This TS-based method can assess at the same time the effect of N P parameters in the EP stability, whereas in the method of eigenvalues and modal analysis, it is not possible. However, this analysis does not provide information about the critical parameter critical load in this case influencing the EP stability but only the critical generator and its most participative states. Thus, the modal analysis does not allow the direct identification of the most sensitive loading directions in the stability of the EPs around a HB point.

In practice, load increments not have a unique loading direction as in the previous study, which results in a valid consideration only for academic interest. All loads are then constantly varying in N P directions, so that it is very important to have a general tool to assess the stability of the EPs, especially operating under stressed conditions of loading. In this context, besides identifying the critical generators states , the SMA-TS approach based on TS identifies the loading directions which are most sensitive to oscillatory instabilities.

Multiparameter analysis of TS allows computing trajectory sensitivities with respect to N P parameters in a power system [ 17 ] at the same time, as explained in Section 3. For this purpose, Figure 6 shows the sensitivity norm SN through the time with respect to the three embedded loads in the system, where the highest peaks indicate the maximum influence of the corresponding load in the oscillatory behavior. The load demand corresponds to the base case as provided in [ 7 ]. The load at bus 6 P L 6 is the next most sensitive and finally the load P L 5.

The results are reported in Table 2 as follows: the first column indicates the load nodes, and columns 2 and 3 present the measured powers in the base case and the active power increment from the base case to the HB point to each loading direction, respectively. It is important to observe in Table 2 that the smallest load increment matches the highest SN value and vice versa, i.

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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 nodes and 46 generators. The transmission components are divided into transmission lines and 83 power transformers.

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Lastly, the system contains 26 capacitive compensators in shunt connection. The unifilar diagram of the power system is shown in Figure 7. 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 to 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 time-domain simulation, which corresponds to solving 91 sensitivity DAE systems, with each one consisting of equations and variables.

Thus, the assessment of the 91 loads is equivalent to solving equations in the same simulation at the same time. However, considering the linearity of the sensitivity systems, the same time-invariant Jacobian matrix is used during the whole time-domain 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 , which resulted as the most sensitive in the sensitivity norm assessment. 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.

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Column 1 Node indicates the most sensitive loads resulted from the TS analysis shown in Figure 8. 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 is 17 times larger than the load at bus ; however, the load at resulted in being more sensitive than the load embedded at bus , as shown in Table 3 , column 3.

It must be observed that the most sensitive loads loads — provided a major change in the critical eigenvalue and thus in the SSS. The same increment in the most sensitive loading directions buses — 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 SMA-TS approach has been successfully proved by determining that the most sensitive loads indicate the shorter ways toward the small-signal instability of the electric power systems. In this chapter an alternative approach for monitoring the Hopf bifurcations along variations in multidimensional loading directions by using a time-domain method is presented, which is based on trajectory sensitivities. SMA-TS allows to identify the critical loading directions that steer the system to Hopf bifurcation points.

Such an approach was tested in the 9-buses, 3-generators system as well as in buses, generators system. Regardless of the number of sensitivity parameters and system dimensions, SMA-TS 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. Furthermore, the approach can handle a very large integration step to drastically reduce the computational effort.

Lastly, its application is suitable for real-time monitoring and security assessment in energy management systems. The authors thank the University of Guanajuato for providing the financial support with PFCE resource for this publication. All the authors of this chapter declare to have no any conflict of interest related with any person, company, institution, etc. Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.

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We are IntechOpen, the world's leading publisher of Open Access books. Built by scientists, for scientists. Our readership spans scientists, professors, researchers, librarians, and students, as well as business professionals. Downloaded: Abstract The security assessment of power systems represents one of the principal studies that must be carried out in energy control centers. Keywords small-signal stability equilibrium points selective modal analysis Hopf bifurcation parameter sensitivities. Trajectory sensitivity analysis 3. Sensitivity discretization TS computation is obtained by means of the sequential solution of the nonlinear DAE system 1 and the linear time-varying DAE system 6 and 7.

Linear sensitivity computation Once algebraized both DAE systems 8 — 11 , the Newton-Raphson NR algorithm is used to provide an approximate solution of the algebraized nonlinear systems 8 and 9. 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.

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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. Modal analysis WSCC system In this subsection the conventional modal analysis is employed in order to investigate the small-signal stability of the WSCC power system, whose diagram is given in Figure 1.

Table 1. Selective modal analysis.

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Using automated gating, beads were then identified using Forward Scatter vs. This data was used to evaluate intra-instrument and inter-instrument sensitivity and stability. NOTE: We only request your email address so that the person you are recommending the page to knows that you wanted them to see it, and that it is not junk mail. We do not capture any email address. Skip to main content.

J Immunol May 1, , 1 Supplement Jolene A Bradford. Cytapex Bioinformatics, Inc. Abstract Flow cytometers are standard tools of the immunologist, providing simultaneous measurement of many targets. Back to top. The networks studied here, even when uncorrelated, have a different degree distribution than the ER network, which means that ER random networks are not a good control. Hence, we have to numerically generate the control distributions rather than having access to the analytical expression for the expected rate of each motif. In addition, in experimental settings we do not have access to the whole network from which to determine the motif distribution, we have to do with sub-networks.

These sub-networks do not come from the same network, rather they come from networks sampled from an ensemble of networks with similar properties. To obtain estimates for how to observe evidence for anti-correlation in the degree distribution we need to deal with each of these issues. The overall goal is to distinguish between pairs of networks with anti-correlated, uncorrelated and correlated degree distribution with the same marginal distribution for in- and out-degree.

We considered 13 different motifs that consisted of three connected neurons and gave each motif a numerical label as shown in Figure 7A.

## Sensitivity and stability analysis for nonlinear programming

We determined the number of motifs in each realization of a network with correlated, anti-correlated or uncorrelated degree distribution and took the average. The complexity of a motif corresponds to the number of edges in the pattern, ranging from 2 to 6, which determines how often it is counted in a network. We normalized the counts such that they took values on the order of unity in order to better compare them across motifs.

The mean count as a function of N sub converged to a constant for network sizes between 50— neurons Figure 7B , with more complex motifs requiring larger N sub. Hence, for large enough networks the differences in mean counts across network type can be detected with certainty. Figure 7. Motif 98 is the most sensitive to degree correlations. A There are 13 motifs that involve 3 connected nodes.