See Appendix. Note that Theorem 1 implies that the network (4) reaches an output agreement providing that there exist constant vectors (x¯, η¯) ∈ (Ωn)N ×(Ωe)M satisfying (6), (7), and thus
See Appendix. Changing Shift
See Appendix. Note that in case the controller at a node i ∈ I11 or i ∈ I21 does not have access to the desired output y∗, one can set ui to a constant, namely a nominal value, and incor- porate the node i in the subdynamics of (11) corresponding to the nodes indexed by I12 or I22, respectively. In Theorem 2, the control input u has been designed such that output agreement on a prescribed vector y∗ is achieved for the network. Observe that the “steady-state” control signal u¯ = ξ¯ is primarily determined by the initialization of the system/controller. Next, under the constraint of output agreement (6), we aim to minimize the following quadratic cost function 0 = (J11 − R11)∇H11(x¯11) min = 1 Σ u¯T Q u¯ (16) − G11(B11 ⊗ I)∇He(η¯) u¯ 2 i∈Ic + G11u¯11 + G11δ11 (13b) 0 = (J12 − R12)∇H12(x¯12) − G12(B12 ⊗ I)∇He(η¯) + G12δ12 (13c) 0 = (J21 − R21)∇H21(x¯21) − G21(B21 ⊗ I)∇He(η¯) (13d) + G21u¯21 + G21δ21 0 = (J22 − R22)∇H22(x¯22) − G22(B22 ⊗ I)∇He(η¯) + G22δ22 (13e) where Qi ∈ Rm × Rm is a positive definite matrix for each i, and Ic = I11 ∪ I21. Note that the optimization above determines the steady-state distribution of the control effort
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 = −BGΓ sin(B⊤θ) + uG − δG (26a)
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
See Appendix. All Training and development opportunities for governors (school based and central courses) are via a credit scheme or PAYG, outlined in the centralised CPD brochure and on-lined at the Shropshire Learning Gateway. Credit scheme or PAYG cost.
See Appendix. It appears that contrary to the requirements of section 35.3 of the Commission’s regulations, 18 C.F.R. § 35.3 (2014), SPP failed to file the
See Appendix. A" The classifications and wage rates for the effective period of this Agreement shall be those attached hereto in Appendix "A".
See Appendix. ARTICLEVIII
See Appendix. B" All eligible employees shall, subject to the conditions therein, have the benefit of the various plans for hourly rated employees outlined in Appendix "B".
