—This paper introduces key propositions and their corresponding proofs pivotal in advancing a matrix-based, fully distributed power system state estimator. I. I NTRODUCTION In modern power systems, real-time monitoring and control are essential to ensure reliable and efficient operation [1]. State estimation, a pivotal task in power system management, involves the continuous assessment of the system’s operating conditions using available measurement data [2]. Traditional centralized state estimation approaches, while effective, face significant challenges in scalability and robustness due to the growing complexity and size of power systems [3]. To address these challenges, fully distributed state estimation methodologies have emerged as a promising alternative. This paper focuses on the development of mathematical propositions and proofs essential for the advancement of a fully distributed power system state estimator. Unlike central- ized methods, distributed state estimation leverages the com- putational capabilities of multiple nodes, such as distributed estimators and local measurements from phasor measurement units (PMUs), which are dispersed throughout the power grid. This decentralized approach enhances scalability and improves the robustness and fault tolerance of the state estimation process. The core contributions of this paper are as follows: 1)Proposition and Proof of Positive Definite Symmetric Matrix: We establish that the gain matrix, commonly denoted as A[4], derived from the measurement matrix H, is a positive definite symmetric matrix. This property is fundamental for ensuring the convergence of the state estimation algorithm. 2)Convergence Conditions for Distributed Estimators: We derive conditions under which the Jacobi iterative update matrix, MJOR, achieves spectral radius less than one, ensuring the convergence of the distributed state estima- tion algorithm. Specifically, we show that the optimal relaxation factor, ω, lies within a specific range depending on the minimum eigenvalue of the iteration matrix MJ. 3)Optimization of the Relaxation Factor: We propose an op- timal value for the relaxation factor, ωopt, that minimizes the spectral radius of MJOR. This optimization signifi- cantly enhances the convergence speed of the distributed estimation process. The authors are with the Department of Electrical Engineering, Indian Institute of Technology Kanpur, Kanpur, Uttar Pradesh, India 208016. (email: sehrawat@iitk.ac.in; saikatc@iitk.ac.in;ketan@iitk.ac.in)4)Case Studies and Practical Implications: Through illus- trative examples, we demonstrate the practical application of our theoretical results. We show how the placement of PMUs affects the reducibility of the gain matrix A, and subsequently, the tuning of the relaxation factor for optimal performance in both reducible and irreducible matrix scenarios. The propositions and proofs presented in section II lay the groundwork for future research and development in the area of distributed power system state estimation. II. P ROPOSITIONS AND PROOFS Proposition A1: Matrix Ais a positive definite symmetric matrix. Proof. Given A=HTWH⇒AT= (HTWH)T:= (H)TWT(HT)T:=HTWH ⇒AT=A⇒Ais a symmetric matrix. Ais a positive definite if vTAv>0for any non zero vector v. Let examine A:vTAv :=vTHTWHv := vTHT¯WT¯WHv :=yTy>0⇒Ais positive definite, where y=¯WHv ,¯Wis a diagonal matrix, and its diagonal elements are ¯Wii=√Wii. Proposition A2: LetAbe a symmetric positive definite matrix, BJbe a diagonal matrix, CJ=A−BJ,BJOR=BJ ω, CJOR =A−BJOR,MJ=−B−1 JCJ:=I−B−1 JA, and MJOR=−B−1 JORCJOR. Then, ρ(MJOR)<1if and only if 0< ω <2 1−λMJminwhere λMJminis the smallest eigenvalue of the MJ. Proof. By Theorem 1.56 of [5], it is sufficient to prove the matrix Q=BJOR +BT JOR−Ato be a positive definite for ρ(MJOR)<1.BJOR =BT JOR, since BJOR is a diagonal matrix. Hence, Q= 2BJOR−A. As per Proposition A3 ,Qis positive definite if and only if ¯Q= B−1/2 JQB−1/2 J is positive definite. Now we have ¯Q= 2B−1/2 JBJORB−1/2 J−B−1/2 JAB−1/2 J:= 2B−1/2 JBJ ωB−1/2 J− B1/2 J(I−MJ)B−1/2 J:= (2 ω−1)I+B1/2 JMJB−1/2 J . By Proposition A4 , eigenvalues of B1/2 JMJB−1/2 J are the same as of MJ. We can get eigenvalues of ¯Qbyλ¯Q=2 ω−1 + λMJ,∀ω > 0. If the smallest eigenvalue λ¯Qmin >0, then all the eigenvalues will be λ¯Q>0; this will make ¯Qpositive definite. Hence, we have the condition2 ω−1 +λMJmin> 0⇒0< ω <2 1−λMJmin. Proposition A3: LetBJis non singular diagonal matrix. ThenQis positive definite if and only if ¯Q=B−1/2 JQB−1/2 J is positive definite. Proof. Given, ¯Q=B−1/2 JQB−1/2 J is positive defi- nite, for any non zero vector y, we can have yT¯Qy:= 2 yTB−1/2 JQB−1/2 Jy>0⇒(BT−1/2 J y)TQ(B−1/2 Jy):= (B−1/2 Jy)TQ(B−1/2 Jy):=vTQv>0, Hence Qis positive definite. where v=B−1/2 Jy. Proposition A4: LetB1/2 Jis non singular diagonal matrix. Then eigenvalues of B1/2 JMJB−1/2 J are same as of MJ. Proof. For the eigenvalue λofB1/2 JMJB−1/2 J , we can write B1/2 JMJB−1/2 Jx=λx, where xis corresponding eigenvector. B1/2 JMJB−1/2 Jx=λx⇒MJB−1/2 Jx=λB−1/2 Jx⇒ MJy=λy, hence λis also the eigenvalue of MJ, where y=B−1/2 Jx. Proposition A5: LetAbe a symmetric positive definite matrix and Dis a positive diagonal matrix with diagonal elements of A. Define E=A−DandBJ=D+F, where Fis a diagonal matrix with diagonal element Fii=ηP j|Eij|. Define BJOR =BJ ω,CJOR =A−BJOR, and MJOR = −B−1 JORCJOR. Then, ρ(MJOR)<1,∀ω∈(0,2)andη≥1. Proof. We have MJ=−B−1 JCJ:= (D+F)−1(F−E). The diagonal element of MJisMJii=Fii Dii+Fiiand the non diagonal element is MJij=Eij Dii+Fii.MJii≥MJij because Fii≥P j|Eij|, implying that MJis a diagonally dominant matrix. Hence, by Theorem 1.8 of [6], Re(λMJ)> 0. Imaginary part of the eigenvalues is zero because by Proposition A4 , eigenvalues of MJandB1/2 JMJB−1/2 J are same. Furthermore, it can be observed that B1/2 JMJB−1/2 J:= B−1/2 JCJB−1/2 J is a symmetric matrix so that eigenvalues will be real only. Thus, we have λMJ>0⇒λMJmin>0. By Proposition A2 , the condition over ωforρ(MJOR)<1is0< ω <2 1−λMJmin. Considering λMJmin>0the upper bound of ωis,2 1−λMJmin≥2. Therefore ρ(MJOR)<1,∀ω∈(0,2). Proposition A6: The spectral radius ρ(MJOR)is mini- mum at optimal value of ω, which is given by ωopt= 2 2−(λMJmin+λMJmax), where λMJmin andλMJmax are the smallest and largest eigenvalues of MJ. Proof. We have MJOR=−B−1 JORCJOR:=−B−1 JOR(A− BJOR):=−B−1 JOR(BJ+CJ−BJOR):=−(BJ ω)−1(BJ+ CJ−BJ ω):= (1−ω)I−ωB−1 JCJ:= (1−ω)I+ωMJ. Hence, we can establish a relationship between the eigenvalues ofMJOR andMJ, given by λMJOR= 1−ω(1−λMJ). Variation in different eigenvalues |λMJORi|withωare shown in Fig.1. The trajectory of the spectral radius, which is the largest magnitude of eigenvalues of a matrix, is shown by red dotted lines. It can be observed that the ωoptlies on the intersection of lines −1 +ω(1−λMJmin)and1−ω(1− λMJmax), which gives ωopt=2 2−(λMJmin+λMJmax). Example Ex1: Optimal value of ωfor distributed estimators: For a system with gain matrix Awe can find the ωopt with the help of Proposition A6 . we can further improve the performance by finding the optimal value of ωfor each estimator based on reducibility or irreducibility of A. The irreducibility of matrix Ais contingent upon the placement strategy of the PMUs. By Theorem 1.6 of [6], a square matrix Ais irreducible if and only if its associated directed graph G(A)is strongly connected, where each non-zero element of matrix Acorresponds to a link in G(A). This property is particularly useful for identifying the group of distributed Fig. 1: Trajectory of the spectral radius ρ(MJOR) estimators that contribute to making a portion of the power system observable and obtaining the optimal weighting factor, denoted as ωopt, for that PMU set. Fig. 2: PMU placement Example; Star represent PMU bus Fig. 3: (a) matrix Afor 2(a) (b) matrix Afor 2(b) Case 1. Ais irreducible: For the PMU placement as shown in Fig.2(a), PMUs are placed to ensure system observability. The corresponding matrix Ais shown in Fig. 3(a), and its irreducibility can be verified by examining its associated directed graph G(A), as illustrated in Fig.14. Case 2. Ais reducible: In Fig.2(b), PMUs are optimally placed to ensure system observability. Ais reducible to irre- ducible sub-matrices. With the help of basic matrix operations, a reducible matrix can be written in terms of its irreducible 3 sub-matrices, as shown in Fig. 3(b). We can directly identify A=A11O O A 22 . We can use this property of the matrix Ato tune the relaxation factor ωfor each sub-matrix exclusively as per Proposition A6 , thereby obtaining the optimal convergence. A11pertains to buses 1, 2, 3, 4, and 5, with estimators posi- tioned at buses 1 and 5. Consequently, both of these estimators will share the same ωopt, which can be computed using MJOR derived from A11. Similarly, ωoptfor the estimator situated at bus 7 can be derived using MJOR obtained from A22. The matrices, shown in Fig 3, have directed graph G(A), as shown in Fig.4. In the case of a reducible matrix, the directed graph divides. Fig. 4: (a) G(A)of irreducible matrix A(b)G(A)of reducible matrix A Proposition A7. If the iteration matrix Mis convergent, then δ(m)→δasm→ ∞ , and δ=A−1β, represents the unique solution. Proof. Let an error vector e(m)be defined by e(m)=δ(m)− δ. Similarly, e(m+1)=δ(m+1)−δ, with the help of δ(m+1)= Mδ(m)+B−1β, can be expressed as Mδ(m)+B−1β−δ:= Mδ(m)+B−1Aδ−δ:=Mδm+B−1(B+C)δ−δ:= Mδ(m)+ (I+B−1C)δ−δ:=M(δ(m)−δ):=Me(m). We establish a relation between two consecutive error vectors, i.e, e(m+1)=Me(m), and this can be extended up to the initial error vector as e(m+1)=Me(m)=. . .=Mm+1e(0). Since Mis a convergent matrix ( M∞→ONull matrix), the error vector e(m+1)→0asm→ ∞ ⇒ δ(m)→δ.