Introduction
Modeling of Cascading Failures
However, due to the complexity of the energy system, there is no existing approach that fully reflects the sequential failure process. Specifically, we explicitly model the frequency control dynamics on a fast time scale as part of the cascading process.
Characterization of Cascading Failures
Our results reveal a deep connection between energy redistribution patterns and the distribution of different families of power network topology (sub)trees. In other words, non-disruptive failures cannot exceed the block decomposition limits of the transmission network.
Mitigation of Cascading Failures
We use ˆD(s,t),ˆijˆto denote the PTDF for the fictitious line(s,t) of the post-contingency subgrid of ˆG1, where susceptance is the effective susceptanceB(e)st of the subgrid G2. We first analyze the optimal solutions of the OIA problem (7.4) in the case where the post-contingency network G is a tree.
Power Flow and Cascading Failure Models
Summary of Notations
Throughout this thesis, we use uppercase boldface characters to denote matrices (eg, A) and lowercase boldface characters to denote vectors (eg, p). Depending on the context, we use Ai to denote the ith row vector or the ith column vector.
DC Power Flow
Without loss of generality, we assume that the graph is simple and assign an arbitrary orientation to the edges in E such that if(i, j) ∈ Ethen(j,i) The determinant of the matrix Li j obtained from L by deleting the i-th row and j-th column is given by. Now suppose that Ai j can be expressed in terms of weighted spanning trees by Cramer's rule. Note that the spanning forestsH andTE, T (ik,jjˆ)andT(ij,ˆ j k) are the spanning forests of the network before the contingency, not just the islands after the contingency. Recall that A−F is defined in terms of the inverse reduced Laplacian matrix L−F = C−FB−FC−FT for the postcontingent network. A subset F ⊆ E of edges is called an acut setof G if removing all edges in F breaks the connection of the graph. The block decomposition of graphG is unique, and there are efficient algorithms for finding all blocks of graphG that run in O(|N |+|E |)time and space on a single processor, or run in O(log|N |)time and O (| N |+|E |)in space on O(|N |+|E |) processors [85]. The line outage distribution factor (LODF) describes the impact of line outages on power flows in the network after the contingency. As in Theorem 3.2, each term of (3.2) also carries clear graphical meanings: (a) The numerator of (3.2) quantifies the impact of stopping ˆpropagation of legs through all possible trees connecting ˆetoe, counting the orientation. b) The denominator of (3.2) sums over all trees involving G that do not pass through ˆe =(i,ˆ jˆ), and each tree of this type specifies an alternative path through which power can flow if the line ˆe is interrupted. The converse part of the Simple Cycle Criterion asserts that Keeˆ , 0 “if” there is a simple cycle containing both and ˆe. To extend fault localizability to the case of an uncut F outcome, we use (3.6d) in Theorem 3.5 to express the GLODFKFin terms for the LODF and PTDF submatrices K−F F and DF F. Indeed, the following example shows that the converses of theorem 3.9 do not hold, i.e. KkF can be zero with a non-zero probability when several lines trip simultaneously. Such counterexamples can occur when the outcome set F is such that the block decomposition of the surviving network has more blocks than the original network (in example 3, they are 4 and 1, respectively). We provide the localization result for uncut faults based on the post-contingent network in the following theorem. In particular, the expression for ˆDF in terms of the pre-contingency network A has two components. The generalized LODF KeFeˆ is equal to the PTDF for line e with the pair of buses ˆi, jˆ of the post-contingency network G˜ = (N,E \ F), i.e. KeFeˆ = De,Fˆ. To build on the presentation of Theorem 7.3, we develop a construction of the post-contingent flow anomalies f ∈Rm. Let f := (feˆ,eˆ ∈ E)and ftie := (feˆ,eˆ ∈ Ftie) be the precontingency branch currents on the lines inside the island G and on the connecting lines, respectively. We therefore assume without loss of generality that the precontingency branch flows feˆ , 0 for all triggered lines ˆe ∈ F0 (otherwise remove ˆfromF0 and the surviving øG). The effect of branch currents on the island G can be modeled by additional injections at buses j(e):ˆ. Ebthe unique block decomposition of G.2 We say that a blockEk is on a simple path between bus jˆand a participating bus i ∈ N if there exists a simple path between bus ˆjan and busi withαi > 0 that contains a linee∈ Ek. Conversely, f˜e , fe applies to all lineseinEk “if” Ek is on a simple path between bus jˆand a participating bus. A non-bridge block is said to be a separate block if there is an unintersected vertex in this block that is a participating bus. The expressions for ˆDF (4.11a) and (4.11b) in terms of pre- and post-contingency network trivially coincide when F. When both F , ∅ and Ftie , ∅, the effect of their perturbations on post-contingency can branch flows are interpreted. in terms of the pre-contingency network A or the post-contingency network A−F, via the expressions for both KF and ˆDF. According to the expression (4.11a) for ˆDF, the post-contingency network integrates both effects: changes in post-contingency branch flows are the sum of the impact of injection adjustments under proportional control factor αk. In addition, we switch off three transmission lines to create more non-bridge blocks to better illustrate the diagonal block structure of the LODF matrix. In both the DC and AC cases, the global effect of a bridge failure is clearly visible in the first column (since only one line is a bridge) of the LODF matrices in figure. For non-bridge faults, the LODF matrix in the DC cabinet shows a clear diagonal block structure. Moreover, the LODF within a block for both cases are similar, indicating that the LODF calculated from the DC model can be a good approximation for the AC model. A case study on the IEEE 118-bus test network confirms the block diagonal structure predicted by our theory, even when the system is under full AC flow equations. It would be interesting to understand this structure and develop bounds for the distance between DC and AC predictions. Larger interface networks require special topological subnet structures to ensure fault localization. In this section, we evaluate the fault localization performance of the three interface networks studied in the previous section for the IEEE 118 bus network. In particular, we show in the next section that the optimal solutions to the OIA problem exhibit a local and progressive pattern. In fact, power redistribution can also be analyzed in a post-emergency network. Recall that in Chapter 3 we relate the LODF for the pre-emergency network and the PTDF for the post-emergence network. The following proposal shows how De,ˆijˆ, PTDF can be decomposed for lines and bus pair ˆi,jˆ in different subnets. Note that in this chapter we focus on the power grid interface network, where subnets are connected by two buses. 5.1, we compare LODF Keeˆfor the original network and the modified network with different interface networks. Note that the limit depends only on the susceptance of the transmission lines for the complete bipartite network and is therefore valid for each pair of the triggered line. We note that the within-subnetwork LODF distributions for the series, parallel, and full bipartite interface networks are very similar, all lower than the original network. The CCDF LODF for all pairs of controlled line and triggered line is shown in the figure. We observe that the network in which the subnets are connected to any of the three interface networks achieves higher fault localizability, similar to the DC model. L| −k+1 tie lines will be turned off, which is the number of balancing areas in the network. We provide analytical results that characterize the topological patterns of the optimal solutions to the OIA problem. Similarly to Corollary 7.5 in the case of a tree post-contingency network, the optimal injection adjustments are also located around the endpoints of the error for general networks. Bridge block decomposition and unified controller have recently emerged as two important tools for network reliability. However, to connect to the bridge block decomposition, we may need to disable some PUC link lines. We refer to such a network as a tree-partitioned network since the balancing zones are connected in a tree structure described by the block decomposition of its bridge. In practice, the balancing zones on which the UC operates are usually connected by multiple connecting lines in a network structure. Our method aims to select a subset of connecting linesT ⊂ L to switch off, so that the remaining network balancing areas are connected in a tree topology, i.e., the reduced graph GP(E \T) is a tree . We note that our strategy can preserve the N−1 security standard even though the balancing zones are connected in a tree structure. Given an initial non-critical failure B(1), if a region is not connected to B(1), then in equilibrium x∗(1) we have∗j(1)=0 for all j ∈ Nl. According to our scheme, failures that disconnect the system are treated exactly the same as failures that do not, provided they are non-critical. Furthermore, the impact of a system-disrupting failure is localized and appropriately mitigated in the corresponding areas as well. Specifically, tighter thresholds allow faster detection of critical failures so that the system experiences only a short period of instability. The iterative release process may follow predefined rules set by the system operator to prioritize different objectives. In practice, these two types of restriction removal can be combined and performed iteratively, according to predefined rules of the system operator. A warning is triggered at 10 seconds, at which point the controller can relieve the load. Second, we sample the collection of all subsets that make up the transmission lines of the IEEE 118-bus test network. Next, we illustrate the improvement of the proposed approach compared to AGC under different congestion levels. 6.8(a), which shows the CCDF of the number of generators whose operating points are adjusted in response to initial failures. Hines, “Dynamic Modeling of Cascade Fault in Power Systems,” IEEE Transactions on Power Systems, vol. Stochastic analysis of cascade failure dynamics in power networks,” IEEE Transactions on Power Systems , vol. Wang, “An improved OPA model and blackout risk assessment,” IEEE Transactions on Power Systems, vol. Equation (7.3) suggests that the system anomalies can be modeled equivalently by the DC current equations over the post-contingency network with injections αes,t+∆p. For each injection adjustment ∆p that satisfies the above constraints, the flow deviations ∆f after the contingency can then be calculated based on (7.3). A disadvantage of the OIA problem introduced so far is that it does not explicitly limit flows after contingencies to meet line capacity limits. To compensate for such imbalances, the optimal response at every other node depends on its predecessor nodes toward the failure endpoints. Assuming a connected post-contingent network and ce(fe)= Be−1fe2 as a cost function, the optimal injection adjustment p∗i at nodei , s,t for the optimal solution p∗from (7.4) satisfies the following properties: An implicit but important component of the above results is , that optimal injection adjustments are localized around the line failure, providing computational gains. We now compare the performance of OIA and OIA-LL with the more traditional OLS approach. Both OIA and OIA-LL achieve similar results in terms of relative injection adjustments and the proportion of nodes with injection adjustment, while OIA-LL requires nodes from a wider region to participate in the mitigation process. Enforcing line constraints in OIA-LL means that it is also outside the Pareto bounds of OIA, but control still remains local, at the expense of slightly higher control costs (note the scale of the y-axis). On the other hand, OLS achieves the smallest injection adjustments, but results in the largest amount of nodes and transmission lines affected after failures. The curve is generated by simulating with β > 0.3 for OIA so that the average number of overloaded lines remains below 1%. The key point here is that OLS is far from the Pareto frontier of OIA, thus emphasizing that OIA achieves a better trade-off between localization responses and the size of injection adjustments. Haarla, “Two-level probabilistic risk assessment of cascading outages,” IEEE Transactions on Power Systems, vol. Vittal, “A corrective switching algorithm for overload and voltage violation relief,” IEEE Transactions on Power Systems, vol. Terzija, “Two-step spectral clustering controlled islanding algorithm,” IEEE Transactions on Power Systems, vol.Laplacian Matrix
Cascading Failure Model
Failures in Power Systems: Non-cut Outages
Block Decomposition
Distribution Factors
Line Failure Localization: Non-cut Outages
Conclusion
Proofs
Failures in Power Sysgtems: Cut Set Outages
Islanding Model
Bridge Outage
Cut Set Outage
Case Studies
Conclusion
Proofs
Interface Networks and Failure Localization
Power Redistribution
Interface Networks and LODFs
Case Study
Conclusion
Failure Mitigation: Adaptive Network Response
The Bridge-block Decomposition and the Unified Controller
Proposed Control Strategy: Summary
Localizing Non-critical Failures
Controlling Critical Failures
An Illustrative Example
Case Studies
Conclusion
Failure Mitigation: Local Injection Response
Problem Formulation
Theoretical Analysis
Case Study
Conclusion
Proofs
![Figure 1.1: The sequence of events, indexed by the circled numbers, that lead to the Western US blackout in 1996 from [1, 2].](https://thumb-ap.123doks.com/thumbv2/123dok/10412202.0/13.918.288.640.112.348/figure-sequence-events-indexed-circled-numbers-western-blackout.webp)
