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To overcome the above challenges, this paper proposes a Bayesian-inference framework that allows us to simultaneously urban areas by crew members can be costly, time-consuming, estimate the topology and the state of a three-phase, unbalanced and labor-intensive, which makes it impossible to rely on for power distribution system.

Specifically, by using the very limited online applications. This is performed through an techniques are, in general, more focused on transmission adaptive importance sampling procedure that greatly alleviates systems. Some examples include a robust Huber estimator the computational burden of the traditional Monte-Carlo MC - proposed by Mili et al.

The simulations conducted on the IEEE bus test topology errors; a traveling-wave-based technique initiated by system and an unbalanced bus system reveal the excellent performances of the proposed method. Korkali and Abur [3], [4] to locate the source of topology change caused by disturbances; a mixed-integer quadratic Index Terms—Topology estimation, power distribution system, programming considered by Caro et al. As for the simultaneous detection of multiple outages, an offline-trained model is suggested by Zhao et al.

On one hand, it serves as a pre- [7] and a neural-network-based approach was introduced by Krstulovic et al. Yet, considering that requisite for the reliable and efficient operations and plannings the distribution system is, typically, radially operated in an of modern power distribution systems where the deployment unbalanced three-phase structure with low observability, these techniques cannot be naturally extended to the distribution Y.

Xu, J. Valinejad and L. Mili are with the Bradley Department of Electri- systems [9]. To overcome the above difficulties, more and more re- M. Apart from the literature focusing solely rkali1 llnl. More specifically, Deka et al.

Zheng is with the College of Electrical Engineering, Sichuan University, [13] propose to utilize a spanning-tree-based graphical model Chengdu, China e-mail: zongsheng56 This work was supported, in part, by the U. National Science Foundation under Grant , by the Scientific Research Startup Fund for Introducing Similarly, the topology and the line parameter joint estimation Talents of Sichuan University under Grant , and by the United are explored by Park et al.

First, the distribution system model is identification [9]. To alleviate these difficulties, some researches which might be impractical for an online application to a adopt a DC model [6], [17], [18] or a linearized model [9], large-scale system [15], [18], [30], we propose to merge [11], [12], [19]. Some works simplify the three-phase structure an AIS scheme into the Bayesian-inference framework by ignoring the mutual coupling between phases with a single- [31], for the first time, in the distribution system topology phase distribution system model [11], [12], [15], [16], [20].

The weights inevitably sacrifice model accuracy. Second, the observability of the AIS further facilitate the recovery of the Bayesian in the distribution system is typically insufficient since the posteriors that quantify the confidence of the estimation. Although the deployment of measurement search algorithm, but also outperforms the traditional devices e. Subsequently, optimal sensor placement is effective tool for online applications.

Alternatively, The performance of our proposed method has been analyzed different measurement devices are also advocated in the liter- through simulations that are carried out on an IEEE test ature, such as smart meters [20], probing technique [27], ping feeder and a real utility-scale system.

These simulations reveal measurement [9], or even pseudomeasurement from forecast the excellent performance of the proposed method from the and historical data [9], [28]. Besides, using some publicly standpoint of simulation accuracy and computing efficiency. Third, the distribution system is, in general, radial. In Section III, the background Therefore, it comes as no surprise that the recent topology on importance sampling and adaptive importance sampling are estimation work is simultaneously conducted with outage introduced.

Section IV presents the proposed method. Case estimation [9]. Lastly, the size of the distribution system studies are presented in Section V, followed by the conclusions remains a bottleneck for most of the existing methods, which and future work in Section VI.

Then, we will also formulate Facing these challenges, this paper proposes a new adap- it into the Bayesian inference framework. Model Description the outages, and the power injections of a distribution sys- tem, while considering the latter as byproducts. Accordingly, we obtain makes it applicable to a realistic nonlinear distribution the three-phase voltage with respect to ground at Bus n as system model with a three-phase unbalanced structure.

First, following the notions in [35], let the distribution system as Ybus. Note that here e and x are 2 Switch Status in the Model: Now, let us consider the assumed to be mutually independent. The corresponding joint topology uncertainty brought by the switch statuses. Apart from the full Bayesian variables, Bsm,n , to capture the status of the switch sm,n. Till now, we have completed the presentation of the distribution system model considering its topology uncertainty.

This is especially true for our topology switch statuses, among others. In this paper, we are interested estimation problem, where a group of binomial distri- in solving the topology uncertainty problem raised by the butions representing the status of the switches and another switch statuses as done in [6], [9], and [12].

The detailed group of continuous random variables representing unknown parameter estimation problems for other model components system states are considered simultaneously. This motivates are beyond the scope of this paper.

In practice, although us to leverage the AIS method to recover all the Bayesian some parameters of the power distribution system model e. For Although IS is more widely known in the realm of rare- example, Yu et al.

In the recovery of the Bayesian posteriors using the weights of the same vein, parameter estimation of other model components IS. Then, we will further elaborate a more cost-effective AIS. Bayesian Inference method. Each sample has an associated importance weight given cross-entropy [43], [45] and antithetic variate [40] to further by improve its estimation accuracy and computational efficiency.

On the contrary, our work adopts the IS technique for a typical where w k represents the importance of the sample x k for inverse uncertainty quantification problem, i. A more detailed review of IS applications function q x. For the parameter estimation problem we have is provided in [31]. More specifically, the posterior functions [31]. Similarly, we can obtain the estimated parameters repeatedly evaluated for the recovery of the Bayesian posteri- either via the MAP estimator using 5 or via the mean esti- ors, 4.

Besides, as shown in Step 1, q x is obtained through mator using 10 in the last iteration as the final results.

This an initial guess, which might be quite inaccurate in practice, updating enables us to find a better proposal function that will diminishing the efficiency of the IS scheme. To overcome these allow us to draw more samples from the sample space with shortcomings, we introduce a more advanced AIS scheme high likelihood and, therefore, will increase the estimation next. XU et al. However, in practice, the power injections from the end- users are also unknown.

They represent the unknown states A. Bayesian Formulation of the Topology Estimation in the system that can also influence the model output in the distribution system. Therefore, the operation topology es- metered value of the active and reactive power in the end- timation should account for the outage estimation as well [9].

Currently, eQE are subvectors of the measurement-error vector, e, in one solution in the literature is to seek help from the ping 3. It is clear that a metered value has a smaller error measurement that can check the connectivity of the load to compared to a pseudomeasurement, i. However, to the best of our independent and identically distributed i. Gaussian error knowledge, the ping measurements have not yet been widely with a standard deviation of 0.

Note that in Therefore, we choose not to rely on the ping measurement practice, different types of loads e. For procedure as proposed next. Although we do not address this problem in this mains the Bayesian inference. But, we propose to formulate paper, these statistics should be carefully chosen for a better it as estimation performance.

This will be further s. Also, the forecast errors, eP F l u and eQF , of our method are assumed to be known. As is shown in 14a , the reformulation eliminates the pseu- To be more realistic, let us assume that we only have one domeasurement from the measurement model to avoid to use meter placed in the primary feeder to measure the power of biased data in the outage area.

The equality constraints from in the line, and a small portion e. In order not to waste users that have no meters rely on the forecast data seen the information in the forecast data, we introduce the lower as pseudomeasurements, which are much less accurate. Note bounds, Pb l and Qb l , and upper bounds, Pb u and Qb u , for the that the measured quantities can vary in practice. In our optimized states, Pb and Qb.

For the pseudo-metered buses at framework, although the quantities assumed to be metered the user-ends, these bounds are calculated from the forecast are power measurements, they may equally be voltage or data. Let us take the mean and the standard deviation for the current measurements.

Therefore, the second-stage estima- 6: Draw the proposed sample set, K tion will focus on the correction in the outage area. Till now, we have completed the presentation of the two- Use MAP or mean estimator to approximate pa- stage estimation procedure that enables the topology, outage, rameters via 5 or 10 ; and state joint estimation. Here, we choose the and states, our initial and major goals are the topology and PMC for its simplicity.

In this scheme, we only need to outage joint estimation. It should be noted that the state update the location parameters of the proposal functions for estimation comes as a byproduct of our estimator. Further- the next iteration [31]. These location parameters can be more, since it is well-known that the number of the possible easily obtained from the MAP or the mean estimator for topologies can be approximated as 2Ns , which requires an the recovered Bayesian posteriors at the current iteration.

Yet, we are still the bound. Since it is suggested to have heavy tails for the able to use a MAP or mean estimator to obtain the switch proposal function in [31], pbino can be extremely close to 0 or status by judging the success probability, pbino , for the binomial 1 for some scenarios. To maintain a relatively thicker tails, we distribution of a switch. Note that, if we get a value of can simply set the lower and upper bounds for the value of pbino very close to 0.

Accordingly, we need to state joint estimation for the unbalanced distribution system. For example, for the load modeled with some discrete distributions, the discrete Poisson distribution V.

Furthermore, for some loads that exhibit Using the proposed method, various case studies are con- more complex behaviors, a hybrid technique may be considered [46].

In this ducted on the modified IEEE bus test system and a mod- paper, we solely apply the Gaussian assumption for simplicity; the detailed load modeling issue and its associated forecasting techniques are outside the ified unbalanced bus system.

Their data can be accessed scope of this paper. Thus, we can focus on the switch-status-induced topology, outage, and! The parameters of the model components are assumed to be known based on our discussion in Remark 1. The general framework for implementing our Fig. Topology of the IEEE bus system. Then, various case studies are conducted to validate the performances of the proposed inducing multiple outage areas simultaneously.

Note that in method. For example, since Switch Bayesian 5 is open, Switch 6 is de-energized and, therefore, needs Samples in AIS Inference to be identified as the inestimable switch.

To make a fair comparison, we conduct the estimation times separately Update Model MATLAB to calculate the estimation accuracy2 for all the switch statuses using the AIS with different maximum iteration number, jmax. More specifically, structure that consists of 3-, 2-, and 1-phase, distribution lines approximately 12 out of 13 switches are accurately estimated associated with 91 loads with different types of connections.

Further, once we use the AIS, in Fig. Gaussian distribution with switch position although the estimation time increases. Gaussian assumption, whose standard devi- short time, rendering it applicable for online applications.

Here, we can get a Bayesian posterior unknown system states. Here, let us first use this test system to distribution for this system state.

However, we acknowledge demonstrate the efficiency of the AIS method in the topology estimation by using a sample size much smaller than 8, 2 Here, the estimation accuracy is defined as the ratio of the switch status while providing an accurate estimation result. To make the estimated correctly. Therefore, network.

The line connecting Buses 56 and 61 is three-phase we view the state estimation capability of our method as a while the left two are single-phase. Then, we further increase byproduct—not the main contribution. Again, we repeat the simulations conducted in Section V-A2 with these modified 0. The simulation results are summarized in Table III. Besides, 0. There- 0. Bayesian posterior for the active power of Load The MAP estimate blue circle is at To demonstrate that, let us test the 13 4 performances of our proposed method using different values 7 for the bounds set for pbino , which is briefly mentioned in Section IV-B.

The other settings remain unchanged. It is shown in Table II that the 10 12 upper bounds range from 0. However, Fig. Structure 1 of the modified topology of the IEEE bus system. This justifies setting such a bound. This is important in Bayesian 6 inference since for a nonlinear optimization problem such as the one formulated in 14 , setting a bound gives the algorithm a certain possibility to jump out from a local optimum to 9 better search for a global optimum.

Otherwise, the estimation 11 5 3 accuracy will be inevitably reduced to some degree. Structure 2 of the modified topology of the IEEE bus system. The latter has been investigated by Zhao et al. S with its kV feeder supplying power to approximately which only brings a marginal improvement in accuracy commercial and residential customers.

Here, we place while significantly increasing the computing time. For an online distribution system applica- number of iterations displayed in Fig. Indeed, it can be tion, it is obviously not practical to have an exhaustive search seen that after approximately 5 iterations, the estimation over more than 1 million possible topologies considering the accuracy tends to level off.

Note that the jump in the 11th computing time and the storage burden of the computing units. To make a fair comparison, we still estimation accuracy i. The estimation accuracy still has the potential for fur- The detailed simulation results and the settings are provided ther improvement if more end-users are equipped with in Table IV.

Convergence plot of the proposed method. In general, with a smaller noise, 4 Here, as one reviewer has pointed out that although the computing speed the estimation accuracy increases slightly. This is, indeed, an important issue though the standard deviation of the errors in the forecast that needs to be addressed. One is to first save the OpenDSS data into a. Loads for the ratio of the end-users that are equipped with the the loads, DSSObj. Lines for the network lines, etc. Then, we meters has a major impact on the estimation accuracy.

Here, we found that the computational speed for the second way is much faster general, for this large system, to obtain a good estimation than the first way. Therefore, to maintain a high computational efficiency result, the ratio should not be too low.

In this way, the communication challenge between the simulation and inference blocks can the AIS method and the IS method with the same amount be greatly overcome to guarantee its computing efficiency for the online of the total samples.

It is quite clear that the incorporating application. To illustrate, some of transmission systems do. The simulation the meters without using an advanced meter placement strat- results are shown in Table V. It is demonstrated that even egy, our algorithm already demonstrates quite good estimation when the nonlinearity of the system model is increased, we accuracy as shown in Group 3 in Table IV.

We also believe that still obtain reasonably good estimation results after only 6 if proper sensor placement strategies e. Also, since the ratio. This makes sense since the Bayesian framework has no meter placement strategy is not the focus of this paper, we linear assumption and, in principle, is applicable to nonlinear will not initiate further discussion on it. It Std. Further Discussions ability analysis may become even more challenging if 1 Discussions on Parameter Tuning: In general, parameter the uncertainties brought by the stochastic nature of tuning is almost an inevitable task for the statistical-inference- renewables are considered.

Indeed, in our recent research based algorithm. The same story applies to our proposed AIS on the observability analysis for a stochastic system, we algorithm as well. In our method, the tunable parameters realize that the traditional deterministic-technique-based mainly include: i the upper and lower bounds for pbino , ii observability analysis tool has some limitations in quan- the iteration number, jmax , and iii the sample size for each tifying the observability of a stochastic power system, iteration.

In this paper, we have conducted extensive case which exhibits more complicated phenomena, e. As shown in Table II, the upper recent work [55], [56].

This is also addressed in [57]. In general, with a smaller noise, 4 Here, as one reviewer has pointed out that although the computing speed the estimation accuracy increases slightly.

This is, indeed, an important issue though the standard deviation of the errors in the forecast that needs to be addressed. One is to first save the OpenDSS data into a. Loads for the ratio of the end-users that are equipped with the the loads, DSSObj. Lines for the network lines, etc. Then, we meters has a major impact on the estimation accuracy. Here, we found that the computational speed for the second way is much faster general, for this large system, to obtain a good estimation than the first way.

Therefore, to maintain a high computational efficiency result, the ratio should not be too low. In this way, the communication challenge between the simulation and inference blocks can the AIS method and the IS method with the same amount be greatly overcome to guarantee its computing efficiency for the online of the total samples.

It is quite clear that the incorporating application. To illustrate, some of transmission systems do. The simulation the meters without using an advanced meter placement strat- results are shown in Table V.

It is demonstrated that even egy, our algorithm already demonstrates quite good estimation when the nonlinearity of the system model is increased, we accuracy as shown in Group 3 in Table IV. We also believe that still obtain reasonably good estimation results after only 6 if proper sensor placement strategies e.

Also, since the ratio. This makes sense since the Bayesian framework has no meter placement strategy is not the focus of this paper, we linear assumption and, in principle, is applicable to nonlinear will not initiate further discussion on it. It Std. Further Discussions ability analysis may become even more challenging if 1 Discussions on Parameter Tuning: In general, parameter the uncertainties brought by the stochastic nature of tuning is almost an inevitable task for the statistical-inference- renewables are considered.

Indeed, in our recent research based algorithm. The same story applies to our proposed AIS on the observability analysis for a stochastic system, we algorithm as well. In our method, the tunable parameters realize that the traditional deterministic-technique-based mainly include: i the upper and lower bounds for pbino , ii observability analysis tool has some limitations in quan- the iteration number, jmax , and iii the sample size for each tifying the observability of a stochastic power system, iteration.

In this paper, we have conducted extensive case which exhibits more complicated phenomena, e. As shown in Table II, the upper recent work [55], [56]. This is also addressed in [57]. Also, as shown in Table IV, we only need deserves more careful consideration. Thus, it enabling fast computation for online applications. In general, the optimization-based method can di- [58]. Moreover, it is important to point out that although we rectly formulate the topology estimation problem into a mixed- do not address the data asynchronism issue in this paper, integer program that can be efficiently solved through some the asynchronous data do pollute the measurement quality packages or commercial software.

In general, it demonstrates in practice [59], which can, in turn, bias the estimator. Our a good estimation accuracy and a higher computing efficiency proposed Bayesian method is not robust to the aforementioned than a statistical-inference-based algorithm that relies on the outliers or bad data as its influence function has not yet sampling procedure.

However, the statistical-inference-based been designed to be bounded. Thus, robustifying our proposed algorithm also has its own benefits. Unlike the optimization Bayesian framework would be a worthwhile future effort. Krstulovic, V. Miranda, A. In this paper, we propose an adaptive-importance-sampling- Power Syst. Gandluru, S. Poudel, and A. Power Syst. Under the validity of the assumptions underlying the [10] D. Deka, M. Chertkov, and S. Deka, S. Backhaus, and M. Control Network Syst.

Cavraro, A. Bernstein, V. Kekatos, and Y. As we discussed earlier, the topology estimation problem in [13] D. Park, D. Yu, Y. Weng, and R. Thus, and topology joint estimation framework for time-varying system in the robustification will be addressed in a future work. Kekatos, G. Giannakis, and R. Smart Grid, vol. Sevlian and R. Baldwin, L. Mili, M.

Boisen, and R. Bhela, V. Kekatos, and S. Ardakanian et al. Sevlian, Y. Zhao, R. Rajagopal, A. Goldsmith, and H. Dominion Energy and Dr.

Hao Huang. Furthermore, careful —, Aug. Xygkis, G. Korres, and N. Liu et al. Mili, G. Steeno, F. Dobraca, and D. Costa, and K. Korkali, H. Lev-Ari, and A. Liao, Y. Weng, C. Tan, and R. Methods Appl. PMAPS , Singh, E. Manitsas, B. Pal, and G. Caro, A. Conejo, and A. Zamani and M. Zhao, J. Chen, and H. Power Symp. Smart NAPS , Grid, vol. Bugallo et al. Donmez and A. Peppanen, M. Reno, R. Broderick, and S. Network Sci. IEEE Trans. Augusto, M. Do Coutto Filho, J. Pires, L. Mili, and F. Gandhi and L.

Signal Process. Cavraro, E. Kaipio and E. GlobalSIP , SIAM, Xu, L. Mili, X. Chen, M. Korkali, and L. Bradley Department of Electrical and Computer [38] Y. Korkali, and X. Petra, C. Petra, Z.

Zhang, E. Constantinescu, and M. Anitescu, Falls Church, VA. Laboratory, Livermore, CA, and power engineer [40] Q. Chen and L. His research interests include power system uncertainty quantification, un- [41] W. Li and G. El-Laham, V. Elvira, and M. Leite da Silva, and V. Gonzalez-Fernandez, A. Leite da Silva, M. Da Rosa, and V. Leite da Silva and A. Greater Washington, D. He is also pursuing Mele, R. His current research interests include power [47] O.

Guillin, J. Marin, and C. Graphical Stat. Elvira, L. Martino, D. Luengo, and M. Cornuet, J. Marin, A. Mira, and C. Dugan and T. Meeting, Boston, MA, in He is currently a Research [51] T. Theodoro, P. Barbosa, M. Tomim, A. Xu et al. Rajicic and A. Mili, J. Valinejad, and Y. Zheng et al. Power Power Systems and Clean Energy. His research has focused on power system planning for enhanced resiliency and sustainability, risk management of complex systems to catastrophic failures, robust estimation and control, nonlinear dynamics, and bifurcation theory.

He is the co-founder and co-editor of the International Journal of Critical Infrastructure. She received her Ph.