The present paper deals with the stability problem of a synchronous generator connected to an infinite bus system. Use of conventional Power System Stabilizer (PSS) will definitely improve the performance of the system by assuring stable operating conditions but it takes more time to tune it and also its non-optimal damping in entire operating conditions is unwanted. A new approach to yield better stable conditions for the entire system is discussed here. An Adaptive Power System Stabilizer derived from model reference adaptive control (MRAC-ASPR) is a better solution for stability problems due to the critical oscillations in the rotor of the synchronous machine under transient faults. The design of Model Reference Adaptive Control - Almost Strictly Positive Real is discussed in the latter sections of the paper.
A power system is a complex system with sophisticated combination of multiple electrical and mechanical components. In general, these elements are highly nonlinear in their behavior which makes the system further more complex. And also, power system operation is characterized by a wide range of operating conditions, random load changes and various unpredictable transient disturbances. Supplementary excitation control, commonly referred to as Power System Stabilizer (PSS) has become an important means to enhance the damping of low frequency oscillations, i.e., dynamic or steady-state stability. The coordination of power system stabilizers for improved dynamic performance of multi-machine power systems, and in particular methods for determination of PSS parameters, have drawn much attention. The common background of a PSS is as fallows.
2. Power system stabilizer (PSS) and its function
The Excitation System Stabilizer[2] is designed to provide effective voltage regulation under open and short circuit conditions. The Power System Stabilizer is designed to provide damping of the rotor oscillations whenever there is a transient disturbance whereas the input signal for Power System Stabilizer is derived from speed / frequency or accelerating power or a combination of three signals of a synchronous machine, while the stabilizer is designed to have zero output in steady state. The output is limited in order not to adversely affect the voltage control. The major factors that contribute to the instabilities are Loading of the generators (or) tie line, power transfer capability of transmission line, power
factor of the generator (leading power factor operation more complicated than lagging power factor operation) and AVR gain. Solution to the
problem of Oscillation Instability[4] is to provide damping for generator oscillations. This is done by power system stabilizer (PSS) with supplementary controllers in the excitation system. The oscillatory instability can be viewed as stability of the operating point, subjected to small random perturbations which are always present. In a practical system, the various modes of oscillation can be grouped into 3 categories. Intra-plant modes with 1.5 to 3 Hz, local modes with 0.8 to 1.8 Hz and Inter-area modes with 0.2 to 0.5 Hz.
2.1 Purpose of design of PSS
Power system stabilizers are designed mainly to stabilize local and inter - area modes (0.2 to 2 Hz). The instability arises due to the negative damping torque caused by fast acting exciter under operating conditions. The objective of PSS is to introduce additional damping torque without affecting the synchronizing torque.
3. Structure of Power System Stabilizer
The block diagram of the Power System Stabilizer used in practice is shown in fig. 1.1
Filter (s)
T(s)
sTï·
1+sTï·
u Å« Å« Vs
Washout Dynamic Torsional Limiter
Circuit Compensator Filter
3.1 Washout Circuit
This is provided to eliminate steady - state bias in the output of power system stabilizer which will modify the generator terminal voltage. The PSS is expected to respond only to transient variation in input signal (say rotor speed) and not to the dc offsets in the signal. This is achieved by subtracting from it the low frequency components of the signal obtained by passing the signal through a low pass filter.
3.2 Dynamic Compensator
This is generally made up to two lead - lag stages[1]
Where Ks is gain of PSS and the time constants T1, T2, T3, T4 are chosen to provide phase lead for the input signal in the range of frequencies that are of interest (0.1 to 3Hz). In general, dynamic compensator can be chosen with the following transfer function
N(s) = 1+a1s+a2s2+ …………apsp
D(s) = 1+b1s+b2s2+ …………bpsp
3.3 Torsional Filter
A filter section is added to suppress frequency components in the input signal of the PSS that could excite undesirable interactions. The two major criteria in the design of torsional filter are as fallows. One is to minimize phase lag of the filter in the low frequency range (1 to3 Hz). The second is that the maximum change in damping of any torsional mode is less than some fraction of the evolved torsional damping.
3.4 Limiter
When load rejection takes place, the Automatic Voltage Regulator acts to reduce the terminal voltage when PSS action[3] calls for higher value of the terminal voltage (due to increase in speed or frequency). The output of PSS must be limited to prevent the PSS acting to counter the action of Automatic Voltage Regulator. It may even be desirable to trip the PSS in case of load rejection.
4. Advantages of Adaptive PSS Over Conventional PSS
Dynamic stability is one of the basic control problems of the synchronous generator in the modern world. The oscillations of the generated active power due to the transient conditions result in the reduction of power transfer capabilities of transmission lines. Linear control theory is used in the design of Conventional Power System Stabilizer (CONV-PSS) [1], [2], [5]. This conventional PSS provides satisfactory operation in wide range of operating conditions. In spite of being well established, a CONV-PSS shows some disadvantages, like Time consuming tuning of the CONV-PSS and non-optimal damping in the entire operating range by varying the loading, also the synchronous generator dynamic characteristics vary, the fact due to which the stabilizer determined in the nominal operating point does not assure the optimal damping in the entire operating range.
Among various modern control theory concepts, robust and adaptive control methods are the most frequently used approaches used in the design of PSS. The use of adaptive control is recommended for the improvement of the dynamic stability of synchronous generators because the dynamics of the adaptive control is much faster than the dynamics of the load and the fluctuations in the synchronous machine. So, it can provide better dynamics for the system under control.
Closed loop adaptive control methods can be divided into Self Tuning Control (STC) and Model Reference Adaptive Control (MRAC). The majority of the adaptive realizations of the power system stabilizers are based on the use of STC. Compared to STC, MRAC has certain advantages, of which lower computing complexity of the adaptation algorithm and the speed of the adaptation process are major advantages. The latter is especially important for the improvement of transient stability. To fallow MRAC approach, synchronous generator has to fulfill some strict conditions, but unfortunately it does not allow simple MRAC applications. So, MRAC approaches are applied with the help of some intermediate approaches like Model Reference Adaptive Control - Full State Access (MRAC-FSA), Model Reference Adaptive Control - Adaptive Observer (MRAC-AO), Model Reference Adaptive Control - Almost Strictly Positive Real (MRAC-ASPR).
Implementation of MRAC-FSA is found to be difficult because of unmeasurability of the necessary state space variables of the system. For the MRAC-A0 implementation one should also be familiar with the structure of the controlled plant. The basic problem of the MRAC-A0 implementation is to choose an input-output description of the controlled plant having a structure for which perfect model following can be achieved. MRAC-ASPR is more recent than previously mentioned adaptive approaches. This approach is an output feedback method which requires neither full state feedback nor adaptive observers. Other important properties of this algorithm are as follows: they are applicable to non-minimum phase systems and to multiple inputs and outputs systems, the order of the controlled plant may be much larger than the order of the reference model, ease of implementation.
5. Mathematical Model Of A Synchronous Generator
The synchronous generator connected to an infinite bus is a multivariable nonlinear dynamic system. A simplified linearized third order model is used for the analysis and design of control systems for synchronous machines.
Where
Tm - mechanical torque [pu]
Te - electrical torque [pu]
W - rotor speed [pu]
- rotor angle [rad]
E'q - voltage behind transient reactance [pu]
Efd - field excitation voltage [pu]
Vt - terminal voltage [pu]
H - inertia constant [s]
D - damping coefficient representing the total lumped damping effects from the damper windings [pu]
Wr - synchronous speed [rad/sec]
T'do - direct axis transient open circuit time constant [s]
K1….K6 - linearization parameters
Vt,ref - reference terminal voltage [pu]
KAVR - exciter and voltage controller gain
TAVR - exciter time constant [s]
s - Laplace variable
- small signal incremental quantities
By varying the operating point, the parameter values K1 through K6 also vary. The effects of the machine loading on the synchronous generator dynamic characteristics can be evaluated by the simplified linearized model Eigen value loci analysis. Synchronous generator dynamics changes because of Automatic Voltage Regulator (AVR) with voltage control loop. Fig. 2 shows a simple AVR with an exciter.
Where Vt,ref represents the reference terminal voltage [pu], KAVR represents the exciter and the voltage controller gain and TAVR represents the exciter time constant [s]. The time constant depends on the selected exciter and the voltage gain is selected so that KAVR < . In this way the sufficient damping for the voltage controlled loop is assured.
There are four eigenvalues for the linearized model of synchronous generator with voltage control system, out of which the dominant complex conjugate eigenvalues are selected for dynamic stability analysis. They are directly linked to the period and damping ration of the rotor angle oscillation. The results of a 160MVA turbo generator are gathered from [1], [2]. 3]. The dependence of dominant complex conjugate eigenvalues on loading is shown in Fig. 3.
The individual curves correspond to the constant reactive power. The actice power P is varied from 0.0 to 1.2 [pu] at the reactive power Q values from 0.0 to 1.2 [pu] in steps of 0.1 [pu]. The instability occurs at high values of P and increases with the decrease of the reactive power. From the analysis of the effect of different loadings on the synchronous generator dynamic characteristics, we can conclude that the variations in the machine dynamics are considerable and therefore an implementation of adaptive power system stabilizer will be meaningful.
6. Power System Stabilizer Design
6.1 Conventional Power System Stabilizer
CONV-PSS generates a stabilizing signal to modulate the reference of the Automatic Voltage Regulator (AVR) loop. The basic inputs of the CONV- PSS are shaft speed, generator active power or bus frequency. Fig. 4 shows one of the possible CONV-PSS control structures, where VIB represents the Infinite bus vole age [pu], f1b is the infinite bus frequency [pu], P is the generator active power [pu], yp is the PSS input [pu] and Up is the PSS output [pu].
CONV-PSS is a fixed structure controller consisting of a gain in series with lead-lag networks and a washout filter.
The parameters of the PSS are selected such that the controlled plant will be having increased damping in the rotor oscillations. The main disadvantage of CONV-PSS is that it doesn't provide optimal damping in all operating conditions.
7. Adaptive Power System Stabilizer
Simplified version of MRAC-ASPR i.e., Model Reference Adaptive Control - Almost Strictly Positive Real is used for the stabilization of synchronous generator.
The plant to be controlled is represented by the fallowing equations
Where Xp(t) is plant state vector, up(t) is control vector and Yp(t) is plant output vector. Ap, Bp and Cp are corresponding state vector matrices.
The Reference Model is represented by the fallowing equations
Where xm(t) is model state vector, um(t) is model command vector and ym(t) is model output vector. Am, Bm and Cm are corresponding state vector matrices. Reference model is assumed to be stable.
The output tracking error is defined as
The control Up(t) for the plant output vector Yp(t) can be approximated well. The output of the reference model Ym(t) without explicit knowledge of Ap, Bp and Cp will be generated by the adaptive algorithm as fallows:
The necessary condition which makes the system asymptotically tracked is that there exists a Ke such that for any Um which is step command the fallowing equation should me always positive real.
In this case the controlled plant is said to be a strictly ASPR. If the controlled plant is not ASPR, the augmenting of the plant with the feed forward compensator such that the augmented plant is ASPR is suggested . The block diagram of the synchronous generator stabilization system consisting of the PSS, based on the MRAC-ASPR is shown in Fig. 5.
The benefit of the control structure shown in Fig. 5 if compared to other adaptive structures is a very simple realization of the adaptation mechanism. The presented MRAC-ASPR is essentially simplified, because of the constant command signal, a reference model is namely not required. The variations of the synchronous generator loading can be seen as a perturbation of the controlled plant parameters.
8. Results
A simplified linearized model of synchronous generator is used for the analysis and design of an adaptive power system stabilizer. But the simplified model does not include all the effects of the synchronous generator which can restrict the use of adaptive control, basically designed for linear models.
The simulation results shown below were taken from a 160 MVA turbo synchronous generator mentioned in [1], [2].
Fig. 6 - 8 shows the behavior of the system for small-signal disturbances. Figures clearly show that the system with adaptive PSS provides improved damping in all operating conditions.
Fig. 9 - 11 shows the behavior of the system under transient fault conditions. For this type of analysis the synchronous generator system is connected to the infinite bus through two symmetrical transmission lines. At t = 1 sec, a three phase to ground fault occurs and the corresponding fault sequence simulations are as shown in figures below. Benefits of adaptive PSS can be easily understood by examining the simulations.
9. Conclusions
The problems of dynamic and transient stabilities of a synchronous generator are examined in this paper and because of characteristic operation of synchronous generator, the use of linearized model for the analysis and design of adaptive control system is possible. The use of conventional power system stabilizer does not guarantee the dynamic and transient stability when compared to the adaptive power system stabilizer at all operating conditions. So the use of adaptive power system stabilizer implemented with model reference adaptive control based on almost strictly positive real is recommended for better stable conditions. This control mechanism is a simplified stabilization mechanism with unsophisticated realization which guarantees dynamic and transient stability of a synchronous generator. Investigations were going on with different stabilization structures in combination of MRAC - ASPR to provide better dynamic and transient stability conditions for a synchronous generator system.
