The ability of synchronous machines of an interconnected power system to remain synchronized after being subjected to a small disturbance is known as small-signal stability that is a subclass of phase angle-related instability problems. It depends on the ability to maintain equilibrium between electromagnetic and mechanical torques of each synchronous machine connected to the power system. The change in electromagnetic torque of a synchronous machine following a perturbation or disturbance can be resolved into two components
(i) a synchronizing torque component in phase with rotor angle deviation and
(ii) a damping torque component in phase with speed deviation.
Lack of sufficient synchronizing torque results in “aperiodic” or non-oscillatory instability, whereas lack of damping torque results in low-frequency oscillations.
Rotor angle stability is of two types, viz. small signal stability (due to small disturbance in the power system) and transient stability (due to large disturbance in the power system). Small signal stability is the ability of the power system to be in a steady-state after a small disturbance. The instability due to this is mainly attributed to insufficient damping torque. While transient stability is associated with the ability of the power system to maintain synchronism when subjected to large disturbances like line fault, bus fault, generator outage, etc. The instability arising due to this is the result of insufficient synchronizing torque.
Small signal instability is due to insufficient damping torque leading to low-frequency electromechanical oscillations in the system, which is oscillatory in nature. During Low-Frequency oscillations, mechanical kinetic energy is exchanged between synchronous generators of the inter-connected system through tie lines. Most of these oscillatory modes in normal power system state are well damped. However, they get excited during any small disturbance in the system and lead to oscillation in power system parameters like rotor velocity, rotor angle, voltage, currents power flow, etc., Due to oscillation in parameters, protection equipment may undesirably operate leading to cascade tripping in the power system.
Therefore, it is necessary to detect such modes and initiate corrective actions to ensure system reliability and security. Among these parameters, the rotor velocity of the generators and the power flow in the network are most important. The rotor velocity variation causes fatigue to the mechanical parts of the turbine-generator system. The power flow oscillations may amount to the entire rating of a power line when they are superimposed on the stationary line flow and would limit the transfer capability by requiring increased safety margins.
CLASSIFICATION OF POWER SYSTEM STABILITY
VARIOUS MODES IN THE SMALL SIGNAL (OSCILATORY) STABILITY
The power system is highly nonlinear in nature due to non-linearity in generators and their controllers and other devices. It can be represented in state-space form. Based on the solution of state-space representation using various methods, eigenvalues, oscillatory modes, and the participation of various states of the system in that mode is calculated. Based on the participation factors obtained analysis, the modes are classified as:
1. Local modes
2. Intra-area modes
3. Inter-area modes
These modes are classified after obtaining similar results across the globe. The mode in which one generator oscillates against another within a plant are called local modes, the modes in which a group of generators oscillates against another group of generators in a different area connected by weak tie lines are called inter-area modes. While within a defined area if a generator forms two coherent groups and oscillate then it is defined as an intra-area mode.
The commercial software available for small signal stability analysis however uses the perturbation technique in order to form a system matrix. In steady-state, all the differential equations have their LHS as zero if the values for the variables on RHS are put as per the current operating point of the system. A small change is made in each state one by one to obtain the elements of the system matrix.
OSCILLATORY INSTABILITY INCIDENTS
Though there have been many incidents related to LFO, not an in-depth study has been performed to see the real reasons behind many of these incidents. Some of the incidents and the lessons learned are summarized below to give an understanding of the underlying problem.
Noteworthy incidents related to LFO include:
· United Kingdom (1980), frequency of oscillation about 0.5 Hz.
· United Kingdom (1980), frequency of oscillation about 0.5 Hz.
· Taiwan (1984, 1989, 1990, 1991, 1992), frequency of oscillation 0.78 – 1.05 Hz.
· West USA/Canada, System Separation (1996), frequency of oscillation 0.224 Hz.
· Scandinavia (1997), frequency of oscillation about 0.5 Hz.
· China Blackout on 6 March (2003), frequency of oscillation around 0.4 Hz.
· US Blackout on 14 August (2003), frequency of oscillation about 0.17 Hz.
· Italian Blackout on 28 September (2003), frequency of oscillation about 0.55 Hz.
· Power System oscillations experienced in Indian Grid on August 2014
Power System oscillations experienced in Indian Grid on August 2014
It can be observed that from the below event occurred during Jan 2014 of the western grid system was not oscillating initially which is the normal operating condition. After a few seconds, the spontaneous oscillation was observed and after SPS (Special System Protection Schemes) operation it again comes back to normal operating condition. The oscillatory period here is what small-signal stability is. It shows that system parameters oscillate during the period and again come back to the new equilibrium point.
Frequency of the various nodes in western grid measured from PMUs during oscillation on 28th January 2014.
Most of these incidents involved a low frequency of oscillation in the range of 0.1 to 0.7 Hz that is considered the most serious and could lead to widespread blackouts. Apart from this, oscillatory incidents in power systems in Ontario-Canada, Sri Lankan, Malaysia, and Bangladesh are also reported in the literature. Most of the incidents had happened due to faults triggered by some disturbances such as a tree contacting with a transmission line, some component failure, faults in transmission lines, etc. Because of the faults, these lines have been disconnected from the grid. Then some other lines in the network have been overloaded and sagged on trees causing more earth faults. Those incidents have been generated sequential line tripping and generator tripping causing oscillation in power. The tripping of transmission lines significantly modifies the characteristics of the remaining grid with longer distance (greater equivalent impedance) for the power flow and consequent higher stability risk. And also the modified grid may have less damping compared with the original grid. The weak tie lines and the nature of the longitudinal structure are some of the causes for low-frequency oscillations. The concentration of outputs to major power plants with insufficient reserve margins, heavy flow across transmission interfaces due to seriously imbalanced regional power and pumped storage units were in pumping mode operation are common causes for low-frequency oscillation observed in some of the cases above mentioned.
With the heavy tie-line power, low-frequency electromechanical oscillation modes have been captured in the cases mentioned above, and decreasing the tie-line power flow made those modes disappeared.
MODELING AND ANALYSIS
Low-frequency oscillation study requires dynamic modeling of most of the power system components. Once the mathematical model is available different methodologies can be applied to study the system oscillatory behavior in the low-frequency range. Eigenvalue analysis, time-domain simulation, and Prony analysis are used among researchers. Though each method of these methods has its own merits and demerits, eigenvalues and time-domain simulations are typically used among the utilities to get a complete understanding of system oscillatory phenomena.
Dynamic modeling of power system includes a set of differential and algebraic equations (DAE). Low-frequency oscillation studies can be done in two ways depending on the interest. If the interest is to capture the local behavior related to an area or particular power plant, then that area of the power plant can be modeled in detail and the rest of the system with simple models. If the interest is to capture both local and global modes such as inter-area mode each and every machine in the system and their associated controllers should be modeled in detail. It is important to include loads, controllers, and other power system components that would influence the LFO. A general mathematical model of the power system is given below.
Where x is a vector of state variable; y is a vector of algebraic variables; l and p are uncontrollable and controllable parameters, respectively. Machine and control dynamics will be included in the differential equations while basic load flow and other network equations will be included in algebraic equations. We are going to discuss only about Eigenvalue analysis, leaving the time domain simulation and Prony analysis. If you need more information please go through the references mentioned.
EIGENVALUE ANALYSIS
The small-signal stability or LFO study of the system can be determined by system eigenvalues at an operating point. The relative participation of state variables and their contribution in certain oscillation modes is given by the corresponding elements in the right and left eigenvectors. Hence, a combination of left and right eigenvectors yields a participation factor matrix. The participation factor matrix can be used to identify the dominant state variable in a particular mode.
The following steps are followed in studying the LFO of power systems.
Step I: Finding equilibrium or operating point
Step II: Linearization DAE model around the equilibrium point
Step III: Forming the reduced system state matrix
Step IV: Finding eigenvalues, eigenvectors, and Participation matrix
In order for the system to be stable or oscillation free, all the eigenvalues should be located in the open left half-plane. This means that the real part of the eigenvalues should be negative and the damping ratio should be positive with more than a pre-specified value according to utilities‟ practice (typically damping ratio should be higher than 0.05). If at least one of the eigenvalues has a positive real part the system is said to be unstable. More specifically, in oscillatory unstable cases, a pair of complex eigenvalues will appear with a positive real part.
SIMULATION USING DIGSILENT POWERFACTORY SOFTWARE
A Sample Transmission system, considered for the Small signal stability (Eigenvalue) Analysis. Two cases have been performed.
Case-1 Normal Operation. (No Outages)
Case-2 Contingency Operation (2 No of 400kV line Outages)
The system is stable in both cases since the real part of the Eigen value of each mode is negative. With the help of the software, In addition to the Real and Imaginary part of the Eigenvalue, we will get Magnitude, Angle & Damping ration for each mode
CONCLUSION
The traditional approach to addressing the low-frequency oscillation problem is to equip PSS in the machines which have the tendency to damp out power oscillations. However, the present power systems are too complex as many utilities around the world are interconnected with each other to deliver reliable and cheap power from environmentally clean resources. Moreover, the introduction of competition had invited many generating plants to be connected to the power system and started to dispatch power. PSS in some cases founds not sufficient and even detrimental, this has open the door for a number of FACTS controllers applied to add damping on weak modes. The remedial measures for oscillation damping can be classified into two broad categories, one at the operational level and the other one is at the planning stage.
Operational level approaches for power system oscillation damping include re-tuning excitation control system and PSS. Re-dispatching of generators and adjusting of load changes can also be considered. At the operational level, load shedding can also be used as the last line of defense to damp low frequency of oscillation Planning level: At the planning stage, a number of damping controllers can be considered for implementation. New PSS, FACTS controllers Superconducting Magnetic Energy Storage (SMES) and flywheel are some of them
REFERENCES
1. Power System Stability and Control by P.Kundur.
2. Modern Power System Analysis by Xi-Fan Wang, Yonghua Song, Malcolm Irving.
3. Understanding low frequency oscillation in power systems by K. Prasertwong, N. Mithulananthan and D. Thakur.
4. POSOCO Report on Power System oscillations experienced in Indian Grid on 9th, 10th, 11th and 12th August 2014.
