can be used to compute the fast electromagnetic transient with larger step size, so that it makes simulation potentially faster than conventional time domain EMTP like simulation.
For the understanding and control of the interactions among FACTS, custom power device and the utility system, there has been increasing needs for convenient dynamic phasor models that allow fast and accurate transient simulation and improve power quality solutions. However, the model analysis of the FACTS and custom power is challenging because of their hybrid nature, as they incorporate both continuous-time dynamics and discrete events. Therefore, a further demand is imposed on solution algorithms in order to maximize computational speed and numerical efficiency.
In this study, the generalized averaging is applied to obtain the TCSC and DVR phasor dynamics models. The algorithms are applied to the 3 generator 9 bus WECC systems with one TCSC and one DVR installed. The TCSC and DVR models are derived based on the time varying Fourier coefficients that capture the phasor dynamics of the TCSC and DVR. This study indicates that the dynamic phasor model has potential advantages to enhance the power system stability as well as power quality solutions.
PHASOR DYNAMICS IN POWER SYSTEM
The extensive dynamic phasor modeling of large modern power system is to be based on generalized averaging theory, in which a periodic waveform x (•) can be approximated on the line interval (t-T, t) with a Fourier series form (Gilead 2002):
(1) |
Where, ω = 2πT and Xk (t) is the Fourier coefficient, varies with the time, is called dynamic phasor, which is determined by the following averaging operation,
(2) |
Where, 〈x〉k(t) is used to denote the averaging operation. A key factor for dynamic phasor development is that the derivative of k-th Fourier coefficient is given by the following expression:
(3) |
This formula can be easily verified using (1) and (2) and integration by parts. Differential equations governing their evolution and derived from the original model and harmonic balancing equations are to be obtained in periodic steady state. Theoretical framework and performance analyzing are to be developed in the form of averaged model involving Fourier coefficients.
The definitions given in (1) and (2) will now be generalized for analysis three phase system. The standard notation, where a = ej(2π/3) and. Then the time domain equation can be written as (Gilead, 2002),
(4) |
where, A is the square transformation matrix. The inverse transform of A can be written as A-1 = AH. The coefficients of (4) are as follows:
(5) |
Equation 5 defines 3-phase dynamic phasor at frequency kws. Now the derivatives of k-th Fourier coefficient of Eq. 5 can be written as:
(6) |
It is observed that Eq. 6 is a vector generalization of Eq. 3.
For the validation of the dynamic phasor models of the large modern power system, faster numerical simulation with larger integration steps will first be made through solving differential equation and then compared the dynamic phasor with detailed time domain EMTP simulation as well as standard quasi steady-state transient stability studies using conventional transient stability software.
FACTS AND CUSTOM POWER IN THE SYSTEM
The 3 machines 9 buses WECC system is used to illustrate transient stability and power quality solution using FACTS and custom power device as shown in Fig. 1.
The generalized average models of non-linear loads, transmission lines and 6th order generator model are considered in this system as in (Sanders et al., 1991). In this paper, the dynamic phasor models of FACTS device, TCSC and custom power device, DVR, has been developed. TCSC and DVR are installed in transmission line between bus B8 and B9 and distribution line between bus B6 and load L1, respectively. While TCSC is connected to the system to enhance the transient stability by dampening out power oscillations, DVR is used to improve dynamic response of power quality at transmission or distribution level.
TCSC model dynamics: Dynamic phasor models of TCSC are based on the representation of voltages and currents as time varying Fourier series. Figure 2 shows the circuit diagram of single phase TCSC and its transient waveforms. The system is composed of a fixed capacitor in parallel with a thyristor control reactor. In case of 3-phase TCSC each phase is composed same as single phase TCSC.
The capacitor voltage v and reactor current i are natural state variables and the state-space model of TCSC is:
(7) |
Where, q is a switching function of thyristor that denotes q = 1, when thyristor on and q = 0, when thyristors are off. The dynamics of the phasors V1 and I1 are the fundamental Fourier coefficient for v and i
Fig. 1: | Basic circuit configuration of 9 bus WECC system including TCSC and DVR |
Fig. 2: | (a) TCSC circuit diagram (b) Transient waveforms |
Applying averaging theory of (2) for k = 1, state-space models of (7) become in the form of (3):
(8) |
According the generalized theory (6), Eq. 8 becomes converted into three phase system, as follows:
(9) |
If 〈qv〉1 depended only on the instantaneous values of V1, I1, Il and the control variables, then Eq. 9 would be already state-space form and ready to use and the model of the TCSC interfacing with network could be represent as shown in Fig. 3.
Fig. 3: | Dynamic phasor equivalent circuit of TCSC interfacing with network |
However, if Eq. 9 fails to be a state-space model, then switching calculation must be needed for equivalent circuit of TCSC. Therefore, the 〈qv〉1 of Eq. 8 can be written as follows (Kundur 1994, Hannan and Chan 2004),
(10) |
where, α is a control variable, which depends on the reference conduction angle σ* and phase angle ξ to the peak line current Il, by:
(11) |
where, as τ depends not only phasors and control variables but also the peak value of actual inductor current and phase shift φ. Therefore, τ is the function of current phasor I1,
(12) |
where σ is the prevailing conduction angle of Thyristor Controlled Reactor (TCR) branch sinusoidal line current. With this assumptions equation (10) can written as:
(13) |
While the models (8) and (13) are seems to be sufficiently accurate for TCSC operation, it need some refinement for extend its applicable range. The 〈qv〉1 in (13) lead the calculation of sinusoidal voltage across the capacitor.
Fig. 4: | Basic circuit configuration of DVR |
This associated sinusoidal steady-state can be determined by setting derivatives to zero in (8) i.e., . Therefore, a simplified model can be obtained by putting the value of dynamic phasor I1 into (8),
(14) |
Now if we consider (14) is an accurate model for TCSC state-space form, because of derivatives of V1 expressed entirely as the function V1, Il and control variable. Then the phase shift φ in (12) can form as:
(15) |
Using (15) when evaluating →<qv>1 in (14), we obtain an accurate dynamic phasor model of TCSC with V1 variable. This is a significant simplification.
DVR model dynamics: DVR is a series reactive power compensation device that generates an ac compensating voltage and injects it in series with the supply voltage through an injecting transformer. The main components of a DVR are voltage source inverter, injection transformer, energy storage device and control system. The output of the inverter is connected to an L-C low pass filter, thus to eliminate high frequency harmonics due to switching actions of the inverter. The power circuit configuration of DVR is shown in Fig. 4.
From the circuit configuration, the measured quantity includes [VL]abc, [Vinj]abc, [ib]abc and Vdc. The DVR required injection voltage obtained by subtracting the measured load voltage [VL]abc, from ideal voltage template. The measured DVR injection voltage [Vinj]abc, should be transformed to line to phase voltage and resulting voltage is the output L-C filter capacitor voltage [Vc]abc. The inverter output current [ib]abc, is measured and the DVR injection current [iinj]abc after the L-C filter is also required to performed the control algorithm. The injection current [iinj]abc is transform from the line to phase current and phase current is name as [ic]abc.
The inverter output voltage [Vb]abc is obtained in terms of inverter modulation index [m]abc and the measured dc-link capacitor voltage as:
(16) |
By applying the Kirchhoffs current and voltage law, the state-space model of the DVR power system equations are obtained as:
(17) |
(18) |
(19) |
By applying to the three-phase differential equations, the system average state-space model is derived in the synchronous d-q frame and in components form as:
(20) |
(21) |
Equation 21 can be written by putting the value of [Vb]dq from Eq. 16 as follows:
(22) |
In order to simplify the analysis procedure an assumption has been made that the dc bus parallel resistor is very large, Rdc ? infinity i.e. there is no loss at the dc bus. Therefore, average state-space model of dc voltage in the synchronous d-q frame is:
(23) |
Fig. 5: | Dynamic phasor equivalent circuit of DVR interfacing with network |
Fig. 6: | Dynamic phasor equivalent dc circuit of DVR interfacing with ac network |
The input variables are the modulation indices [m]dq and the output variables are the series injection phase voltage [Vc]dq. The other state variables are inverter output ac current [ib]dq, dc bus voltage Vdc and the series injection current are considered as perturbation.
Dynamic phasor model of Eq. 20, 21 and 23 are derived from time domain model using (3) and (6), in which it will turn out accurate dynamic model of DVR as follows:
(24) |
(25) |
(26) |
Generalized averaging synchronous reference frame of ac and dc side of DVR model (25) and (26) interfacing can be represent as shown in Fig. 5 and 6. Where YDVR and YdcDVR are admittances of model (25) and (26).
Therefore, in a steady state the dynamic phasor model 25) and (26) can be complete equivalent circuit of DVR model. However, it may include PWM control, detail modulation index for a better accuracy, but it will increase more complexity.
RESULTS AND DISCUSSION
This section present the results obtained from dynamic phasor models and EMTP simulation of 3 machine 9 bus system network as shown in Fig. 1, where generator 3 is a slack bus and 3 constant power loads of each 250 MW and 125 MVar were considered. The rating of generator 1, 2 and 3 are 588 MVA, 588 MVA and 5880 MVA, respectively. The field excitation voltage and mechanical power of generators are assumed to be a constant. The injecting transformers of the system with rated 588 MVA and 5880 MVA, 500/13.8 kV and a leakage reactance of 0.01 p.u. were considered. The numerical value of network elements and generator models parameters are used as in (Hannan and Chan 2004).
In TCSC, the switching elements are considered as anti-parallel thyristors. The fixed capacitor and the thyristor controlled reactor of TCSC are assumed to be values of 1.5 μF and 6.8 mH, respectively. The DVR of the system consists of a voltage source inverter, 3 phase transformer rated at 375 MVA, 500/13.8 kV with leakage reactance of 0.1 p.u and a dc capacitance. The optimum size of dc capacitance value is assumed to be 175 μF. A low pass passive filter of reactance and capacitance value as 0.1 p.u and 1.32e-3 p.u., respectively are considered for attenuation of high order harmonics generated by voltage source inverter.
Fig. 7: | Generator rotor angle transient a) With and without TCSC b) Without TCSC EMTP and dynamic phasor c) With TCSC EMTP and dynamic phasor |
Fig. 8: | Generator rotor speed variation a) With and without TCSC b) Without TCSC EMTP and dynamic phasor c) With TCSC EMTP and dynamic phasor |
For creating transient, a balanced 3-phase to ground fault occurred between DVR and load L1 at 0.3 sec for duration of 100 ms. The EMTP simulation result of subtransient reactance of generator rotor angle without TCSC and with TCSC connected to the system is shown in Fig. 7a. In Fig. 7a, it can be seen that transient stability increased with the aids of TCSC and finally it become almost steady state value at 3 sec. In Fig. 7b and c, the dynamic phasor models behind subtransient reactance of generator rotor angle without and with TCSC connected are closely hug with the EMTP simulation.
Figure 8 shows the generator rotor speed variation of EMTP and dynamic phasor models without and with TCSC connected to the system. The spectrum analysis of this case suggest that TCSC responds well to increase transients stability as well as dynamic phasor models hug the EMTP simulation with quite satisfactory.
In dynamic phasor modeling, the network differential equations are replaced by a set of algebraic equations at a fixed frequency, thus the simulation time reducing dramatically. Figure 9 illustrates the electrical power at generator 1 during transient.
Fig. 9: | Electrical power in generator terminal a) Solid line EMTP b) Dotted line dynamic phasor |
Fig. 10: | Generator terminal voltage a) With and without DVR b) With DVR EMTP and dynamic phasor |
In particular implementation, it has found that the phasor model is typically 2 or 3 time faster than detail time domain simulation. It can also be seen that with the abrupt transient of EMTP simulation, the phasor models are still able to provide satisfactory agreement of tracking between the waveforms.
Figure 10a illustrates the terminal voltage at bus 1, it is found that without DVR connected, the terminal voltage sag down more than 50% due to 3-phase fault created at 0.3 sec for a duration of 100ms, where fault impedance X/R ratio equal to 1. However, with connected DVR, the sag down voltage is compensated and returns to near its rated voltage. Figure 10b shows the generator terminal voltage of dynamic phasor model and EMTP simulation. We considered here only fundamental components of dynamic phasor model. It can also be seen that there is a spike at the end of the fault in both EMTP and dynamic phasor models. This is due to the switching of voltage source inverter. In this case dynamic phasor model also excellently match with time domain simulation.
In order to simulate generator swing dynamics, Fig. 11a shows that the phase A voltage at load 3 is modulated at 10° phase shift during 0.2 s to 0.24 s between DVR output voltage, [Vb]abc and network voltage, where firing angle is 60°.
Fig. 11: | Voltage at load 3 EMTP and dynamic phasor simulation a) Without DVR b) With DVR |
The 3-phase to ground fault occurred at 0.215 s for a duration of 10 ms. It can be seen from Fig. 11a that due to modulation there are some harmonics in both the EMTP and dynamic phasor simulation. It can also be seen that the rms value of dynamic phasor models recover at the end of fault period, where as in case of EMTP, fault recovering process has to wait until zero crossing of the phase voltage occurs. In Fig. 11b, the sag down voltage is compensated by DVR and with the aid of low pass LC filter of DVR, voltage harmonics are also eliminated. However, in both cases, without and with DVR, the rms values of dynamic phasor model are still able to provide satisfactory agreement of tracking between the waveforms of the time domain and dynamic phasor models.
CONCLUSIONS
Dynamic phasor models capture transients in fundamental harmonic coefficients of the system network. In this paper, the dynamic phasor models of FACTS device, TCSC and custom power device, DVR, were developed using the methodology of average generalized technique. Their performance has been fully evaluated using a modified 3 machines 9 bus WECC system. Test results obtained from the dynamic phasor simulation and detail time domain simulation illustrated that dynamic phasor models can faithfully represent the transient behaviors of the switching devices and can be used to analyze the transient operations of model power systems with embedded FACTS and custom power device. While the accuracy of dynamic phasor simulation is close to the detail time domain simulation, it has a faster simulation speed due to the use of larger integration steps.
ACKNOWLEDGMENTS
The authors gratefully acknowledge the support of the Hong Kong Polytechnic University under Project G-T732 and A-PA2L.
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