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4.1. Simulation Results The initial condition of the VRB, is determined by an initial state of charge, SOC(0), and corresponding concentrations, given by c2,5(0) = c ̄SOC(0), c3,4(0) = c ̄(1 − SOC(0)), (45) where SOC(0) = 0.1 for the charging and SOC(0) = 0.9 for discharging (i.e., 10% and 90% initial respective state of charge) Note, equal concentrations for the tank and the cell implies a zeroed initial conversion factor, X(0) = 0, since the state of charge at the inlet is equal to the state of charge at the outlet. The initial state, x(0) = (x1(0), x2(0)), is then given by x1(0) = x2(0) = c2(0)c5(0). (46) c3 (0)c4 (0) The current input, I, is considered as a non-ideal charging current with 25% fluctuation and is implemented as a pseudo-random square wave I = (1 + ki)Is, (47) with nominal current, Is = 20A for charging and Is = −20A for discharging, with random variation, 135 ki ∈ −0.5 0.5, sampled every 600s. All other simulation parameters and constraints are as in Table 1. Stabilisability (via controllability tests, see, e.g., [12]) was confirmed for each pair of system vertices (Aζ,j,Bζ,j) for j = 1...,N. The controller gains are then obtained via (29), with Q(ρ) = Qs = diag{1, 1, 5 × 103}, R(ρ) = Rs = 1 × 104 designed to place significant importance on the 140 tracking performance, X → Xs, i.e., operational efficiency, whilst not neglecting the pump energy losses due to high flow rates. The existence of the transformation V such that the matrix Λ in (42) is Schur was verified, which ensures closed-loop system boundedness of the overall system using the tracking controller with integral action (25)–(39). We simulated the proposed control scheme (as shown in Figure 2) to achieve a conversion 145 factor setpoint of Xs = 0.14, whilst charging the VRB from 10% to 85% SOC. The controller was simulated with the feedback gains computed using a convex combination (36), (37) as shown in Figure 3. The VRB was then simulated for the discharging scenario of 85% to 10% SOC, as can be seen in Figure 4. As shown, the proposed control scheme is capable of charging and discharging the VRB under a fixed conversion factor when physically viable (i.e., considering SOC under applied 17PDF Image | Electrolyte Flow Rate Control Vanadium Redox Flow Batteries
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