A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> Algorithm 6 1: Prediction updates Calculatex <s> Phasor measurement units (PMUs) have the advantage of providing direct measurements of power states. However, as the number of PMUs in a power system is limited, the traditional supervisory control and data acquisition (SCADA) system cannot be replaced by the PMU-based system overnight. Therefore, hybrid power state estimation taking advantage of both systems is important. As experiments show that sampling phase errors among PMUs are inevitable in practical deployment, this paper proposes a distributed power state estimation algorithm under PMU phase errors. The proposed distributed algorithm only involves local computations and limited information exchange between neighboring areas, thus alleviating the heavy communication burden compared to the centralized approach. Simulation results show that the performance of the proposed algorithm is very close to that of centralized optimal hybrid state estimates without sampling phase error. <s> BIB001 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> Algorithm 6 1: Prediction updates Calculatex <s> This paper reviews signal processing research for applications in the future electric power grid, commonly referred to as smart grid. Generally, it is expected that the grid of the future would differ from the current system by the increased integration of distributed generation, distributed storage, demand response, power electronics, and communications and sensing technologies. The consequence is that the physical structure of the system becomes significantly more distributed. The existing centralized control structure is not suitable any more to operate such a highly distributed system. Hence, in this paper, we overview distributed approaches, all based on consensus ${+}$ innovations, for three common energy management functions: state estimation, economic dispatch, and optimal power flow. We survey the pertinent literature and summarize our work. Simulation results illustrate tradeoffs and the performance of consensus ${+}$ innovations for these three applications. <s> BIB002 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> Algorithm 6 1: Prediction updates Calculatex <s> This paper studies distributed estimation of unstable dynamic random fields observed by a sparsely connected network of sensors. The field dynamics are globally detectable, but not necessarily locally detectable. We propose a consensus+innovations distributed estimator, termed Distributed Information Kalman Filter. We prove under what conditions this estimator is asymptotically unbiased with bounded mean-squared error, smaller than for other alternative distributed estimators. Monte Carlo simulations confirm our theoretical error asymptotic results. <s> BIB003 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> Algorithm 6 1: Prediction updates Calculatex <s> In this paper, we address the distributed filtering and prediction of time-varying random fields represented by linear time-invariant (LTI) dynamical systems. The field is observed by a sparsely connected network of agents/sensors collaborating among themselves. We develop a Kalman filter type consensus + innovations distributed linear estimator of the dynamic field termed as Consensus+Innovations Kalman Filter. We analyze the convergence properties of this distributed estimator. We prove that the mean-squared error of the estimator asymptotically converges if the degree of instability of the field dynamics is within a prespecified threshold defined as tracking capacity of the estimator. The tracking capacity is a function of the local observation models and the agent communication network. We design the optimal consensus and innovation gain matrices yielding distributed estimates with minimized mean-squared error. Through numerical evaluations, we show that the distributed estimator with optimal gains converges faster and with approximately 3dB better mean-squared error performance than previous distributed estimators. <s> BIB004 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> Algorithm 6 1: Prediction updates Calculatex <s> In contrast to the traditional centralized power system state estimation methods, this paper investigates the interconnected optimal filtering problem for distributed dynamic state estimation considering packet losses. Specifically, the power system incorporating microgrids is modeled as a state-space linear equation where sensors are deployed to obtain measurements. Basically, the sensing information is transmitted to the energy management system through a lossy communication network where measurements are lost. This can seriously deteriorate the system monitoring performance and even lose network stability. Second, as the system states are unavailable, so the estimation is essential to know the overall operating conditions of the electricity network. Availability of the system states provides designers with an accurate picture of the power network, so a suitable control strategy can be applied to avoid massive blackouts due to losing network stability. Particularly, the proposed estimator is based on the mean squared error between the actual state and its estimate. To obtain the distributed estimation, the optimal local and neighboring gains are computed to reach a consensus estimation after exchanging their information with the neighboring estimators. Then, the convergence of the developed algorithm is theoretically proved. Afterward, a distributed controller is designed based on the semidefinite programming approach. Simulation results demonstrate the accuracy of the developed approaches under the condition of missing measurements. <s> BIB005 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> Algorithm 6 1: Prediction updates Calculatex <s> In this brief, we study the detectability issues in the context of distributed state estimation problems for a class of locally undetectable sensor networks. First, we introduce a novel detectability condition, i.e., weightedly uniform detectability (WUD), which is a sufficient condition to prove that the error covariances of the consensus filtering are uniformly bounded even though the local sensor nodes are undetectable. Different from the existing detectability (or observability) conditions, our condition includes the interacting weights which could further optimize the lower detectability Gramian bound. Hence, a new weights selection method is derived in term of the criterion of WUD. This new rule of selecting weights provides a new framework for distributed state estimation. The advantages of this approach lead to a better performance in estimation without extra computational burden to the filtering process. Finally, an example shows the effectiveness of the proposed method. <s> BIB006 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> Algorithm 6 1: Prediction updates Calculatex <s> In this paper, a novel distributed dynamic state estimation (DSE) method for real-time monitoring of power systems is implemented. In modern large-scale power grids, the number of deployed meters and the frequency of collecting data have remarkably increased. Such a growth in the spatiotemporal size of collected data overwhelms the existing monitoring system with a centralized star structure. Streaming of data from all meters of the network to the central control center also increases communication latency. To overcome these challenges, the power system is partitioned into subsystems with local estimators, and the DSE process is distributed among local estimators. The distributed DSE is hosted at regional control centers and utilizes distributed extended Kalman filtering based on internodal transformation theory to estimate the dynamic states of power systems. The local estimators only require data within their own subsystem, and information is exchanged only between neighboring subsystems. The proposed distributed DSE is implemented on a 68-bus test system, and its accuracy is demonstrated. <s> BIB007 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> Algorithm 6 1: Prediction updates Calculatex <s> The purpose of this paper is to explore the applicability of linear time-invariant dynamical systems with polytopic uncertainty for modeling and control of islanded dc microgrids under plug-and-play (PnP) functionality of distributed generations (DGs). We develop a robust decentralized voltage control framework to ensure robust stability and reliable operation for islanded dc microgrids. The problem of voltage control of islanded dc microgrids with PnP operation of DGs is formulated as a convex optimization problem with structural constraints on some decision variables. The proposed control scheme offers several advantages including decentralized voltage control with no communication link, transient stability/performance, PnP capability, scalability of design, applicability to microgrids with general topology, and robustness to microgrid uncertainties. The effectiveness of the proposed control approach is evaluated through simulation studies carried out in MATLAB/SimPowerSystems Toolbox. <s> BIB008
Error calculations Calculate q i,k via (7); 6: Estimation updates Make the pseudo-observation z i,k via (7); 6) Algorithm 6. "Consensus+Innovation" Filtering: By resorting to the pseudo-observation, a distributed estimator of pseudostate G k x k , a linear transformation of target states, is embedded into the fusion process based on transformed measurements, which results in the following algorithm: where q i,k describes the weighted consensus error of predicted pseudostates. In above algorithm, some critical matrices are defined as where † stands for the MoorePenrose pseudoinverse. The formulas L ij k and K i,k can be found in BIB004 , and the corresponding pseudocode is provided in Algorithm 6. In comparison with the distributed information Kalman filter developed in BIB003 , the algorithm for the timeinvariant case can converge to a bound only requiring global detectability and network connectivity. Generally speaking, the calculation complexity and communication burden are two important indexes for above-discussed filtering algorithms. For Kalman-based algorithms, the calculation complexity at each instance is mainly dependent on the number of consensus steps, the inversion operation with O(n 3 ) for an n × n square matrix, and the multiplication operation with O(nsm) for an n × s matrix multiplying an s × m matrix. Meanwhile, the communication burden relies on the number of transmitted data. Assuming that x k ∈ R n , one needs to implement 3 times of the matrix inversion operation, 12 times of the matrix multiplication operation, and the transmission of Θ i n(n + 2)(L + 1) data for Algorithm 1, and 2Θ i + 2, 4Θ i + 6, and 2Θ i n for Algorithm 2. As such, Algorithm 1 possesses a low calculation complexity and Algorithm 2 has a low 1) Algorithm 1 needs to carry out the L-step consensus calculation at each instant, Algorithm 3 depends on the cross-covariance of filtering errors, and Algorithm 4 has to execute complex calculation to obtain the filter gains; 2) Algorithm 3 and Algorithm 4 need to exchange crosscovariances and global error covariances, respectively; and 3) Algorithm 5 is highly dependent on global error covariances, which reduces its scalability (see Table I for more details). Finally, recalling Algorithms 1 and 2, the selection of weight π ij shows a considerable impact on filtering performance, and thereby the optimized weight design receives the persistent investigation (see BIB006 ,  , and the references therein). In the past few years, Algorithm 1 with L = 1 is employed in BIB007 to implement the dynamic state estimation for real-time monitoring of power systems. It should be pointed out that distributed estimators using local data effectively overcome the challenges from both communication latency and communication burden existing in a central case. Under the variational inference framework, a simplified version of Algorithm 2 is developed in BIB001 for distributed hybrid power state estimation, where an auto-encoder technique is adopted to reduce the data dimensionality in mixed measurements subject to phase errors. Algorithm 3 is employed to estimate the operating condition of renewable microgrids with reliable channels in  and packet losses in BIB005 , and the conditions of corresponding convergence related on the Laplacian matrix are also analyzed simultaneously. For large-scale power networks, the merit of Algorithm 4 is clearly shown in BIB008 and the estimated state is utilized to compensate the information loss in distributed control due to the multirate nature of systems. Finally, according to a weighted least squares cost, a "consensus+innovation" based algorithm is proposed in BIB002 to estimate the vector of bus angles and converges almost surely to the centralized least squares estimator when the assumption of connectivity is satisfied. 