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we can then parameterise the system (9) in an LPV state space representation1 dx(t)= 0 0 x(t)+ ρ1 u(t)+ 0 w(t), (11) dt 0 ρ ρ ρ 55 where u(t) := Q(t) and w(t) := I(t). If we consider an Euler discretisation with sampling period τ for a particular system described by some continuous function x ̇ = f (x, u) to be equal to x+ = x + τ f (x, u), then the following provides a discrete-time LPV model for the system in (11) x(k + 1) = 234 1 0 x(k) + τρ1 u(k) + 0 w(k), (12) 0 1+τρ2 τρ3 τρ4 A(ρ) B(ρ) E(ρ) The dynamic system model in (3) has been embedded in an LPV state space description, whereby the varying parameters ρ1, . . . , ρ4 are dependent on the states of the system. In our practical setup we can assume that the varying parameter is measured as described in the following. 2.3. Varying Parameter Update from Measurements We assume that the two half cells of the battery are fully balanced and provide details for reconstructing the varying parameters from measurements based on the following model: (in prac- tice, the SOC imbalances can be caused by the crossover of vanadium ions which can be monitored and corrected by remixing the positive and negative electrolyte) c2 + c3 = c4 + c5 = c ̄ and c2 = c5 , (13) c2 + c3 c5 + c4 where c ̄ is the total concentration. From (6) and (13) we have: c2t,2 =x1, c2c,2 (c ̄− ct,2)2 (c ̄− cc,2)2 which can be solved explicitly to give ct,5 = ct,2 = c ̄(√x1 − x1), c ̄(√x2 − x2) cc,5 =cc,2 = 1−x , =x2, c5 =c2,c3 =c4, (14) 1−x1 ct,3 = ct,4 = c ̄− ct,2, cc,3 =cc,4 =c ̄−cc,2, (15) 2 1Note that this parameterisation is not unique and has been chosen to address specific controller design prereq- uisites (see Section 3). 7PDF Image | Electrolyte Flow Rate Control Vanadium Redox Flow Batteries
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