Grouping of $N-1$ Contingencies for Controller Synthesis: A Study for Power Line Failures

The $\mathcal{H}_{\infty}$ and $\mathcal{H}_{2}$ norms of a transfer function $T$ are denoted by $\|T\|_{\mathcal{H}_{\infty}}$ and $\|T\|_{\mathcal{H}_{2}}$, respectively. These norms are defined in Section II-C. The transfer function from disturbance to performance output of the closed-loop system of a plant $P$ and controller $K$ is denoted by $\mathcal{F}(P,K)$.

A power system can be modeled by a system of nonlinear differential-algebraic equations (DAEs). The differential equations describe the dynamics of, for example, the generators, while the algebraic equations represent the network equations. We consider a linear time-invariant (LTI) model where the DAE has been linearized around an operating point and the network equations have been eliminated. We model the operating point as fixed for some period of interest, noting that it will change on a slow time scale as conditions such as loads throughout the network change. Note that this LTI model can be formed from the DAE model regardless of whether the power system has synchronous generators, inverter-based resources (such as renewable energy sources), or a mix of the two. See, for example, [19] and [20].

We assume the LTI model is of the form

where $x\in\mathbb{R}^{n}$ is the deviation of the power system state from the operating point, $w\in\mathbb{R}^{n_{w}}$ is the disturbance, and $u\in\mathbb{R}^{m}$ is the control input.

In a power system with synchronous generators, for example, the LTI model can be constructed such that $x$ consists of the following: $\theta$, the vector of generator rotor angles, with the average of rotor angles removed; $\dot{\theta}$, the vector of deviations of generator rotor frequencies from the operating frequency; and additional states accounting for fast electrical dynamics [2]. An LTI model with this interpretation can also be constructed for power systems with inverter-based resources using virtual inertia techniques. A review of such techniques can be found in [21].

The contingencies we consider are single-line failures. For a power system, we can construct a graph where buses are nodes and lines between buses are edges. A line failure is represented by the removal of the corresponding edge from the graph. For simplicity of modeling, we consider line failures that do not disconnect the graph. This ensures that the states of the model have the same interpretation across all contingencies. Under these assumptions, only the $A$ matrix differs between contingencies. Thus, the dynamics of contingency $i$ can be represented as

Note that under contingency $i$, the state vector $x$ represents the deviation of the power system state from the operating point of contingency $i$.

With this model, the event where contingency $i$ occurs is where the dynamics change from nominal dynamics $(A_{0},B_{w},B_{u})$, to contingency dynamics $(A_{i},B_{w},B_{u})$, and the operating point changes from the nominal operating point to the operating point under contingency $i$. Since we handle only single-line failures, we assume that this change in dynamics happens once.

We model the controller as a state-feedback LTI controller. In particular, for a a plant $P$ as in equation (1), the controller $K$ is of the form

where $x_{k}\in\mathbb{R}^{n_{k}}$ is the controller state, $x$ is the state of the plant $P$, and $u$ is the control input to plant $P$. Matrices $A_{k}$, $B_{kx}$, $C_{k}$, and $D_{kx}$ are all tunable controller parameters.

This is a centralized controller, with all entries of the control input $u$ being a function of all entries of the plant state $x$. Further, we assume this centralized communication topology does not change when a line failure occurs; i.e. line failure does not result in communication failure.

With this controller, the closed-loop system of $P$ and $K$ can be written as

where

The control design problem is to find suitable parameters $A_{k}$, $B_{kx}$, $C_{k}$, and $D_{kx}$, where we define “suitable” with respect to a performance objective, defined in the next subsection.

We consider the control problem of suppressing effects of disturbances. In our experiments in particular, we consider the effect of disturbances on inter-area oscillations. To measure performance of a controller, we first add a virtual output $z$ to the closed-loop system, as follows, for some $\mathcal{C}$ and $\mathcal{D}$.

We refer to the transfer function from disturbance $w$ to performance output $z$ of this closed-loop system of plant $P$ and controller $K$ by $\mathcal{F}(P,K)$. Then, we measure disturbance suppression using the $\mathcal{H}_{\infty}$ norm of $\mathcal{F}(P,K)$, where a smaller norm means more suppression, with the optimal value being $0$. The $\mathcal{H}_{\infty}$ norm is defined as

where $\bar{\sigma}$ denotes the maximum singular value of a matrix [22]. This metric represents the maximum possible amplification of an input disturbance. Methods involving $\mathcal{H}_{\infty}$ control are widespread in power systems control literature, e.g. in [8, 23, 24].

In this work, we focus on control design and performance measurement using the $\mathcal{H}_{\infty}$ norm. However, we note that this norm can be replaced by other norms, as long as the same norm is used consistently throughout. Thus, in our experiments, we will also consider the $\mathcal{H}_{2}$ norm. The $\mathcal{H}_{2}$ norm is defined as

In the case where the disturbances $w$ are stochastic, the squared $\mathcal{H}_{2}$ norm corresponds to the variance in performance output $z$ at steady-state [2].

For our experiments, we use a performance output previously used in [2] for suppressing inter-area oscillations. This performance output is defined with respect to inertia, which may be either inertia of a synchronous generator, or virtual inertia of a virtual synchronous generator. In particular, let $\tilde{\theta}$ be the deviation of the generator rotor angles from the average angle, $\dot{\theta}$ be the deviation of rotor frequencies from the operating frequency, and $M_{G}$ be the diagonal matrix of generator inertias. Then we define the performance output of the system to be

We investigate three metrics to measure distance between contingencies. Due to the number of pairwise distances scaling in the square of the number of contingencies, we choose metrics with reasonable computation times. We apply these metrics to the plants in feedback with the nominal controller. A metric may be refined for the needs of a particular application. For example, if an application is sensitive to low-frequency differences between plants, one could apply a low-pass filter to the frequency response creating a weighted frequency response metric.

To group the $M$ contingencies into $k$ clusters using the selected metric, we consider several existing graph or metric clustering algorithms. These algorithms only require pairwise distances between points, not the points themselves (as might be required in, for example, k-means to average points), allowing the clustering algorithm to be chosen independently from the metric.

We consider three types of clustering: k-centers [25], k-medoids [26][27], and divisive clustering [28]. These methods optimize for different objectives and have varying scalability.

For a set of $M$ points and a metric $d(i,j)$, the k-centers problem is to find clusters and centers of each cluster such that

is minimized, where $c(i)$ gives the index of the center of the cluster that point $i$ is in. Note the center of a cluster is one of the points in the cluster. This method minimizes the maximum distance from a point to its cluster center.

While finding an optimal solution to this problem is NP-hard, there exist approximation algorithms with time complexity $O(Mk)$ that can achieve a maximum distance to center within twice the optimal value [25].

This clustering method is useful for robust group controller design techniques that view the radius of a group (the maximum distance between a point in the group and the center of the group) as the radius of an uncertainty set that covers plants in the group.

The k-medoids problem is to find clusters and centers of each cluster to minimize

Note that algorithms to minimize this objective can also be used to minimize $\sum_{i}\tilde{d}(i,c(i))^{2}$ by setting $d=\tilde{d}^{2}$.

Algorithms of various approximation performance and time complexity are available, such as an algorithm with time complexity $O(k(M-k)^{2})$ presented in [26] and one with time complexity $O(Mk)$ presented in [27].

Divisive clustering is an iterative clustering algorithm that starts with all points in one cluster and repeatedly divides the worst performing cluster until $k$ clusters are reached [28]. We choose the worst performing cluster as the one with the highest average distance of points in the cluster to the cluster center. This cluster is divided using another clustering algorithm, like one for k-centers or k-medoids.

While we divide clusters until we reach $k$ clusters, one might also divide until a certain performance criterion is achieved, if such a criterion is available and fast enough to compute.

The IEEE 39-bus system is a 10-machine, 39-bus, 46-line system representing the New England power system. Of the 46 lines, there are 11 whose removal will disconnect the underlying graph. We assume these line failures are handled separately, and only consider the remaining $M=35$ lines as contingencies. These lines are labeled in Fig. 2. In this example we study the performance of the algorithm presented in Section III with several metrics and clustering methods, for varying numbers of groups.

We run Algorithm 1 on all nine combinations of the metric and clustering methods in Section III for numbers of groups $k=1,\dots,20$ and compare performance across metric and clustering methods versus number of groups. Figure 3 summarizes the average scaled $\mathcal{H}_{\infty}$ norm over contingencies versus number of groups for each grouping method. As a baseline, we include the cost of the nominal controller applied to the contingencies. All tested grouping methods perform near optimal on every contingency when partitioning the contingencies into 20 groups. For numbers of groups under 10, we observe four well-performing methods: PSN k-medoids, PSN divisive k-medoids, SR k-medoids, and FR divisive k-medoids. Additionally, for this system, we observe that designing just one controller to handle all contingencies performs better on average than continuing to use the nominal controller after a contingency occurs.

While Figure 3 summarizes mean cost of grouping algorithms, mean cost may not be the only performance characteristic of interest. Figure 4 plots the scaled closed-loop $\mathcal{H}_{\infty}$ norm for each contingency when applying the step response metric with k-medoids clustering for numbers of groups 1 through 20. This shows the distribution of the cost on each contingency for this grouping method for groupings into various numbers of groups. The right-most column displays the scaled closed-loop $\mathcal{H}_{\infty}$ norm when applying the nominal controller to each contingency. There are several contingencies on which the nominal controller achieves a scaled $\mathcal{H}_{\infty}$ norm of under 0.5. There are also several contingencies on which the nominal controller achieves a scaled $\mathcal{H}_{\infty}$ norm of over 1.0, which is worse than the performance achieved by one controller designed to handle all contingencies. For low numbers of groups, the scaled $\mathcal{H}_{\infty}$ norms achieved by the step response and k-medoids grouping method does not decrease consistently across contingencies. Instead, some contingencies see a large decrease in scaled $\mathcal{H}_{\infty}$ norm while the performance on other contingencies improves only slightly.

Figure 5 shows the change in cost on each contingency as the number of groups increases when using the step response metric and k-medoids clustering algorithm. The cost of the nominal controller is also shown, but cutoff for those contingencies where the nominal controller’s scaled $\mathcal{H}_{\infty}$ norm is above 1.1 since, as seen in Figure 4, the nominal controller’s scaled $\mathcal{H}_{\infty}$ norm goes as high as 4.5 for one contingency on this system. As expected, the cost on any particular contingency is generally non-increasing in the number of groups. Across most contingencies, there is a large improvement in performance between 5 and 8 groups, and improvements in performance become more incremental as the number of groups rises after that. Since additional groups means additional computation time to design the extra controllers, this information could be used by an operator to refine the number of groups to use in the future. The scaled $\mathcal{H}_{\infty}$ norms of some lines such as 3, 4, 25, 27, and 34 drops to near-optimal after 2 groups, suggesting these lines are split into smaller groups by the clustering algorithm at low numbers of groups. Note that while the scaled $\mathcal{H}_{\infty}$ norm of the nominal controller on some contingencies is high enough that it is not displayed in this figure, on a few contingencies such as lines 8, 9, 10, 11, 13, 18, and 19, the nominal controller achieves a level of performance that exceeds that of the step response and k-medoids grouping method at 11 groups. This suggests one direction to improve this grouping algorithm would be to use the nominal controller itself as the basis for one group, since the nominal controller is pre-computed.

For illustration of the groupings generated by this grouping method, we examine two groupings in closer detail: the grouping generated using the step response metric, k-medoids clustering, and a choice of 7 groups (depicted in Figure 6); and the grouping generated using the frequency response metric, divisive k-medoids clustering, and a choice of 7 groups (depicted in Figure 7). These two groupings have similar mean scaled $\mathcal{H}_{\infty}$ norm cost.

While the two groupings are generated using both different metrics and different clustering algorithms, there are several similarities between the groupings. Both have a group consisting of only lines 1 and 2. Group 7 of the step response & k-medoids grouping consists of lines 30, 39, 41, 42, 43, and 44, and group 4 of the frequency response & divisive k-medoids grouping is almost the same, only differing through the addition of line 6. Both groupings also have a “catch-all” group: group 4 in the step response & k-medoids grouping, and group 2 in the frequency response & divisive k-medoids grouping. Finally, both groupings isolate lines 4 and 25 into groups of size 1.

Compared to the nominal controller, both of these groupings improve significantly upon the highest $\mathcal{H}_{\infty}$ (not scaled) norms: the highest $\mathcal{H}_{\infty}$ norm seen using the nominal controller is $0.202$, when applied to line 25, while under the step response & k-medoids grouping it is $0.152$, when applied to line 37, and under the frequency response & divisive k-medoids grouping it is $0.152$, when applied to line 3. The five highest $\mathcal{H}_{\infty}$ norms seen using the nominal controller are 0.202, 0.190, 0.181, 0.179, and 0.177 on lines 25, 4, 3, 34, and 27 respectively. These same contingencies see the $\mathcal{H}_{\infty}$ norms of 0.151, 0.149, 0.146, 0.149, and 0.149 respectively using the step response & k-medoids grouping, and $\mathcal{H}_{\infty}$ norms of 0.151, 0.149, 0.152, 0.149, and 0.149 respectively using the frequency response & divisive k-medoids grouping. Under both groupings, the $\mathcal{H}_{\infty}$ norms on these lines is reduced significantly.

Some aspects of the groupings are intuitive. For example, lines 1 and 2 are in the same group in both groupings, which is unsurprising since the two lines are connected by a bus with no other lines, loads, or generators. Additionally, many groups in both groupings consist of lines that are close together in the power system. In fact, only group 4 in the step response & k-medoids grouping, and groups 2 and 4 in the frequency response & divisive k-medoids are disconnected. It is these disconnected groups where the utility of this automated grouping method is demonstrated. It may not be obvious to a human operator that line 35 might be put with lines 7, 8, 9, etc., as done in group 4 of the step response & k-medoids grouping, instead of with lines 27, 28, and 34. We also note that several lines are isolated into groups of size one by each grouping: the step response & k-medoids grouping isolates lines 3, 4, and 25; and the frequency response & divisive k-medoids grouping isolates lines 4 and 25. While line 3 is not isolated in the frequency response & divisive k-medoids grouping, it is the line on which this grouping performs the worst. This suggests that the removal of any of lines 3, 4, or 25 causes unique changes to the power system dynamics. This type of analysis could be useful to a power system operator for identifying contingencies which may need further consideration.

Table I summarizes the average time required to evaluate each metric once an Intel Xeon Skylake 6230 at 2.1 GHz. Running the clustering algorithm may require all $M(M-1)/2$ pairwise distances. For the IEEE 39-bus system, this means computing all pairs of distances on a single thread using the frequency response metric takes 55.63 seconds, using the step response metric takes 28.95 seconds, and using the perturbation spectral norm metric takes 0.1039 seconds. This can be parallelized to bring the computation time for the distance matrix down by a constant factor.

Over all controllers synthesized for these IEEE 39-bus system experiments, the time to synthesize a single controller ranged between a minimum of 33.771 seconds and a maximum of 512.746 seconds, with a mean of 138.864 seconds. With these times, Algorithm 1 could be re-run—using appropriate parallelism for control design—every 15 minutes to account for changing load and generation levels, even with the worst case controller design times. Note that this is heavily dependent on the controller design algorithm used, and in these experiments we used methods readily available in Matlab. Other works such as [7] and [8] present methods for computationally efficient $\mathcal{H}_{\infty}$ controller design.

We also apply our grouping method to the larger IEEE 68-bus system. Of note, the perturbation spectral norm metric does not perform well on this system, even though it did on the 39-bus system. This 68-bus system, extracted from data presented in [32], consists of 16-machines, 68 buses, and 86 lines. For further description of this system, we refer to [33]. Of the 86 lines in this system, there are 18 lines whose removal disconnects the graph, and 4 more lines whose removal results in a lack of a power flow solution. We exclude these lines and consider only the remaining $M=64$ lines as contingencies.

We compute groupings using the step response, frequency response, and perturbation spectral norm metrics with the k-medoids clustering algorithm based on the results on the IEEE 39-bus system and compare the mean cost versus the number of groups in Figure 8. The mean scaled $\mathcal{H}_{\infty}$ norm approaches 0 (optimal) for all three methods faster relative to the number of contingencies than they did on the IEEE 39-bus system. However, while the perturbation spectral norm metric performed better than the other metrics on the IEEE 39-bus system, it significantly under-performs compared to the step and frequency response metrics on this system. Both the step response and frequency response metrics with k-medoids outperform the nominal controller on average over the contingencies with just 3 groups and perform close to optimally on average at 15 or more groups. This is a significant reduction from the 64 controllers that would need to be designed to handle each contingency individually.

Figure 9 shows the cost applying the step response and k-medoids grouping method to each contingency in the IEEE 68-bus system for various numbers of groups. As in the IEEE 39-bus system, the grouping method tends to create groups on which controllers do near-optimal, and a remaining large group whose performance slowly improves as the number of groups increases. Note that while the scaled $\mathcal{H}_{\infty}$ norm should be greater than or equal to 0, we see some slightly negative values due to controller design variance in systune.

The five highest $\mathcal{H}_{\infty}$ norms (not scaled) using the nominal controller on each contingency in this system are 0.262, 0.249, 0.148, 0.138, and 0.130. Under the grouping into 10 groups with the step response and k-medoids grouping algorithm, these $\mathcal{H}_{\infty}$ norms become 0.120, 0.123, 0.101, 0.102, and 0.109 respectively. The overall highest $\mathcal{H}_{\infty}$ norm seen under this grouping is 0.123. This is less than half of the worst-case $\mathcal{H}_{\infty}$ norm from continued use of the nominal controller.

Due to the larger state dimension when modeling this system, the time required to compute the distance between two contingencies increases significantly: the frequency response metric takes 0.653 seconds to evaluate compared to 0.0935 seconds on the IEEE 39-bus system; the step response metric takes 0.3522 seconds compared to 0.0487 seconds on the IEEE 39-bus system; and the perturbation spectral norm metric takes 0.000959 seconds compared to 0.000175 seconds on the IEEE 39-bus system. While the perturbation spectral norm is fast to compute, its performance does not match that of the other metrics tested on the IEEE 68-bus system.

In this section, we repeat the experiments of Section IV-A using $\mathcal{H}_{2}$ control instead of $\mathcal{H}_{\infty}$ control. To design the controllers, we use systune configured for $\mathcal{H}_{2}$ control. The rest of the control design configuration is the same as in Section IV-2. Other than the change in control design and performance measurement, our testing methodology is the same as in Section IV-A. Note that these changes do not affect the distance metrics or grouping.

Figure 10 summarizes the average scaled $\mathcal{H}_{2}$ norm over contingencies versus number of groups for each grouping method. The SR k-medoids, FR k-medoids, FR divisive k-medoids, FR k-centers, FR divisive k-centers, and PSN divisive k-medoids methods perform the best for numbers of groups under 9. These methods outperform the nominal controller at 4 or more groups. All methods perform roughly the same at 10 or more groups. The PSN metric, with both k-centers and divisive k-centers, performed erratically, not achieving a monotonic decrease in mean scaled $\mathcal{H}_{2}$ norm that would be desired.

Figure 11 plots the scaled closed-loop $\mathcal{H}_{2}$ norm for each contingency when applying the perturbation spectral norm metric with divisive k-medoids clustering for various numbers of groups. For two and three numbers of groups, the scaled $\mathcal{H}_{2}$ norms are above 1, indicating variance in the results of the control design method. The mean drops rapidly between from three to four groups, but the performance on many individual contingencies remains high until eight groups and above. Between eight and twelve groups, the groupings perform poorly on one contingency. Further analysis shows that this contingency corresponds to the removal of line 3 in all of these groupings. The removal of line 3 was found to cause unique changes to the power system dynamics in Section IV-A.