University of Cambridge

The Role of Hopf Bifurcations in Stability Analysis of Power Systems

1. Introduction

Electrical power systems are very complex nonlinear systems that may show various complicated events not been thoroughly examined until now [1]. Power systems, in the past two decades, have been working under much more stressed conditions than in the past. This is mainly because of the ecological pressures on transmission development, enlarged electricity use in heavy load parts, novel models of system loading for the deregulated electricity market, and so forth. As a consequence of these stressed situations, a power system may reveal new kinds of dynamic unstable performances like slow voltage drops, or even voltage collapse [2-5]. In recent years, voltage stability and voltage collapse phenomena have turned into very significant concerns in power system analysis and control [6,7]. Voltage collapse is an inherently nonlinear phenomenon, which is related to bifurcations from nonlinear dynamic systems point of view[1, 8]. Hence, the necessity of dynamically analyzing the power system has developed widely in recent years. One of the limitations to the operational security of power systems is oscillatory instability [9], i.e., an inherently nonlinear phenomenon related to bifurcation from the viewpoint of nonlinear dynamic systems. Considering Hopf bifurcation theory, large amount of research has been carried out to understand and analyze the mechanism of this type of instability [10-13].

Bifurcation points out a change in the number of candidate operating conditions of a nonlinear system when a parameter is quasistatically changed [14]. The candidate operating condition may be an equilibrium point, a periodic solution, or other invariant subset of its limit set, irrespective of its stability characteristics [14]. The parameter  which is being varied is recognized as the bifurcation parameter. A nonlinear dynamical system may exhibit a range of different bifurcations when one or more parameters are changed. A leading feature of a bifurcation is the direction, or stability, of the bifurcation.

Apparently, an exact modeling of power system equipment, including generators, loads, and regulators is fundamental in order to accurately understand and reproduce voltage instability [15, 16]. Besides, flexible AC transmission systems (FACTS) have been recognized as capable solutions to improve the stability of power system [17, 18].

The bifurcation theory provides a set of mathematical procedures for nonlinear differential algebraic equations (DAE) being sufficient for studying power systems, which are characteristically modeled as a set of nonlinear DAE. Above all, the bifurcation theory is usually identified as a useful method for voltage stability assessment [19–21].

Power networks are composed of nonlinear parts, such as synchronous generators (supplying real and reactive power for the networks), load buses (representing the consumers), and transmission lines for power distribution. The dynamics of a system like this is represented by swing equations of synchronous generators, dynamical load differential equations (e.g., induction machines), and algebraic equations to characterize the network nonlinear loads [23]. With changing the load conditions, the set of nonlinear differential equations goes through qualitative variations at bifurcation points. Hopf bifurcation and saddle node bifurcation are known as the typical bifurcations in power systems [9].

It is the aim of this paper to study the Hopf bifurcations in power systems. A variety of phenomena that may result in Hopf bifurcation are taken into account and their identification and various methods to analyze and control them are considered.

The organization of the paper is in this manner: Hopf bifurcations in dynamic systems are explained in Section 2 and different types of them together with mathematical equations are offered. In Section 3, different events in power systems, which may cause Hopf bifurcations, are studied and in Section 4, identifying and controlling of Hopf bifurcations in different power systems with various equipment and models are explained. Finally, there are conclusions in Section 5.

2. A brief description of Hopf bifurcations in dynamic power systems

Hopf bifurcations do not cause any changes in the number of equilibrium points [24]. These types of bifurcations are demonstrated by a complex conjugate pair of eigenvalues for the equilibrium point (x0; y0; ?0) of lying on the imaginary axis of the complex plain [9]. As the parameter ? changes, the complex conjugate pair moves away from the imaginary axis, either to the right or to the left of the axis causing stable or unstable limit cycles (system oscillations) to appear or disappear. This kind of oscillatory instability in nonlinear systems is related to the Hopf Bifurcation (HB)[1]. The positions of the eigenvalues near the Hopf bifurcation point are shown in Fig. 1 [1].

Fig. 1. the locations of the eigenvalues near the Hopf bifurcation point

Considering Fig. 1, it can be explained that a Hopf bifurcation occurs at a point where the system has a simple pair of purely imaginary eigenvalues (?=±j ?) and no other eigenvalue with zero real part. Hopf bifurcations show the birth or the obliteration of period orbits known as limit cycles [25]. A limit cycle is an isolated periodic solution of a nonlinear system x = f (x) [1]. At this point, a periodic solution is a function x(t) that satisfies and has the characteristics of x(t +T) = x(t) for all t, where T is the period of the periodic solution. Hence, we can say that a limit cycle is a closed curve in n-dimensional space [25]. The stability of the equilibrium point at an HB is lost when it interacts with a limit cycle [26]. Near this bifurcation, it is expected that either stable or unstable limit cycles to exist. Regarding the bifurcation value parameter and based on the stability of the limit cycles and where they occur, Hopf bifurcations can be either subcritical or supercritical [26, 27]:

Subcritical HB: In this kind of HB, an unstable limit cycle, which exists before the bifurcation, gets smaller and eventually disappears as it combines with a stable equilibrium point at the bifurcation. After the bifurcation, the equilibrium point becomes unstable resulting in growing oscillations.

Supercritical HB: In this case, a stable limit cycle is produced at the bifurcation, and a stable equilibrium point becomes unstable with increasing amplitude oscillations that are finally attracted by the stable limit cycle. Fig. 2 shows the normal forms for Hopf bifurcation in both supercritical and subcritical cases.

The type of the Hopf bifurcation can be confirmed by either perturbation techniques founded on the multiple scales method, or numerical method based on the response of the perturbed system [28]. The instability mechanism of the subcritical Hopf bifurcation is depicted in Fig. 3 (a) where annihilation of the operating point causes oscillatory diverging. In addition, the instability mechanism of the supercritical Hopf bifurcation is illustrated in Fig. 3 (b). It is noticed that in this case the operation varies from an operating point to stable oscillations [29].

From an engineering viewpoint, none of the above cases are acceptable in view of the fact that both of them give an unstable operating point with oscillations after this bifurcation [1]. The necessary condition for a HB is the existence of equilibrium with purely imaginary eigenvalues.

When the Hopf bifurcation occurs, the Jacobian matrix of the system has a simple

Fig. 1. Normal forms for Hopf bifurcation: (a) supercritical; (b) subcritical

Fig. 3. Instability mechanism of the Hopf bifurcation for a typical ?= 0.2; (a) Subcritical, (b) supercritical

pair of purely imaginary eigenvalues and there are no other eigenvalues on the imaginary axis. At the point of Hopf bifurcation, the power system may face with undamped oscillations [9]. Based on dynamic system stability theory, it is known that the behavior of a given nonlinear system, as shown in (1), at a critical point x0, is similar to that of the same system linearized at x0, as depicted in (2) [30]:

                                           (1)

                                     (2)

J(x0, ?0) = Df(x0, ?0) is the Jacobian matrix of the system at x0. If J(x0, ?0) presents only eigenvalues with negative real part, the original nonlinear system is locally stable at x0. The Hopf bifurcation occurs at an operating point whose J(x0,?0) has one, and only one, pair of purely imaginary eigenvalues, and all others having non-zero real part [31]. At this point ?0 will correspond to the bifurcation value of the system.

As defined in [32], a Hopf bifurcation happens when in the differential system x´= f(x, ?) the following conditions are satisfied:

(a)    f(x0, ?0) = 0

(b) Df (x0, ?0) has a simple pair of purely imaginary eigenvalues ?(?0) = ± j? and no other eigenvalues with zero real part.

(c)  d[Re ?(?0)] / d? ? 0

where ? is the bifurcation parameter—a parameter for changes in the structure of the system under study, Df is the Jacobian matrix of the mismatch function f, and subscript ‘0′ refers to a bifurcation point. Condition (c) assures the transversal crossing of the imaginary axis. The sign of (c) confirms if there is a birth or death of a limit cycle at (x0, ?0) [33]. Power systems are usually modeled using a set of differential and algebraic equations (DAEs):

x´= f(x, y, ?)

0 = g(x, y, ?)                                          (3)

where x are state and y are algebraic variables. To study the Hopf bifurcation conditions in (3), initially, the equilibrium variable values (x, ?) need to be calculated. This is done by solving the load flow equations for a given ?=?0 [29]:

0 = f(x, y, ?)

0 = g(x, y, ?)                                           (4)

The dynamic state matrix Js is then computed as follows; provided that gy is nonsingular, i.e., singularity-induced bifurcations are abolished:

Js(x, y, ?) = fx ? fy(gy)-1gx                      (5)

A Hopf bifurcation can be found by solving the set of equations:

F(X, ?) = 0

det [ j?I – Js (X, ?)] = 0                          (6)

Function F(x, y, ?) in (6) stands for the functions f and g in the load flow equations (4), and X=(x, y). The second condition in (6) means the same as (b) in the general Hopf bifurcation conditions.

3. Various phenomena leading in Hopf bifurcations

Hopf bifurcations occur when a parameter in the system is varied and they cause a pair of complex eigenvalues cross the imaginary axis, which may lead in unstable oscillations in the system rooted in the type of the Hopf [9]. In power systems, many phenomena may result in Hopf bifurcations. Improper tuning of the control parameters in generation units can cause Hopf bifurcation [34-36]. Nonlinear loads may cause Hopf bifurcation as well [37]. In [11], [38], [39] an investigation is performed about a disturbance on the midwestern segment of the US interconnected power system and the consequential oscillations caused by line tripping. These analysis confirm that the event was obviously related to a Hopf bifurcation. In the following, a brief description of several important phenomena, which lead in Hopf bifurcations, is presented.

3-1) SSR

SSR is a phenomenon in power systems in which the dynamic bifurcation “Hopf bifurcation” theory can be applied [28]. Zhu et al. [40], making use of this theory, studied a SMIB power system with SSR in which the dynamics of the AVR and damper windings are neglected. In their effort, a prediction of supercritical Hopf bifurcation is considered. The bifurcation analysis has been used by Nayfeh et al. [41] to study the complex dynamics of a heavily loaded SMIB power. In their work, the dynamic effects of d-axe and q-axe damper windings are taken into account while that of the AVR is ignored. Based on the results of their work, by increasing the compensation factor the operating point loses its stability through supercritical Hopf bifurcation. Furthermore, the effect of electrical machine saturation on SSR is examined by Harb et al. [42]; concluding that the saturation of generator, by shifting the Hopf bifurcation point to smaller compensation level, slightly makes the positively damped region smaller. In addition, it slightly shifts the secondary Hopf bifurcation to smaller compensation level.

3-2) Hydro-turbine Governing Systems With Saturation

In [43], by means of simulations, the simple nonlinear state equations of the PI governing system considering saturation for hydro-turbines are set up and next, the Hopf bifurcation behavior in various conditions is studied. The results of their work express that a hydro-turbine governing system may have unexpected limit cycle oscillations when the parameters of the governor satisfy certain conditions. Computer simulations have been applied to imitate the occurrence of oscillatory behaviors in a hydroelectric power plant [44]. Hopf bifurcation phenomenon has never been considered as one of the possible reasons for the oscillatory behaviors. In [43] it is suggested that the Hopf bifurcation can be thought as one of the probable events causing sustained oscillations. Mansoor et al. [44] present additional verification to the above conclusion through recorded oscillatory behaviors. Daijian Ling and Yang Tao [43] perform a new analysis approach to investigate the stability of hydraulic turbine governing systems employing the bifurcation theory of nonlinear dynamical systems.

3-3) Static Var Compensator (SVC)

The effects of a Static Var Compensator (SVC) devices on the stability of a simple “single-machine dynamic-load” system have been revealed in [6]. It is demonstrated that the SVC is able to improve the loadability of the system but results in a Hopf bifurcation. In addition, a multi-parameter bifurcation study is accomplished to examine the effect of system parameters on the Hopf bifurcation.

3-4) Changes in load conditions

The changes in load conditions or the mechanical energy input for synchronous machines in a power system can cause Hopf bifurcation, which must be avoided [23]. K. Kobravi et al. [23], applied the MATCONT package to study the Hopf bifurcation in a simple power system. The MATCONT predicts the existence of a Hopf bifurcation point with regard to both Q1 and Pm. Figure 4 shows the Hopf bifurcation diagram for the power system with respect to Pm and Q1 [21].

3-5) Converters

The Hopf bifurcation study of a hysteretic current mode controlled Luo DC-DC converter has been presented in [45]. It has been illustrated that as the control parameters are changed, the nominal periodic orbit experiences a Hopf bifurcation. It has also been concluded from the experimental results that the margin of system stability decreases when the load resistance increases [45].

An analysis on the sliding mode control and nonlinear phenomena in SEPIC converter have been performed in [46], where the sum of two inductor currents have been taken as control variable to design the sliding mode control in SEPIC, obtain the equivalent control and differential equations on the sliding surface, and investigate the stability of the equilibrium point by calculating of the eigenvalues of Jacobian matrix. Using numerical simulation from the exact model, it has been demonstrated that the equilibrium point may undergo a Hopf bifurcation as the current reference increases. In Figure 5, a bifurcation diagram with current reference as the bifurcation parameter is depicted [46]. The work done in [46] can provide guidelines to the design and application of SEPIC converters.

3-6) Lorenz-type chaotic systems

Yang Qigui and Liu Mengying [47], with accurate symbolic calculation and a absolutely mathematical study, utilized the first Lyapunov coefficient to analyze the Hopf bifurcation of the Lorenz-type chaotic system with whole parameter space completely. Therefore, it is shown that this system has three Hopf bifurcation points, at which those parameters meet a particular condition, the Hopf bifurcation points are supercritical, satisfying another particular conditions, the Hopf bifurcation points are subcritical.

3-7) Time delays

In [48], the distribution of the roots of the associated characteristic equation has been analyzed and the stability of trivial equilibrium is studied. It is proved that when the delay passes through critical values, Hopf bifurcation takes place from trivial equilibrium. Numerical integration technique is applied to study the effect of time delay on autonomous and non-autonomous systems, respectively, with the aid of given system parameters. The results reveal that in an autonomous system when time delay increases to certain value, a stable periodic solution happens from equilibrium point through Hopf bifurcation, which confirm the validity of the analytical method applied in [44]. For a non-autonomous system, the results prove an incensement in the amplitude of stable period motion, which may cause some complex dynamical behaviors like quasiperiodic motion, and the system may get out of control as the time delay increases.

Fig. 4. The Hopf bifurcation diagram for the power system with respect to Pm and Q1

3-8) Tabu learning models

A study on tabu learning single neuron model has been performed in [49] using the frequency domain approach having the benefit of not needing many mathematical computations and not being as intricate as analyzing the model in the time domain. Considering the memory decay rate as the bifurcation parameter, it has been demonstrated that when this parameter passes through a critical value, a Hopf bifurcation arises. The stability and direction of the bifurcating periodic orbits have been evaluated by drawing the amplitude locus L1 and the locus ?(j?) in a neighborhood of a Hopf bifurcation point. Totally, there are four differential equations in the system with respect to a two-neuron model with a linear proximity function. Applying these four differential equations, the Hopf bifurcation in time domain is studied in [50]. For a two-neuron model with a quadratic proximity function, there are totally six differential equations.

3-9) Congestion Control Algorithms

In [51], an analysis for a class of end-to-end network congestion control algorithms with communication delays is performed. When the communication delay is selected as a parameter, it is proved that there exists bifurcation (Figure 6). The methods to determine the stability of bifurcation periodic solutions and the direction of the Hopf bifurcation are also developed in [51].

Gaurav Raina [52] presents the fundamental calculations to verify the stability and asymptotic forms of solutions that are bifurcated from steady state in a nonlinear delay differential equation with a single discrete delay. The results are applied to study the loss of local stability in a choice of congestion control algorithms utilized over a single link.

4. Identification and control of Hopf bifurcations in power systems

Power system models exhibit a wide range of nonlinear phenomena that have been studied by bifurcation theory. Specifically, the appearance of oscillations in power system models are referred to Hopf bifurcations, which arise when the control gains or machine loading exceed critical values. The bifurcation problem of differential algebraic system (power system dynamic voltage stability) has been considered in [53] through unreduced Jacobian analysis based on singular perturbation.

In [54], the dynamics of a basic power system model is analyzed while the field voltage of the generator is subject to a time delay. Based on this analysis, when the time delay exceeds a critical value ?c through a Andronov-Hopf bifurcation, a nominally stable operating point can be destabilized.

A new method is introduced in [55] to calculate Hopf bifurcation boundaries for DAE systems that can model power systems dynamics. The basic scheme is depicted in Figure 7. This method has been applied to a number of example power systems to reveal its efficiency and applicability.

Fig. 5. Bifurcation diagram with current reference as the bifurcation parameter

 By utilizing this technique, we can systematically determine all three main local bifurcation boundaries for a general DAE problem. It is a direct method and keeps the DAE form. Thus, it preserves the sparsity of the power system data structure.

Complex nonlinear phenomena in power system models have been widely studied in the past. Although extensive attention has been paid to the investigation of power system models in the absence of time delays ([56-60]), not as much notice has been devoted to study power system dynamics in the presence of time delays excluding [61].

A novel algorithm, which calculates the minimum and maximal Hopf bifurcations and load flow feasibility boundary points as part of an ordinary procedure has been proposed in [62] and has been tested and validated by means of numerical simulations.

In [24], the bifurcation analysis of a detailed power system model composed of an aggregated induction motor and impedance load supplied by an under-load tap-changer (ULTC) transformer and an equivalent generator and transmission system has been presented. The model of the test system is presented in Figure 8 and the system parameters are summarized in Table 1 [24].

 Various modeling levels with their relevant differential-algebraic equations are considered to determine the minimum dynamic model of the system for bifurcation studies of power systems. An aggregated model of a realistic load is employed to demonstrate the ideas proposed in the paper. This paper [24] provides a comprehensive bifurcation investigation of different order models of aggregated induction motors fed by ULTC transformers. It is shown that when oscillatory modes associated with Hopf bifurcations are of interest, high order models must be utilized [24]. Moreover, it is proved that when bifurcation studies are performed to investigate the existence of limit cycles, which are associated with Hopf bifurcations, detailed generator, ULTC transformer and induction motor models should be employed. However, it is not necessary to use detailed dynamic transmission line models. Considering detailed transmission models may be advantageous in reducing computational complexity due to algebraic constraints [24].

   

Fig. 6. State and phase plot of a system with a delay more than reference value

Fig. 7. Basic scheme of continuation method

Fig. 8. Aggregated load model of the test system [24]

Table 1. Aggregated Load Data (100MVA, 4KV Base)

Variable

Value (p.u.)

Rs

0.07825

Xls

0.8320

Xm

16.48

Rr

0.1055

Xlr

0.8320

H

0.1836 s

?e

2?60 s-1

G

0.06047

B

0.03530

Xth

0.2

Tt

5 s

V2

1

Different local periodic solutions may give different classes of storage patterns or memory patterns, and arise from the different equilibrium points of neural networks (NNs) by employing Hopf bifurcation method. In [64], a bidirectional associative memory NN with four neurons and multiple delays has been considered and using the normal form theory and the center manifold theorem, investigation of its linear stability and Hopf bifurcation has been done. An algorithm has been applied to determine the direction and stability of the bifurcated periodic solutions.

The distribution of the roots of a general fourth-degree exponential polynomial equation was studied in [63] and it was discovered that under certain conditions, when the sum of the delays changes, the zero solution loses its stability and a Hopf bifurcation happens. Furthermore, utilizing the normal form theory and center manifold reduction, the stability and direction of the Hopf bifurcation are determined. It has been proved that based on some conditions, the direction of Hopf bifurcation and stability of the bifurcating periodic solutions can be determined by sign.

In [64], a novel normal form in nonlinear control systems is developed to control the Hopf bifurcation and simple expressions for stability coefficients of the Hopf bifurcation are attained throughout the design of simple control rules.

An index has been proposed in [30] to identify Hopf bifurcation points in power systems. Two models are employed to develop the method. Once the network dynamics are modeled, systems susceptible to sub-synchronous resonance are identified. Alternatively, if interactions between generators and the network are not in the center of attention, Hopf bifurcations may be identified as a result of generator oscillations. This index [30] is based on the calculation of a reduced set of eigenvalues, which makes the algorithm useful even for large power systems.

Two different types of power system models have been focused in [30]: one for angular stability analysis and another for sub-synchronous resonance (SSR) analysis. Two indices have been evaluated and investigated to identify Hopf bifurcations—HBI and IND. On the subject of the HBI index, for both model types, it is capable of detecting Hopf bifurcations, provided that the frequency is updated at every step. Another HBI characteristic is the need for tracking more than one eigenvalue to guarantee stable operation for the system under study [30].

The IND index identifies the bifurcation point properly for both studies, without needing to know in advance the critical system mode. This is possible because a cluster of eigenvalues is evaluated at each operating point. As a result, while the system parameter is changed, the critical pair of eigenvalues is included into this cluster [30]. To verify the bifurcation point in studying angular stability, two other indices have been computed. The results have been proved to be reliable, enabling one to identify correctly the bifurcation point for any types of analysis.

5. Conclusions

In this paper,  a study on Hopf bifurcations is presented for power system stability analysis employing of the results of so far developments achieved in this field. Hopf bifurcations in dynamic systems are explained and mathematical equations are presented. Various phenomena in power systems that may result in Hopf bifurcations are taken into account. In addition, identifying of Hopf bifurcations are investigated and different methods to analyze and control them are studied. Furthermore, various modeling levels with their respective differential-algebraic equations are studied and Hopf bifurcation analysis of different order models of system is performed considering induction motors and impedance load conditions supplied by under-load tap-changer transformers (ULTC). The results of this study demonstrate the fundamental role of identification and analysis of Hopf bifurcations in power system stability assessment.

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