Cooperative Distributed State Estimation: Local Observability Relaxed
Cooperative Distributed State Estimation: Local Observability Relaxed Le Xie, Dae-Hyun Choi, and Soummya Kar
Abstract—This paper explores a fully distributed state estimation algorithm in multi-area interconnected power systems. By iteratively exchanging information with neighboring control areas, all the balancing authorities (control areas) can achieve an unbiased estimate of the entire interconnection’s states. Compared with existing literature on distributed or hierarchical state estimation, the novelty of the proposed approach lies in that: (1) no assumption is needed for all the areas to be locally observable; (2) communication structure can be different than the physical topology of the power network; and (3) no central coordinator is needed for the state estimation in each local control area to achieve provable convergence of the entire system’s states with the centralized estimation. The performance of the proposed algorithm is illustrated in the IEEE 14-bus system.
I. I NTRODUCTION
T
HIS paper is motivated by the fact that electric power industry is undergoing profound changes as our society emphasizes the importance of a smarter grid in support of sustainable energy utilization. Technically, enabled by the advances in sensing, communication, and actuation, power system estimation and control are likely to involve many more fast information gathering and processing devices (e.g. Phasor Measurement Units) [1]. Institutionally, electricity industry deregulation has led to the creation of many regional transmission organizations (RTOs) within a large interconnected power system [2]. Both technical and institutional drivers suggest the need for more distributed estimation and control in power system operations [3]. The main subject of this paper is a fully distributed approach to multi-area state estimation in interconnected power systems. State estimation is one of the key modules in the energy management system (EMS) in power system control centers [2]. The state estimation module converts redundant meter readings and other available information into an estimate of the state of an interconnected electric power system [4]. While large power systems such as the eastern interconnection are usually operated by several independent RTOs, advanced applications such as wide area monitoring and control require the state of the entire system available to all the RTOs. [5]. This creates the need for a more decentralized approach to state estimation in large interconnections.
Revised manuscript submitted in February 2011. This work was supported in part by Texas Engineering Experiment Station. The authors greatly appreciate the financial help. L. Xie and D. Choi are with the Department of Electrical and Computer Engineering, Texas A&M University, College Station TX, USA
[email protected], [email protected] S. Kar is with the Department of Electrical Engineering, Princeton University, Princeton NJ, USA [email protected]
Several approaches to a more decentralized state estimation have been proposed. Early work in [6] and [7] a hierarchical method was proposed. The local state estimation results at the first level are coordinated at the second level. The starlike hierarchical methods are also discussed in recent literature such as [5] [8]. However, as number and the sampling rate of measurements increase, these hierarchical methods suffer from communication bottleneck and reliability problems inherent in systems with one coordination center. A parallel and distributed state estimation is envisioned in [9]. By exploring the naturally decoupled characteristic of the weighted least square (WLS) estimation problem, the centralized estimation problem is decomposed into each area’s local estimator with a coupling constraints optimization technique to ensure convergence of the boundary buses’ estimate. The numerical test cases show the speeding up of the computational time, as well as the acceptable accuracy of the distributed algorithm. However, local observability is always assumed in the aforementioned algorithms. In other words, all the local control areas must have enough redundancy to compute the local decoupled weighted least square estimation (excluding the boundary bus measurements). This assumption may not always holds true due to (1) the increasing vulnerability of measurements subject to temporal bad/malicious data, and (2) the emergence of smaller control areas such as micro-grids. In this paper the possibility of a fully distributed static state estimation algorithm without assuming local observability is exploited. A fully distributed state estimation for power system dynamics are discussed in [10]. However no analytical study was conducted for provable convergence of the distributed state estimation algorithms with the centralized estimation. Starting from one of the authors’ recent work in [11], we propose in this paper an iterative distributed state estimation scheme, under which the local control areas begin with their own estimates of the system, communicate their estimates with neighboring areas, and eventually all converge to the centralized state estimation result for the entire interconnection. In summary, the main contribution of this paper is twofold: •
We propose a distributed fast algorithm that does not require either local observability or a central coordinator. As long as the whole interconnection is observable and the communication graph is connected (not necessarily need to be the same topology with the physical networks), all the local areas’estimate of the entire system’s states will converge to the centralized estimation. Also for the proposed algorithm, there is no extensive matrix inversion computational burden.
2
•
We analytically show the convergence rate of the proposed distributed state estimation algorithm with the centralized WLS state estimation.
The rest of the paper is organized as follows. In Section II we introduce the preliminaries and problem formulation for a distributed state estimation. In Section III we present a fully distributed algorithm for multi-area state estimation, and we prove the convergence with the centralized state estimation. In Section IV an illustrative case study is performed in an IEEE 14-bus four-control-area system. We show that even some of the control areas are locally unobservable, by following the proposed distributed algorithm each control area will form an unbiased system-wide state estimation. The concluding remarks and future work are discussed in Section V.
The smallest eigenvalue λ1 (l) is always equal to zero, with √ 1/ N 1N being the corresponding normalized eigenvector. The multiplicity of the zero eigenvalue equals the number of connected components of the network; for a connected graph, λ2 (L) > 0. This second eigenvalue is the algebraic connectivity or the Fiedler value of the network; see [13], [14], [15] for detailed treatment of graphs and their spectral theory. Kronecker product: Since, we are dealing with vectors, most of the matrix manipulations will involve Kronecker products. For example, the Kronecker product of the N × N matrix L and IM will be an N M × N M matrix, denoted by L ⊗ IM . B. Multi-area State Estimation in Power Systems
II. P ROBLEM F ORMULATION A. Notation and Preliminaries For completeness, this subsection sets notation and presents preliminaries on algebraic graph theory and matrices to be used in the sequel. Preliminaries: We denote the k-dimensional Euclidean space by Rk . The k × k identity matrix is denoted by Ik , while 1k , 0k denote respectively the column vector of ones and zeros in Rk . The operator · applied to a vector denotes the standard Euclidean 2-norm, while applied to matrices denotes the induced 2-norm, which is equivalent to the matrix spectral radius for symmetric matrices. Throughout, we assume that all the random objects are defined on a common measurable space, (Ω, F). Also, all inequalities involving random variables are to be interpreted a.s. (almost surely.), see [12]. Spectral graph theory: We review elementary concepts from spectral graph theory. For an undirected graph G = (V, E), V = [1 · · · N ] is the set of nodes or vertices, |V | = N , and E is the set of edges, |E| = M , where | · | is the cardinality. The unordered pair (n, l) ∈ E if there exists an edge between nodes n and l. We only consider simple graphs, i.e., graphs devoid of self-loops and multiple edges. A graph is connected if there exists a path1 , between each pair of nodes. The neighborhood of node n is Ωn = {l ∈ V | (n, l) ∈ E}
(1)
Node n has degree dn = |Ωn | (number of edges with n as one end point.) The structure of the graph can be described by the symmetric N × N adjacency matrix, A = [Anl ], Anl = 1, if (n, l) ∈ E, Anl = 0, otherwise. Let the degree matrix be the diagonal matrix D = diag (d1 · · · dN ). The graph Laplacian matrix, L, is L=D−A (2) The Laplacian is a positive semidefinite matrix; hence, its eigenvalues can be ordered as 0 = λ1 (L) ≤ λ2 (L) ≤ · · · ≤ λN (L)
(3)
1 A path between nodes n and l of length m is a sequence (n = i0 , i1 , · · · , im = l) of vertices, such that, (ik , ik+1 ) ∈ E ∀ 0 ≤ k ≤ m − 1.
We assume there are a total of N regions in an interconnected power system, with each region n corresponds to a geographically non-overlapping control area. Each control area is responsible for the measurements in this region, and is capable of communicating with its immediate neighboring areas. In other words, the physical and communication network graph among the control areas is assumed to be identical in this paper. The measurement model of the multi-area state estimation can be formulated as: zn = hn (x) + en
(4)
where zn corresponds to measurements vector in control area n (real power, reactive power, voltage magnitudes of buses, etc). Vector x corresponds to the state variable of the entire interconnected power system (voltage magnitudes and voltage phase angles of all the nodes in the system). Based on the static power flow equations, if a measurement is located inside area n, then it is only a function of state variables xn corresponding to area n, which is a subset of x. Error vector en corresponds to the measurement error for the measurements in area n. The observation error vectors are assumed to be zero mean. In this paper, we analyze the linearized DC power flowbased state estimation problem as our first step. With the assumptions that (1) all the bus voltage magnitudes are close to 1.0 per unit; (2) all the transmission lines are lossless; and (3) the nodal voltage phase angle difference is small, the system state vector x becomes the vector of voltage phase angle at all buses: θ. Therefore the measurement model becomes: zn = Hn θ + en
(5)
The centralized state estimation problem is to find the optimal estimate of θ to minimize the weighted least square of measurement error: minimize J(θ) = rT R−1 r s.t. r = z − Hθ
(6) (7)
where R = cov([e1 · · · eN ]T ). The following assumption on R is imposed throughout: Assumption (E.0): The measurement noises are independent across the areas, i.e., R is a block-diagonal matrix of the form R = diag(R1 , · · · , RN ), where Rn is the covariance
3
of the noise vector en . Also, we assume that Rn is positive definite for each n. We denote the optimal solution of problem (6)-(7) as θˆlc . In the next section we will describe a distributed algorithm, through which each control area n’s estimate of system-wide state converges to the centralized estimate θˆlc . III. D ISTRIBUTED A LGORITHM In this section we introduce and analyze the distributed estimation algorithm CSE for static state estimation in power systems. The algorithm CSE may be viewed as a variant of a larger class of distributed state estimation algorithms introduced in [11]. The goal of this section is to show that for linear observation models, it is possible to design totally distributed iterative schemes, such that, each control area converges almost surely 2 (a.s.) to the centralized least squares estimator of the state θ. In particular, we show that the CSE algorithm leads to such convergence at each control area, under the assumption of global observability and connectivity of the inter-control area communication network. Before proceeding to the details, we observe that the CSE scheme introduced in this paper can be modified suitably (1) to yield more general centralized estimates at each control area, for example, the maximum likelihood estimate; (2) to deal with nonlinear observation models; (3) to operate in unpredictable environments with random inter-control area communication failure or transmission noise. We refer the reader to [11] for such general treatment of distributed estimation schemes, however, for clarity of presentation, we restrict ourselves to the linear observation model and the least squares estimator as the performance metric in this paper. Recall the observation model at the n-th control area: zn = Hn θ + en
(8)
We make the following assumption on global observability: Assumption (E.1) - Global Observability: The matrix G G=
N
HnT Hn
(9)
n=1
is full-rank. Remark 1 Assumptions (E.0)-(E.1) imply that the weighted Gramian N G= HnT Rn−1 Hn (10) n=1
is also invertible. Centralized least squares estimator: Consider the local transformations for each control area n: zn = Rn−1/2 zn , H n = Rn−1/2 Hn 2 Note
(11)
here, that all estimates (centralized or distributed) are random objects, being a functional of the random observations zn . Hence, any meaningful convergence of such estimate sequences needs to be interpreted in a probabilistic sense. In this paper, all convergence results are shown to hold in the almost sure (with probability one) sense, which means convergence for all sample paths or instantiations.
−1/2
where Rn is the (unique) square root of the positive definite matrix Rn−1 . Denote the matrices H and H by T
T
T
T
T H T = [H1T H2T · · · HN ], H = [H 1 H 2 · · · H N ]
(12)
Under (E.1), the centralized weighted least squares estimate of θ is given by, T −1 T −1 T θˆlc = H H H z=G H z (13) Clearly, the centralized computation of θˆlc requires the knowledge of all the observation matrices Hn , the covariances Rn , and observations zn at the center. We now present the distributed CSE algorithm, where by inter-control area data exchange, each control area (having access to its local observation matrix and observation only) in the network is able to construct the estimate θˆlc . Starting from some initial deterministic estimate of the states (the initial states may be random, we assume deterministic for notational simplicity), x ˆn (0) ∈ RM , each control area generates by a distributed iterative algorithm a sequence of estimates, {ˆ xn (i)}i≥0 . The state estimate x ˆn (i+1) at the n-th control area at time i+1 is a function of: its previous estimate; the communicated estimates at time i of its neighboring control areas; and the local observation zn . Algorithm CSE: Based on the current state x ˆn (i), the exchanged data {ˆ xl (i)}l∈Ωn , and the (transformed) observation zn , we update the estimate at the n-th control area by the following distributed iterative algorithm: x ˆn (i + 1) = x ˆn (i) − a b (ˆ xn (i) − x ˆl (i)) l∈Ω
T −H n
n zn − H n x ˆn (i)
(14)
In (14), a, b > 0 are constant step-sizes. Algorithm (14) is distributed because for control area n it involves only the data from the control areas in its neighborhood Ωn . The iterations in (14) can be written in compact form as: T x(i) − DH z − DH x ˆ(i) x ˆ(i + 1) = x ˆ(i) − a b(L ⊗ IM )ˆ (15) T
T ˆTN (i) is the vector of control area Here, x ˆ(i) = x ˆ1 (i) · · · x
T states (estimates) and z = zT1 · · · zTN . The Laplacian matrix L captures the topology of the control area network . We also define the matrices DH and DH as T T (16) DH = diag H 1 · · · H N T T T (17) DH = DH DH = diag H 1 H 1 · · · H N H N We refer to the recursive estimation algorithm in (15) as CSE. The following assumption on the connectivity of the intercontrol area communication network is assumed: Assumption (E.2) - Connectivity: The inter-control area communication network is connected, i.e., λ2 (L) > 0.
4
A. Convergence Analysis: CSE We start with the following technical lemma, which is key to subsequent convergence analysis of CSE.
Proof: The first assertion of the corollary follows from (24)-(25). The second assertion is due to the fact, that ra is minimized at a∗ .
Lemma 2 Let b > 0. Then, the symmetric matrix J = (b(L ⊗ IM ) + DH ) is positive definite iff (E.1)-(E.2) hold.
IV. C ASE S TUDIES
The proof is omitted and is a consequence of Lemma 4 in [11]. We now state the main convergence result: Theorem 3 Consider the CSE under (E.1)-(E.2). Let the constant a satisfy 2 0
Abstract—This paper explores a fully distributed state estimation algorithm in multi-area interconnected power systems. By iteratively exchanging information with neighboring control areas, all the balancing authorities (control areas) can achieve an unbiased estimate of the entire interconnection’s states. Compared with existing literature on distributed or hierarchical state estimation, the novelty of the proposed approach lies in that: (1) no assumption is needed for all the areas to be locally observable; (2) communication structure can be different than the physical topology of the power network; and (3) no central coordinator is needed for the state estimation in each local control area to achieve provable convergence of the entire system’s states with the centralized estimation. The performance of the proposed algorithm is illustrated in the IEEE 14-bus system.
I. I NTRODUCTION
T
HIS paper is motivated by the fact that electric power industry is undergoing profound changes as our society emphasizes the importance of a smarter grid in support of sustainable energy utilization. Technically, enabled by the advances in sensing, communication, and actuation, power system estimation and control are likely to involve many more fast information gathering and processing devices (e.g. Phasor Measurement Units) [1]. Institutionally, electricity industry deregulation has led to the creation of many regional transmission organizations (RTOs) within a large interconnected power system [2]. Both technical and institutional drivers suggest the need for more distributed estimation and control in power system operations [3]. The main subject of this paper is a fully distributed approach to multi-area state estimation in interconnected power systems. State estimation is one of the key modules in the energy management system (EMS) in power system control centers [2]. The state estimation module converts redundant meter readings and other available information into an estimate of the state of an interconnected electric power system [4]. While large power systems such as the eastern interconnection are usually operated by several independent RTOs, advanced applications such as wide area monitoring and control require the state of the entire system available to all the RTOs. [5]. This creates the need for a more decentralized approach to state estimation in large interconnections.
Revised manuscript submitted in February 2011. This work was supported in part by Texas Engineering Experiment Station. The authors greatly appreciate the financial help. L. Xie and D. Choi are with the Department of Electrical and Computer Engineering, Texas A&M University, College Station TX, USA
[email protected], [email protected] S. Kar is with the Department of Electrical Engineering, Princeton University, Princeton NJ, USA [email protected]
Several approaches to a more decentralized state estimation have been proposed. Early work in [6] and [7] a hierarchical method was proposed. The local state estimation results at the first level are coordinated at the second level. The starlike hierarchical methods are also discussed in recent literature such as [5] [8]. However, as number and the sampling rate of measurements increase, these hierarchical methods suffer from communication bottleneck and reliability problems inherent in systems with one coordination center. A parallel and distributed state estimation is envisioned in [9]. By exploring the naturally decoupled characteristic of the weighted least square (WLS) estimation problem, the centralized estimation problem is decomposed into each area’s local estimator with a coupling constraints optimization technique to ensure convergence of the boundary buses’ estimate. The numerical test cases show the speeding up of the computational time, as well as the acceptable accuracy of the distributed algorithm. However, local observability is always assumed in the aforementioned algorithms. In other words, all the local control areas must have enough redundancy to compute the local decoupled weighted least square estimation (excluding the boundary bus measurements). This assumption may not always holds true due to (1) the increasing vulnerability of measurements subject to temporal bad/malicious data, and (2) the emergence of smaller control areas such as micro-grids. In this paper the possibility of a fully distributed static state estimation algorithm without assuming local observability is exploited. A fully distributed state estimation for power system dynamics are discussed in [10]. However no analytical study was conducted for provable convergence of the distributed state estimation algorithms with the centralized estimation. Starting from one of the authors’ recent work in [11], we propose in this paper an iterative distributed state estimation scheme, under which the local control areas begin with their own estimates of the system, communicate their estimates with neighboring areas, and eventually all converge to the centralized state estimation result for the entire interconnection. In summary, the main contribution of this paper is twofold: •
We propose a distributed fast algorithm that does not require either local observability or a central coordinator. As long as the whole interconnection is observable and the communication graph is connected (not necessarily need to be the same topology with the physical networks), all the local areas’estimate of the entire system’s states will converge to the centralized estimation. Also for the proposed algorithm, there is no extensive matrix inversion computational burden.
2
•
We analytically show the convergence rate of the proposed distributed state estimation algorithm with the centralized WLS state estimation.
The rest of the paper is organized as follows. In Section II we introduce the preliminaries and problem formulation for a distributed state estimation. In Section III we present a fully distributed algorithm for multi-area state estimation, and we prove the convergence with the centralized state estimation. In Section IV an illustrative case study is performed in an IEEE 14-bus four-control-area system. We show that even some of the control areas are locally unobservable, by following the proposed distributed algorithm each control area will form an unbiased system-wide state estimation. The concluding remarks and future work are discussed in Section V.
The smallest eigenvalue λ1 (l) is always equal to zero, with √ 1/ N 1N being the corresponding normalized eigenvector. The multiplicity of the zero eigenvalue equals the number of connected components of the network; for a connected graph, λ2 (L) > 0. This second eigenvalue is the algebraic connectivity or the Fiedler value of the network; see [13], [14], [15] for detailed treatment of graphs and their spectral theory. Kronecker product: Since, we are dealing with vectors, most of the matrix manipulations will involve Kronecker products. For example, the Kronecker product of the N × N matrix L and IM will be an N M × N M matrix, denoted by L ⊗ IM . B. Multi-area State Estimation in Power Systems
II. P ROBLEM F ORMULATION A. Notation and Preliminaries For completeness, this subsection sets notation and presents preliminaries on algebraic graph theory and matrices to be used in the sequel. Preliminaries: We denote the k-dimensional Euclidean space by Rk . The k × k identity matrix is denoted by Ik , while 1k , 0k denote respectively the column vector of ones and zeros in Rk . The operator · applied to a vector denotes the standard Euclidean 2-norm, while applied to matrices denotes the induced 2-norm, which is equivalent to the matrix spectral radius for symmetric matrices. Throughout, we assume that all the random objects are defined on a common measurable space, (Ω, F). Also, all inequalities involving random variables are to be interpreted a.s. (almost surely.), see [12]. Spectral graph theory: We review elementary concepts from spectral graph theory. For an undirected graph G = (V, E), V = [1 · · · N ] is the set of nodes or vertices, |V | = N , and E is the set of edges, |E| = M , where | · | is the cardinality. The unordered pair (n, l) ∈ E if there exists an edge between nodes n and l. We only consider simple graphs, i.e., graphs devoid of self-loops and multiple edges. A graph is connected if there exists a path1 , between each pair of nodes. The neighborhood of node n is Ωn = {l ∈ V | (n, l) ∈ E}
(1)
Node n has degree dn = |Ωn | (number of edges with n as one end point.) The structure of the graph can be described by the symmetric N × N adjacency matrix, A = [Anl ], Anl = 1, if (n, l) ∈ E, Anl = 0, otherwise. Let the degree matrix be the diagonal matrix D = diag (d1 · · · dN ). The graph Laplacian matrix, L, is L=D−A (2) The Laplacian is a positive semidefinite matrix; hence, its eigenvalues can be ordered as 0 = λ1 (L) ≤ λ2 (L) ≤ · · · ≤ λN (L)
(3)
1 A path between nodes n and l of length m is a sequence (n = i0 , i1 , · · · , im = l) of vertices, such that, (ik , ik+1 ) ∈ E ∀ 0 ≤ k ≤ m − 1.
We assume there are a total of N regions in an interconnected power system, with each region n corresponds to a geographically non-overlapping control area. Each control area is responsible for the measurements in this region, and is capable of communicating with its immediate neighboring areas. In other words, the physical and communication network graph among the control areas is assumed to be identical in this paper. The measurement model of the multi-area state estimation can be formulated as: zn = hn (x) + en
(4)
where zn corresponds to measurements vector in control area n (real power, reactive power, voltage magnitudes of buses, etc). Vector x corresponds to the state variable of the entire interconnected power system (voltage magnitudes and voltage phase angles of all the nodes in the system). Based on the static power flow equations, if a measurement is located inside area n, then it is only a function of state variables xn corresponding to area n, which is a subset of x. Error vector en corresponds to the measurement error for the measurements in area n. The observation error vectors are assumed to be zero mean. In this paper, we analyze the linearized DC power flowbased state estimation problem as our first step. With the assumptions that (1) all the bus voltage magnitudes are close to 1.0 per unit; (2) all the transmission lines are lossless; and (3) the nodal voltage phase angle difference is small, the system state vector x becomes the vector of voltage phase angle at all buses: θ. Therefore the measurement model becomes: zn = Hn θ + en
(5)
The centralized state estimation problem is to find the optimal estimate of θ to minimize the weighted least square of measurement error: minimize J(θ) = rT R−1 r s.t. r = z − Hθ
(6) (7)
where R = cov([e1 · · · eN ]T ). The following assumption on R is imposed throughout: Assumption (E.0): The measurement noises are independent across the areas, i.e., R is a block-diagonal matrix of the form R = diag(R1 , · · · , RN ), where Rn is the covariance
3
of the noise vector en . Also, we assume that Rn is positive definite for each n. We denote the optimal solution of problem (6)-(7) as θˆlc . In the next section we will describe a distributed algorithm, through which each control area n’s estimate of system-wide state converges to the centralized estimate θˆlc . III. D ISTRIBUTED A LGORITHM In this section we introduce and analyze the distributed estimation algorithm CSE for static state estimation in power systems. The algorithm CSE may be viewed as a variant of a larger class of distributed state estimation algorithms introduced in [11]. The goal of this section is to show that for linear observation models, it is possible to design totally distributed iterative schemes, such that, each control area converges almost surely 2 (a.s.) to the centralized least squares estimator of the state θ. In particular, we show that the CSE algorithm leads to such convergence at each control area, under the assumption of global observability and connectivity of the inter-control area communication network. Before proceeding to the details, we observe that the CSE scheme introduced in this paper can be modified suitably (1) to yield more general centralized estimates at each control area, for example, the maximum likelihood estimate; (2) to deal with nonlinear observation models; (3) to operate in unpredictable environments with random inter-control area communication failure or transmission noise. We refer the reader to [11] for such general treatment of distributed estimation schemes, however, for clarity of presentation, we restrict ourselves to the linear observation model and the least squares estimator as the performance metric in this paper. Recall the observation model at the n-th control area: zn = Hn θ + en
(8)
We make the following assumption on global observability: Assumption (E.1) - Global Observability: The matrix G G=
N
HnT Hn
(9)
n=1
is full-rank. Remark 1 Assumptions (E.0)-(E.1) imply that the weighted Gramian N G= HnT Rn−1 Hn (10) n=1
is also invertible. Centralized least squares estimator: Consider the local transformations for each control area n: zn = Rn−1/2 zn , H n = Rn−1/2 Hn 2 Note
(11)
here, that all estimates (centralized or distributed) are random objects, being a functional of the random observations zn . Hence, any meaningful convergence of such estimate sequences needs to be interpreted in a probabilistic sense. In this paper, all convergence results are shown to hold in the almost sure (with probability one) sense, which means convergence for all sample paths or instantiations.
−1/2
where Rn is the (unique) square root of the positive definite matrix Rn−1 . Denote the matrices H and H by T
T
T
T
T H T = [H1T H2T · · · HN ], H = [H 1 H 2 · · · H N ]
(12)
Under (E.1), the centralized weighted least squares estimate of θ is given by, T −1 T −1 T θˆlc = H H H z=G H z (13) Clearly, the centralized computation of θˆlc requires the knowledge of all the observation matrices Hn , the covariances Rn , and observations zn at the center. We now present the distributed CSE algorithm, where by inter-control area data exchange, each control area (having access to its local observation matrix and observation only) in the network is able to construct the estimate θˆlc . Starting from some initial deterministic estimate of the states (the initial states may be random, we assume deterministic for notational simplicity), x ˆn (0) ∈ RM , each control area generates by a distributed iterative algorithm a sequence of estimates, {ˆ xn (i)}i≥0 . The state estimate x ˆn (i+1) at the n-th control area at time i+1 is a function of: its previous estimate; the communicated estimates at time i of its neighboring control areas; and the local observation zn . Algorithm CSE: Based on the current state x ˆn (i), the exchanged data {ˆ xl (i)}l∈Ωn , and the (transformed) observation zn , we update the estimate at the n-th control area by the following distributed iterative algorithm: x ˆn (i + 1) = x ˆn (i) − a b (ˆ xn (i) − x ˆl (i)) l∈Ω
T −H n
n zn − H n x ˆn (i)
(14)
In (14), a, b > 0 are constant step-sizes. Algorithm (14) is distributed because for control area n it involves only the data from the control areas in its neighborhood Ωn . The iterations in (14) can be written in compact form as: T x(i) − DH z − DH x ˆ(i) x ˆ(i + 1) = x ˆ(i) − a b(L ⊗ IM )ˆ (15) T
T ˆTN (i) is the vector of control area Here, x ˆ(i) = x ˆ1 (i) · · · x
T states (estimates) and z = zT1 · · · zTN . The Laplacian matrix L captures the topology of the control area network . We also define the matrices DH and DH as T T (16) DH = diag H 1 · · · H N T T T (17) DH = DH DH = diag H 1 H 1 · · · H N H N We refer to the recursive estimation algorithm in (15) as CSE. The following assumption on the connectivity of the intercontrol area communication network is assumed: Assumption (E.2) - Connectivity: The inter-control area communication network is connected, i.e., λ2 (L) > 0.
4
A. Convergence Analysis: CSE We start with the following technical lemma, which is key to subsequent convergence analysis of CSE.
Proof: The first assertion of the corollary follows from (24)-(25). The second assertion is due to the fact, that ra is minimized at a∗ .
Lemma 2 Let b > 0. Then, the symmetric matrix J = (b(L ⊗ IM ) + DH ) is positive definite iff (E.1)-(E.2) hold.
IV. C ASE S TUDIES
The proof is omitted and is a consequence of Lemma 4 in [11]. We now state the main convergence result: Theorem 3 Consider the CSE under (E.1)-(E.2). Let the constant a satisfy 2 0
