International Journal of Electrical Power & Energy Systems
Voltage stability – Case study of saddle node bifurcation with stochastic load dynamics
Highlights
► We provide a better way of understanding voltage stability in power system. ► We examine the effect of stochastic models of the load on the SMIB power system. ► We evaluate Mean First Passage Time (MFPT) of SMIB system for various random load variation.
Introduction
The dynamics of the power systems at any given time is governed by the difference between the mechanical power input and the electrical power demand at the load centers. The demand for electrical power is increasing in the load centers without proportionate increase in the transmission infrastructure due to the increased cost of construction, environmental concerns for building new transmission line and problems of right of way. This forces the power system operators to increase the loading in the existing transmission network and operate the power system closer to its maximum loadability limit [1]. Further small intensity uncontrollable random load perturbations are always present in the system [2]. This impresses the need to understand power system stability in a better way and to develop indices which provide the stability margin of the particular operating point of the power system considering stochastic load dynamics. Similar works on assessment of voltage stability margin using Neural network has been carried out in [3], using P–Q–V curve in [4] and using enhanced look ahead algorithms [5].
In this study, random load perturbation is modeled as the Gaussian white noise. White noise is characterized by the delta correlated autocorrelation function [6]. The white noise model of the power system loads can be validated by the fact that characteristic time of the system is much greater than the correlation time of the stochastic load [7]. The stability margin of power system with stochastic loads can be characterized with the help of Mean First Passage Time (MFPT) of the power system from the stable equilibrium point to reach the closest unstable equilibrium point when perturbed by the small load perturbation [8]. This index enables power system operator to understand the need to control and operate the power system at the most stable operating point for each load setting.
This paper also provides the easy and better way of understanding the effect of increased loading on the stability for Single Machine connected to Infinite Bus (SMIB) system by visualizing the change in the shape of the system potential. The counterpart of the system potential for multidimensional system are the energy functions. Voltage stability of multidimensional system can be explained as the change in the shape of the energy function with increased loading of the system as explained in chapter two of [9].
Section snippets
System potential
The mathematical model for any two dimensional system with displacement (x), damping (γ), potential energy (U(x)), parameters (λ) and the time dependent force (η(t)) is given by
Eq. (1) is the force balance equation of any two dimensional system. The force experienced by the system is equated to the summation of the friction force experienced by the system, force due to the system potential energy (spatial derivative of the system potential energy) and the
Effect of increased loading on system potential
Fig. 2 shows the shape of the system potential as the loading in the transmission line is increased by increasing the mechanical power input to the machine. We can observe the increase in the tilt of the system potential with increasing mechanical power input of the machine.
Stability can be explained as the change in the topological shape of the system potential of the power system where by the stable and unstable equilibrium point of power system approach closer to each other and coalesce with
Energy functions in voltage collapse
Static voltage stability in power system is characterized by the occurence of the saddle node bifurcation at maximum loadability of the power system [9]. The counterpart of system potential in multidimensional power system is the energy functions. The equilibrium points of the energy function are the high (stable) and low (unstable) voltage solutions of the load flow equations [10]. Voltage stability can be explained as the change in the topological shape of the energy function of the power
Mean First Passage Time (MFPT)
The analytical expression for MFPT of the SMIB system as evaluated in [8] is given as follows:where
Extension to calculation of MFPT of multi dimensional power systems
The calculation of MFPT to multi dimensional power system with small dissipation and perturbed with white noise is given by [11],where Hcr is the energy at the closest unstable equilibrium point or saddle point. Hin is the energy at the initial stable equilibrium point. A is the strength of the load perturbation or covariance matrix of the random load perturbation. .
The key issue lie in finding the closest unstable equilibrium point of the multidimensional power
Simulation of Stochastic Differential Equation
Simulation study for the calculation of the MFPT of the system for the various types of random load perturbations was done for the SMIB system. Simulation of the load perturbations on the SMIB system has been carried out using the Fox’s algorithm [15]. Initially the system is assumed to be in the stable state and time the system takes to reach the unstable state is calculated for 3000 runs. The mean of the time taken by the system to move from the stable state to the unstable state gives the
Conclusion
This paper illustrates the need to operate the power system far away from the maximum loadability limit of the system because of the presence of the uncontrollable random load components. The effect of increased loading, in the shape of the system potential and energy function are illustrated. Analysis of the two dimensional power system perturbed by the random load variation is described in this paper. The MFPT of system for various intensities of noise strength, correlation time is tabulated.
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