Effects of generators inertia on the transient stability.
Transient stability is the ability of a synchronous power system to return to stable condition and maintain its synchronism following a relatively large disturbance. The power generation systems are mainly subjected to faults, and hence its extremely important to know the stability conditions of the system (Dou et al, 2013; Karami, 2011; Khani et al., 2012; Mahdad and Srairi, 2013; You et al., 2013; Wadduwage et al, 2013). Paper (Mahmud et al., 2014) presents an approach to design a nonlinear observer-based excitation controller for multi-machine power systems to enhance the transient stability. The controller is designed based on the partial feedback linearization of a nonlinear power system model which transforms the model into a reduced-order linear one with an autonomous dynamical part. Then a linear state feedback stabilizing controller is designed for the reduced-order linear power system model using optimal control theory which enhances the stability of the entire system. The states of the feedback stabilizing controller are obtained from the nonlinear observer and the performance of this observer-based controller is independent of the operating points of power systems. The performance of the proposed observer-based controller is compared to that of an exact feedback linearizing observer-based controller and a partial feedback linearizing controller without observer under different operating conditions. Paper (Karami and Esmaili, 2013) investigates the use of multi-layered perceptron (MLP) neural network (NN) for assessing the transient stability of a power system considering the detailed models for the synchronous machines, and their automatic voltage regulators (AVRs). Two MLP NNs are employed here to estimate the critical clearing time (CCT) and a transient stability time margin (TM), as indicators for measuring power system transient stability for a particular contingency under different system operating conditions. The training of MLP NNs is accomplished using some carefully chosen system features as the inputs and the CCT and/or the TM as the desired targets. In this paper, the required training and/or testing patterns for the neural network are obtained by performing time-domain simulation (TDS) on the New England 10-machine 39-bus test system and the IEEE 16-machine 68-bus test system with fourth-order machines models and their AVRs using the software tool PSAT (Power System Analysis Toolbox), whereas the proposed neural network models are implemented in MATLAB. In addition, a neural network based sensitivity method and principal component analysis (PCA) are employed to reduce the dimension of the input data vectors. The simulation results obtained prove that the trained neural networks give satisfactory estimations for both CCT and TM. In paper (Nguimfack-Ndongmo et al., 2014), a simplified nonlinear method is proposed to enhance the transient stability of multimachine power system by using a Static Synchronous Series Compensator (SSSC). The rate of dissipation of transient energy is used to determine the additional damping provided by a SSSC. The proposed algorithm is based on the direct Lyapunov method. The simplicity of the proposed scheme and its robustness with respect to large disturbances constitute the main positive features. Simulation results in the case of 3-machines power system show the effectiveness of the proposed method under large disturbances.
This paper presents the effects of generators inertia on the power system transient stability. Simulation results demonstrate the great effect of generators inertia on the transient stability.
1. Synchronous Generators Inertia:
Generally, in power systems, a multi-stage turbine drives the generator rotor through a common drive shaft. Figure 1 shows a typical generation system consisting of high-pressure, intermediate-pressure and low-pressure stages. Where, each turbine stage contributes a proportion of the total mechanical driving torque. The drive system can be modeled by a series of rotational masses, to represent the inertia of each turbine stage, connected together by springs, to represent the torsional stiffness of the drive shaft and coupling between the stages. The parameters of Figure 1 are defined as follows: HP, IP, LP: high-pressure, intermediate-pressure and low-pressure section of the turbine; G: generator; Ex: rotating exciter; J: moment of inertia of individual sections; [tau]: external torque acting on a mass; k: stiffness of a shaft section; [delta]: angular displacement of a mass; I, II, III, IV: shaft couplers.
Following equation cab be derived based on the Newton's second law:
J[d[[omega].sub.m]/dt] + [D.sub.d][[omega].sub.m] = [T.sub.t] - [T.sub.e] (1)
Where J is the total moment of inertia of the turbine and generator rotor (kg[m.sup.2]), [[omega].sub.m] is the rotor shaft velocity (mechanical rad/s), [[tau].sub.t] is the torque produced by the turbine (Nm), [[tau].sub.e] is the counteracting electromagnetic torque and [D.sub.d] is the damping-torque coefficient (Nms) and accounts for the mechanical rotational loss due to wind age and friction.
Although the turbine torque [[tau].sub.t] changes relatively slowly, due to the long thermal time constants associated with the boiler and turbine, the electromagnetic torque [[tau].sub.e] may change its value almost instantaneously. In the steady state the rotor angular speed is synchronous speed [[omega].sub.sm] while the turbine torque [[tau].sub.t] is equal to the sum of the electromagnetic torque [[tau].sub.e] and the damping (or rotational loss) torque [D.sub.d][[omega].sub.sm]:
[T.sub.t] = [T.sub.e] + [D.sub.d][[omega].sub.sm] or [T.sub.m] = [T.sub.t] - [D.sub.d][[omega].sub.sm] = [T.sub.e] (2)
Where [[tau].sub.m] is the net mechanical shaft torque, that is the turbine torque less the rotational losses at [[omega].sub.m] = [[omega].sub.sm]. This torque is converted into electromagnetic torque. If, due to some disturbance, [[tau].sub.m]>[[tau].sub.e] then the rotor accelerates; if [[tau].sub.m]<[[tau].sub.e] then it decelerates. The rotor velocity can be expressed as:
[[omega].sub.m] = [[omega].sub.sm] + [DELTA][[omega].sub.m] = [[omega].sub.sm] + [d[[delta].sub.m]/dt] (3)
Where [[delta].sub.m] is the rotor angle expressed in mechanical radians and [DELTA][[omega].sub.m] = d[[delta].sub.m]/dt is the speed deviation in mechanical radians per second. Then, following equation can be obtained:
J[[d.sup.2][[delta].sub.m]/d[t.sup.2]] + [D.sub.d]([[omega].sub.sm] + [d[[delta].sub.m]/dt]) = [T.sub.t] - [T.sub.e] or J[[d.sup.2][[delta].sub.m]/d[t.sup.2]] + [D.sub.d][d[[delta].sub.m]/dt] = [T.sub.m] - [T.sub.e] (4)
Multiplying through by the rotor synchronous speed [[omega].sub.sm] gives:
J[[omega].sub.sm][[d.sup.2][[delta].sub.m]/d[t.sup.2]] + [[omega].sub.sm][D.sub.d] [d[[delta].sub.m]/dt] = [[omega].sub.sm][T.sub.m] - [[omega].sub.sm][T.sub.e] (5)
As power is the product of angular velocity and torque, the terms on the right hand side of this equation can be expressed in power to give:
J[[omega].sub.sm][[d.sup.2][[delta].sub.m]/d[t.sup.2]] + [[omega].sub.sm][D.sub.d] [d[[delta].sub.m]/dt] = [[[omega].sub.sm]/[[omega].sub.m]] [T.sub.m] - [[[omega].sub.sm]/[[omega].sub.m]] [T.sub.e] (6)
Where [P.sub.m] is the net shaft power input to the generator and [P.sub.e] is the electrical air-gap power, both expressed in watts. During a disturbance the speed of a synchronous machine is normally quite close to synchronous speed so that [[omega].sub.m] [approximately equal to] [[omega].sub.sm] and above equation becomes:
J[[omega].sub.sm] [[d.sup.2][[delta].sub.m]/d[t.sup.2]] + [[omega].sub.sm][D.sub.d] [d[[delta].sub.m]/dt] = [P.sub.m] - [P.sub.e] (7)
The coefficient J[[omega].sub.sm] is the angular momentum of the rotor at synchronous speed and, when given the symbol [M.sub.m], allows equation to be written as:
[M.sub.m] [[d.sup.2][[delta].sub.m]/d[t.sup.2]] + [P.sub.m] - [P.sub.e] - [D.sub.m] [d[[delta].sub.m]/dt] (8)
Where [D.sub.m] = [[omega].sub.sm][D.sub.d] is the damping coefficient. Equation (8) is called the swing equation and is the fundamental equation governing the rotor dynamics. It is common practice to express the angular momentum of the rotor in terms of a normalized inertia constant when all generators of a particular type will have similar 'inertia' values regardless of their rating. The inertia constant is given the symbol H defined as the stored kinetic energy in mega joules at synchronous speed divided by the machine rating [S.sub.n] in megavolt-amperes so that:
H = [0.5J[[omega].sup.2.sub.sm]/[S.sub.n]] and [M.sub.m] = [2H[S.sub.n]/[[omega].sub.sm]] (9)
The units of H are seconds. In effect, H simply quantifies the kinetic energy of the rotor at synchronous speed in terms of the number of seconds it would take the generator to provide an equivalent amount of electrical energy when operating at a power output equal to its MVA rating. In Continental Europe the symbol [T.sub.m] is used for mechanical time constant.
2. Test System:
A power system with three generators and nine-bus is considered as test system. Figure 2 indicates the proposed test system. The system data are given in (Sauer and Pai, 1998). The inertia (M=2H) of generators [G.sub.1], [G.sub.2] and [G.sub.3] is 47.28, 12.8 and 6.02 seconds respectively.
3. Simulation Results:
As stated before, this paper aims at showing the effects of inertia on the transient stability. Thus, a 5 seconds three-phase short circuit at bus 6 is applied as a large signal disturbance. The system performance following this disturbance is investigated through Figures 3 to 8. In order to appropriate comparison, two cases are simulated; where, first case considers the nominal inertia of the generators and in the second case, the inertia of the generators is assumed as 50% of the nominal values. Figures 3 to 5 show the speed of generators and Figures 6 to 8 show the rotor angle of generators. According to the results, it is clear that the frequency deviation in case 1 is less than case 2. This is due to high inertia of case 1, which prevents fast changing of frequency in the system. However, in second case, the inertia is low and frequency can rapidly change. The results show that inertia has a significant effect on the system performance and transient stability.
This paper presented the effects of generators inertia on the transient stability of the system. It was shown that by changing the inertia, frequency deviations are highly changed and systems with high inertia are more robust and stable. Simulation results were carried out on a multi machine power system.
Received 25 January 2014
Received in revised form 12
Accepted 14 April 2014
Available online 25 June 2014
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Mojtaba Shirvani, Mostafa Abdollahi, Ali Akbar Dusti, Iman Baghbani
Department of Electrical Engineering, College of Engineering, Boroujen Branch,
Islamic Azad University, Boroujen, Iran
Corresponding Author: Mojtaba Shirvani, Islamic Azad University, Department of Electrical Engineering, Boroujen, Iran. Tel: 933824223812; E-mail: firstname.lastname@example.org
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|Author:||Shirvani, Mojtaba; Abdollahi, Mostafa; Dusti, Ali Akbar; Baghbani, Iman|
|Publication:||Advances in Natural and Applied Sciences|
|Date:||Jun 1, 2014|
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