Selective modal analysis.
Abstract
The security assessment of power systems represents one of the principal studies that must be carried out in energy control centers. In this context, smallsignal stability analysis is very important to determine the corresponding control strategies to improve security under stressed operating conditions of power systems. This chapter details a practical approach for assessing the stability of power system’s equilibrium points in real time based on the concept of trajectory sensitivity theory. This approach provides complementary information to that given by selective modal analysis: it determines how the state variables linked with the critical eigenvalues are affected by the system’s parameters and also determines the way of judging how the system’s parameters affect the oscillatory behavior of a power system. The WSCC 9bus and a 190buses equivalent system of the Mexican power system are used to demonstrate the generality of the approach as well as how its application in energy management systems is suitable for power system operation and control.
Keywords
 smallsignal stability
 equilibrium points
 selective modal analysis
 Hopf bifurcation
 parameter sensitivities
1. Introduction
Smallsignal stability (SSS) is the ability of a power system to maintain synchronism when subjected to small disturbances such as small load and/or generation changes [1]. The analysis of SSS consists of assessing the stability of an equilibrium point (EP), as well as determining the most influential state variables in the stability of the operating point. For small enough disturbances, the system behavior can be studied via the theory of linear systems around an equilibrium point [2]. The stability of an EP is assessed by eigenvalue analysis (eigenanalysis) according to the Lyapunov criterion [3], which states that an EP will be stable in the smallsignal sense, if all system eigenvalues of the system matrix are located on the left side of the complex plane. On the contrary, the EP will be unstable if at least one eigenvalue is located on the right side of the imaginary axis. In this context, the resulting dominant eigenvalue from the eigenanalysis is called the critical eigenvalue, and its association to the state variables is investigated by selective modal analysis (SMA) [4, 5]. Based on the participation factors analysis (PFA), the SMA provides those state variables having the highest influence in the EP stability by means of their coupling to the critical eigenvalue [6, 7]. Since the system eigenvalues are directly related to its dynamic performance, different forms of instability in a power system can be studied by means of welldefined structures of eigenvalues which are called bifurcations.
The theory of bifurcations is a powerful mathematical tool based on eigenanalysis to assess the stability of EPs in nonlinear systems [8]. This theory consists of searching for specific eigenvalue structures associated to different instability forms that appear on power systems [9]. One of the most common local bifurcations that can appear in the power system operation is the Hopf bifurcation (HB), which occurs when the system matrix contains a pair of purely imaginary eigenvalues causing undamped oscillatory behavior [10, 11, 12, 13]. Any parameter variation in the system may result in complicated behavior until the system stability changes. This point, where the stability changes, is defined as a bifurcation point. In this chapter the system loads are changed in order to analyze the stability of the EPs. The maximum load in a specific direction that a power system can provide before the appearance of a bifurcation point establishes the loading limit in that direction. Thus, the loading limit is directly associated with the stability margin of the system. The critical eigenvalue of an EP is used as an index of the stability margin. After a load change, an eigenanalysis permits us to assess the stability margin closeness. A small margin indicates closeness to a bifurcation point (instability).
SSS analysis is very important to determine the corresponding control strategies to improve security under stressed operating conditions of power systems. Control strategies employed in electric power systems are usually tested by means of an assessment of the stability improvement. Thus, the influence of parameters and components in the EP stability provides an insight to achieve the best control. In this context, the participation factors let us know the highest association between state variables and the critical eigenvalue dominating the EP stability [6, 7]. In this way, PFA allows the selection via the associated states to the critical eigenvalue of those components that will provide the best control in EP stability. Although the PFA selects the most sensitive states in the EP stability, it is not possible to identify in a direct form the most influential parameters, e.g., those most sensitive loads influencing the stability. In order to achieve this, the PFA must be combined with other methods providing parameter sensitivity features. In [14, 15] the authors combined the PFA and modal controllability of the weak damping oscillatory modes to obtain an optimal location of static VAR compensators (SVCs). In [16] the authors presented an approach to examine the effect of loads in the system stability by using participation factors and mode shape analysis. In Ref. [7], both a voltage stability analysis and a lowfrequency oscillation analysis were performed by using the SSS analysis in the EPs. As the system faces increased loading conditions, bifurcation points appear, and the participating generators are identified by means of the most associated states obtained from the SMA.
This chapter presents an alternative method based on the trajectory sensitivity (TS) theory [17, 18] to investigate the stability of the EPs by using a timedomain simulation. In this approach which will be referred to as selective modal analysis and trajectory sensitivities (SMATS), the TS were computed with respect to the load parameters; therefore, besides the stability assessment and the participating states, this approach also has the ability to identify those most sensitive load parameters influencing the critical states. The stability assessment consists of just examining the TS oscillations. However, the stability analysis of the EP is never perturbed. The TS oscillations required for the stability assessment are produced by means of the initial condition values selected for the sensitivity variables. The SMATS approach uses an index of sensitivity quantification, which facilitates identifying the influence of the system loads around the HB points. The index allows to rank the power system loads in order to predict the most critical loading directions toward a B point. Its application is suitable to monitoring in real time the power system operation and improve the security. The results of the study cases of two real systems, i.e., WSCC 9bus and an equivalent of the Mexican power system of 190buses, are used to show the performance and applicability of the proposed approach.
2. Trajectory sensitivity theory
An electric power system can be represented analytically by a set of differentialalgebraic equations (DAEs), as given by (1), where
2.1. Analytical formulation
Let
The sensitivities of the dynamic and algebraic state vectors with respect to a chosen system’s parameter,
Lastly, the smooth evolution of the sensitivities along the trajectory (6) and (7) is obtained by differentiating (4) and (5) with respect to
where
3. Trajectory sensitivity analysis
3.1. Sensitivity discretization
TS computation is obtained by means of the sequential solution of the nonlinear DAE system (1) and the linear timevarying DAE system (6) and (7). By applying the trapezoidal rule to algebraize the differential equations, both DAE systems are converted into the following systems of algebraic difference Eqs. (8)–(11).
where
3.2. Linear sensitivity computation
Once algebraized both DAE systems (8)–(11), the NewtonRaphson (NR) algorithm is used to provide an approximate solution of the algebraized nonlinear systems (8) and (9). In this case, the resulting linearized system
The initial guess
Once the states have been computed for a new time step, the TS can be calculated. To do this, the linear timevarying systems (10) and (11) rearranged as indicated in (13) should be evaluated with the recently computed states and solved for the TS
It is very important to note that at each time step
3.3. Multiparameter sensitivity
It is clear that the solution of the linear timevarying system (13) uses the same Jacobian matrix already factored for solving the TS calculation with respect to any
3.4. Sensitivity quantification
As numerically demonstrated in [19, 20, 21, 22, 23], when the system approaches an unstable operation condition, the sensitivities of state trajectories have more rapid changes in magnitudes and larger excursions than the state trajectories. Hence, as the system approaches its stability boundary, the TS approach infinity [19]. In this sense, it is possible to associate the sensitivity information with the stability level of the system for a particular system parameter. For this purpose, an Euclidian norm of the trajectory sensitivity vector, referred to as a sensitivity norm, is proposed in [23] as a measure of proximity to instability. This sensitivity norm also permits the computation of the critical parameters whose variations steer the system much faster to an unstable operation condition. In this chapter, such a sensitivity norm is used to provide a timevarying index of proximity to oscillatory instability for a
where
In this case, the time evolution of sensitivities is necessary to quantify the loads’ influence on the possible occurrence of an oscillatory instability, where the highest values of the sensitivity norms
Note that this sensitivity norm has been successfully applied before for developing suitable approaches to improve the transient stability of power systems, e.g., for the estimation of the critical clearing time of a faulted system [23], the best possible location of FACTS controllers for transient stability enhancement [24, 25, 26], and the thyristorcontrolled series compensator (TCSC) control design to enhance transient stability [27].
3.5. Sensitivity initial conditions
The analysis of a stationary operating point based on this approach is performed by keeping the corresponding EP constant during the whole simulation. Thus, during this simulation period with no perturbation, which is considered from the instant of time
3.6. Approach for smallsignal stability with trajectory sensitivities
The application of the SMATS approach to assess the effect of a set of system’s parameters on the stability of the equilibrium points is summarized as follows:
Step 1. For an arbitrary set of fixed parameters
Step 2. Based on the Schur and the implicit function theorems [28, 29], compute the reduced Jacobian matrix
Step 3. Determine the critical eigenvalues of
Step 4. Compute the sensitivities of associated state variables with respect to the selected system’s parameters (load powers) at the equilibrium point,
Step 5. Quantify the interaction between the system parameters and the associated state variables by using the sensitivity index (15). Since this index is a function of the sensitivities of those state variables directly associated with the oscillatory modes and the critical eigenvalues, it can be used to quantify the effect of the
4. Smallsignal stability analysis
This section presents the analysis of the TS applied on assessing the SSS analysis. The effectiveness of the SMATS approach is numerically tested by analyzing the WSCC 9buses and 3generator systems [7] and a reduced equivalent system corresponding to the Mexican power system consisting of 190buses with 46 generators. For the purpose of the studies presented in this section, the system generators are represented by means of the twoaxis model with a simple fast exciter loop containing max/min ceiling limits. For this case the system loads are represented by means of the constant power load model; however, the generality of the method allows to consider any load model.
4.1. Modal analysis WSCC system
In this subsection the conventional modal analysis is employed in order to investigate the smallsignal stability of the WSCC power system, whose diagram is given in Figure 1. Table 1 presents the modal analysis for different EPs as the active power at bus 5 is increasing.


Associated states  PF 

4.2  −0.5085 ± 7.3001 

1.0, 0.99, 0.18, 0.18, 0.16 
4.3  −0.5395 ± 6.8512 

1.0, 0.99, 0.28, 0.39, 0.35 
4.4  −0.0305 ± 6.1462 

1.0, 0.99, 0.46, 0.99, 0.83 
4.5  0.7064 ± 5.8935 

1.0, 0.85, 0.85, 0.79, 0.38 
4.6  1.5118 ± 5.7190 

1.0, 0.81, 0.80, 0.76, 0.32 
4.7  2.6677 ± 5.5020 

1.0, 0.79, 0.78, 0.74, 0.28 
4.8  5.3798 ± 4.5835 

1.0, 0.77, 0.75, 0.76, 0.26 
SSS and SMA were performed to investigate the different operating points corresponding to the different levels of loading. Eigenanalysis revealed the SSS of EPs, whereas participation factors were used to identify the most associated states to the critical eigenvalue at each EP, as given in Table 1. The first column represents the load changes at bus 5. The second column presents the critical eigenvalue for the EP. The third column presents the most associated states to the critical mode (eigenvalue), obtained by selecting the highest magnitudes of participation factors, which are given in the fourth column in the table.
As the load embedded at bus 5 increased, the stability of the new EP decreased with respect to the previous one. The power system oscillatory instability called the HB was detected when the load changed from 4.4 to 4.5 p.u. The SMA around the HB revealed generator 2 as the most participative in the unstable EP. It is important to observe in Table 1 that generator 2 is the most associated with the critical eigenvalues for all analyzed EPs.
5. Trajectory sensitivity analysis: WSCC system
In order to test the proposed method based on TS to assess the EP stability, operating points before and after the HB point were investigated [30]. As the active power increases at bus 5, as was performed in [7], the system proximity to the bifurcation points is assessed by using the analysis of TS. The rotor angle and speed sensitivities with respect to the load active powers were traced and observed through the time. The TS oscillations provide qualitative information used to investigate the proximity to bifurcation points. Such sensitivity oscillations agree with the critical eigenvalues obtained by the SMA at each EP, as reported in Table 1. Thus, the oscillatory behavior in the TS indicates that the EP is around the HB point and the critical eigenvalue of the EP is complex. The proximity to the HB point is qualitatively assessed by observing the damping of the TS oscillation, i.e., if the TS oscillation is positively damped, the system is operating before the HB (stable EP); however, if the TS oscillation is undamped, the system operates after the HB (unstable EP).
Multiparameter sensitivity is used to assess the load influence around the HB. Such an assessment requires only one simulation to identify the critical loading direction on approaching HB. The TS computation in all cases started from second one onward.
5.1. Stability around the Hopf bifurcation
In order to qualitatively analyze the oscillatory behavior of the EPs around the HB, Figures 2–4 show the TS of the dynamic variables (generator states) with respect to the load embedded at bus 5. The oscillation waveforms and their peak values allow to assess and to determine the EP stability as well as its most associated states because of the loading increase. Figure 2 shows the TS with respect to
It should be noted from the figures that the load demand at bus 5 increases as the peaks of TS are higher. This qualitative information indicates that such a load variation has a major impact in the load angle and speed of generator 2 (red lines) than the state variables associated with generator 3 (blue lines). Note that this line of reasoning also applies for unstable EPs, i.e., after passing the HB point, as can be observed in Figure 4. Thus, a short simulation in time is enough for this approach to determine the EP stability and their most influencing generators.
5.2. Simulation efficiency
In order to observe the effect of the timestep integration
It is important to observe that the big difference between
5.3. Most sensitive loads to Hopf bifurcation
The participation factors provide the change of critical eigenvalues to the change of the states of critical machines
Multiparameter analysis of TS allows computing trajectory sensitivities with respect to
The oscillation of
Node  Δ 


8  100.0  299  399  1.723 
6  90.0  314  404  0.997 
5  125.0  316  441  0.739 
6. Trajectory sensitivity analysis: the Mexican system
In this section, the study consisted of computing the TS norm for 91 loads embedded in a reduced equivalent of the Mexican energy system, which consists of 190 nodes and 46 generators. The transmission components are divided into 180 transmission lines and 83 power transformers. Lastly, the system contains 26 capacitive compensators in shunt connection. The unifilar diagram of the power system is shown in Figure 7.
In order to assess the effect of the system loads on the system’s dynamic performance, the sensitivity norms with respect to 91 loads were computed. Figure 8 shows the effect of these sensitivities on a critically stable EP. Note that the active power demanded by loads connected at buses from 150 to 152 is the most sensitive in the EP stability. Therefore, according to the reasoning used into the previous section, the loading increase in such directions will steer in a faster way the system to a HB than the rest of the system loads. It must be pointed out that the computation of the sensitivity norms for the 91 system loads were carried out by using one sole timedomain simulation, which corresponds to solving 91 sensitivity DAE systems, with each one consisting of 702 equations and variables. Thus, the assessment of the 91 loads is equivalent to solving 63882 equations in the same simulation at the same time. However, considering the linearity of the sensitivity systems, the same timeinvariant Jacobian matrix is used during the whole timedomain simulation, which considerably reduces the computational burden.
Once the critical loads have been identified from Figure 8, it is possible to know the most affected generators by the most sensitive loads. Figure 9 shows the TS with respect to the active power at bus 152, which resulted as the most sensitive in the sensitivity norm assessment. The damped oscillations in the TS indicate that the EP is stable and the operation point is not at a HB, which agrees with the corresponding critical eigenvalue of the EP
In order to validate the load ranking influence via the sensitivity norm, Table 3 shows how the increments in the most sensitive loading directions influence the SSS, as well as the proximity to the HB point. Column 1 (Node) indicates the most sensitive loads resulted from the TS analysis shown in Figure 8. Column 2
Node  Δ 
Δ 


152  0.021 ± 5.02 
172.64  232.64  53.0  225.64 
150  0.018 ± 5.03 
188.24  248.64  54.0  242.24 
151  0.015 ± 5.03 
18.72  78.72  54.0  72.72 
147  0.012 ± 5.05 
104.00  164.00  56.0  160.00 
153  −0.011 ± 8.83 
78.00  138.00  63.0  141.00 
145  −0.015 ± 5.13 
83.20  143.20  72.0  155.20 
120  −0.050 ± 7.85 
308.88  368.88  189.0  497.88 
It is important to outline that the load effect in the EP stability is not only dependent on the magnitude but also on the topologic location of loads. For example, the power demand embedded at bus 120 is 17 times larger than the load at bus 151; however, the load at 151 resulted in being more sensitive than the load embedded at bus 120, as shown in Table 3, column 3. It must be observed that the most sensitive loads (loads 152–147) provided a major change in the critical eigenvalue and thus in the SSS. The same increment in the most sensitive loading directions (buses 152–147) led the system to oscillatory instability due to a HB point, whereas with the increment in the least sensitive loading directions, the system remained stable. Then, the stability margins in the most sensitive loading directions become more reduced; therefore, according to the sensitivity ranking in Table 3, as the most sensitive loads were increased, the appearance of the HB was found faster as can be observed in column 5. Once more the SMATS approach has been successfully proved by determining that the most sensitive loads indicate the shorter ways toward the smallsignal instability of the electric power systems.
7. Conclusions
In this chapter an alternative approach for monitoring the Hopf bifurcations along variations in multidimensional loading directions by using a timedomain method is presented, which is based on trajectory sensitivities. This approach, SMATS, is general and flexible, i.e., the size of the power systems, as well as the complexity of their mathematical modeling, does not represent any restriction. SMATS allows to identify the critical loading directions that steer the system to Hopf bifurcation points. Such an approach was tested in the 9buses, 3generators system as well as in 190buses, 46generators system. Regardless of the number of sensitivity parameters and system dimensions, SMATS requires only one simulation. Such a method keeps constant the Jacobian matrix of the system, requiring only one evaluation and factorization during the whole simulation. The computational effort then consists of performing just one forward/backward substitution at each time step. Furthermore, the approach can handle a very large integration step to drastically reduce the computational effort. Lastly, its application is suitable for realtime monitoring and security assessment in energy management systems.
Acknowledgments
The authors thank the University of Guanajuato for providing the financial support with PFCE 2018 resource for this publication.
Conflict of interest
All the authors of this chapter declare to have no any conflict of interest related with any person, company, institution, etc.
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