# Double-Consensus Based Distributed Optimal Energy Management for Multiple Energy Hubs
## Appendix C Proof of Theorem 1  
\[M\ =\ \left[ \begin{array}{ccccc}R_{pp}&R_{pc}&\mathbf{0}&\mathbf{0}& \mathbf{0}&\mathbf{0}\\ R_{cp}&R_{cc}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\overline{R}&\mathbf{0}&\mathbf{0}&\mathbf{0}\\ A^{p}(I-R_{pp})&-A^{p}R_{pc}&\mathbf{0}&S_{pp}&S_{pc}&\mathbf{0}\\ -A^{pc}R_{cp}&A^{pc}(I-R_{cc})&-O^{pc}(I-\overline{R})&S_{cp}&S_{cc}&\mathbf{0} \\ O^{\hbar c}R_{cp}&-O^{hc}(I-R_{cc})&A^{hc}(I-\overline{R})&\mathbf{0}& \mathbf{0}&\overline{\mathcal{S}}\end{array} \right], \tag{15a}\] \[\Delta\ =\ \left[ \begin{array}{ccccc}\mathbf{0}&\mathbf{0}&\mathbf{0}&I& \mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&I&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{0}&-A^{p}&\mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{0}&-A^{pc}&O^{pc}\\ \mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&O^{\hbar c}&-A^{hc}\end{array}  \right],V\ =\ \left[ \begin{array}{ccccc}V_{1}^{T}\\ V_{1}^{T}\\ V_{1}^{T}\\ V_{1}^{T}\end{array} \right]^{T}\ =\ \left[ \begin{array}{ccccc}\mathbf{0}&\mathbf{0}& \mathbf{1}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{1}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{1}\\ v_{p}&\mathbf{0}&-\kappa^{pp}v_{p}&\kappa^{pc}v_{p}\\ v_{c}&\mathbf{0}&-\kappa^{pp}v_{c}&\kappa^{pc}v_{c}\\ \mathbf{0}&\overline{v}_{c}&\kappa^{cp}\overline{v}_{c}&-\kappa^{hh}\overline{v }_{c}\end{array} \right],\] (15b) \[W^{T}\left[ \begin{array}{c}W_{1}^{T}\\ W_{1}^{T}\\ W_{1}^{T}\\ W_{1}^{T}\end{array} \right]\ =\ \left[ \begin{array}{ccccc}\mathbf{1}^{T}A^{p}&\mathbf{1}^{T}A^{pc}&- \mathbf{1}^{T}O^{pc}&\mathbf{1}^{T}&\mathbf{1}^{T}&\mathbf{0}^{T}\\ \mathbf{0}^{T}&-\mathbf{1}^{T}O^{hc}&\mathbf{1}^{T}A^{hc}&\mathbf{0}^{T}& \mathbf{0}^{T}&\mathbf{1}^{T}\\ w_{p}^{T}&w_{c}^{T}&\mathbf{0}^{T}&\mathbf{0}^{T}&\mathbf{0}^{T}&\mathbf{0}^{T} \\ \mathbf{0}^{T}&\mathbf{0}^{T}&\overline{w}_{c}^{T}&\mathbf{0}^{T}&\mathbf{0}^{T} &\mathbf{0}^{T}\end{array} \right],Q\ =\ \left[ \begin{array}{ccccc}\mathbf{0}&\mathbf{0}&\mathbf{0}& \mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}\\ \mathbf{1}&\mathbf{0}&-\kappa^{pp}&\kappa^{pc}\\ \mathbf{0}&\mathbf{1}&\kappa^{cp}&-\kappa^{hh}\end{array} \right] \tag{15c}\]  
The eigenvalue perturbation approach is employed to analyze the convergence properties of the presented algorithm. Along with Equations (7)-(10), we get  
\[\Theta(k+1)\ =\ (M+\eta\Delta)\Theta(k) \tag{16}\]  
where \(\Theta\ =\ [\lambda_{\ell,p}^{T},\lambda_{\ell,\ell^{\prime}}^{T},\lambda_{\mathcal{S}, \ell^{\prime}}^{T},y_{\ell,p^{\prime}}^{T},y_{\ell^{\prime}}^{T},y_{\mathcal{S},\ell^{\prime}}^{T}]^{T}\). Due to the definition of \(R\), \(\overline{R}\), \(S\) and \(\overline{S}\), it is not difficult to verify that \(R\), \(\overline{R}\), \(S\) and \(\overline{S}\) are all primitive with the maximum eigenvalue \(1\). Based on Lemma 1, there exist four vectors \(w^{T}\), \(v\), \(\overline{w}^{T}\) and \(\overline{v}\), satisfying \(w^{T}R\ =\ w^{T}\), \(Sv\ =\ \overline{w}^{T}\overline{R}\ =\ \overline{w}^{T}\overline{\ }\overline{\ } \overline{\ }\overline{\ }\overline{\ }\overline{\ }\overline{\ }\overline{\ } \overline{\ }\overline{Since the object function is a convex function, it is not difficult to verify \(\kappa^{pc}\kappa^{cp}<\kappa^{hh}\kappa^{pp}\). According to \(Q\), we have the four eigenvalues of \(W^{T}\Delta V\) satisfying \(d\psi_{1}/d\eta\ =\ d\psi_{2}/d\eta\ =\ 0,\,d\psi_{3}/d\eta<0\) and \(d\psi_{4}/d\eta<0\), which means \(\psi_{1}\) and \(\psi_{2}\) do not change against \(\eta\), and \(\psi_{3}\) and \(\psi_{4}\) become smaller when \(\eta>0\). Thus, there exists an upper boundary \(\varsigma\) such that \(0<\eta<\varsigma\). Except for \(\psi_{1}\ =\ \psi_{2}\ =\ 1\), the remaining eigenvalues lie in the open unit disk.