Common use of See Appendix Clause in Contracts

See Appendix. the vectors and matrices MG, AG, θG, and uG, as MG = diag(Mi), AG = diag(Ai), θG = col(θi), uG = col(ui), and δG = col(δi) where i ∈ VG. The vectors and matrices AI , θI , and uI are defined as AI = diag(Ai), θI = col(θi), uI = col(ui), and δG = col(δi) with i ∈ VI . In addition, let AL = diag(Ai), θL = col(θi) and δL = col(δi) where i ∈ VL. Finally, let P = col(Pi), θ = col(θG, θI, θL), and sin(x) := col(sin(xi)) for a given vector x. Then, it is easy to observe that the dynamics of the synchronous generators, the inverters, and the loads can be written compactly as: MGθ¨G + AGθ˙G = −BGΓsin(B⊤θ) + uG − δG (26a) IV. Case study We consider a (fairly) general heterogeneous microgrid which consists of synchronous generators, droop-controlled AI θ˙I ALθ˙L = −BI Γsin(B⊤θ) + uI = −BLΓsin(B⊤θ) + δL − δI (26b) (26c) inverters, and frequency dependent loads. We partition the buses, i.e. the nodes of G, into three sets, namely VG, VI , and VL, corresponding to the set of synchronous generators, inverters, and loads, respectively. The dynamics of each synchronous generator is governed by the so-called swing equation, and is given by: Miθ¨i = −Aiθ˙i + ui − Pi + δi, i ∈ VG, (22) Note that this is the same model as [8], see also [19]. By defining η = BT θ, ωG = θ˙G, ωI = θ˙I , ωL = θ˙L, and θ˙ = ω = col(ωG, ωI, ωL), the network dynamics (26), admits the following representation η˙ = BT ω (27a) MGω˙ G + AGωG = −BGΓsin(η) + uG + δG (27b) AIωI = −BI Γsin(η) + uI + δI (27c) where Pi = Σ Im(Yij)ViVj sin(θi − θj) (23) ALωL = −BLΓsin(η) + δL (27d) Now, let pG = MGωG, HG = 1 pT M −1pG, HI = 1 ωT ωI , {i,j}∈E HL = 1 wT wL, and He = −1T Γcos(η). Then, (27) can be is the active nodal injection at node i. Here, Mi > 0 is the moment of inertia, Ai > 0 is the damping constant, ui is the local controllable power generation, and δi is the local load at node i ∈ VG . The value of Yij ∈ C is equal to the admittance of the branch {i, j} ∈ E, and θi is the voltage angle at node i. Also, Vi is the voltage magnitude at node i, and is assumed to be constant. For the droop-controlled inverters, we consider the follow- ing first-order model