H control of a redox flow battery overpotentials

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H control of a redox flow battery overpotentials ( h-control-redox-flow-battery-overpotentials )

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being the ohmic and activation losses the most important within the VRFB. On the one hand, the ohmic over-potential (ηohm) can be formulated as follow: ηohm =r·j·se, (7) where r is the cell/stack resistance and se is the electrode surface. On the other hand, the activation over-potential (ηact) expression can be formulated by means of the Butler-Volmer equation considering no mass-transfer effect (electrode surface concentrations do not differ from bulk values) [18]:  (1−α)·Fηact −α·Fηact j=j0eR·T−eR·T(8) being j0 the exchange current density at equilibrium and α the change transfer coefficient. The exchange current density depends on the species con- centrationinsidethecellandtherateconstantkθ,andcanbe computed as follows: j = 1 ·􏰎F ·kθ ·x1−α ·xα ·xα ·x1−α􏰏 (9) 0se 1234 As can be noticed, the term ηact only depends on the current density and the exchange current density. On the one hand, the current density is easy to measure, as it is imposed by the user. On the other hand, the exchange current density can be determined by the specie concentration inside the cell. Although it cannot be measured directly, by means of an observer its value can be estimated [19]. However, (8) defines a smooth implicit function ηact (j, j0, α) that cannot be numerically isolated. In this work, considering that the values of α are near to 0.5, it has been decided to define a 3 piecewise function using the linear and hyperbolic sine approximations [20]. Therefore, for low values of |ηact| the linear approximation will be used, and for large values of ηact the hyperbolic sine function. It has been found that the limit of ηact is near 30 mV depending on the value of α. Fig. 1 shows the corresponding 3 piecewise function for α = 0.51, where it is shown the relation between ηact and the current/exchange current density term (j /j0 ). The red trace defines the original Butler-Volmer expression, the blue trace defines the linear approximation considering a value of α = 0.5, and black and green traces define the hyperbolic sine approximation for charging and discharging currents, respectively. Considering that the linear approximation can be simplified taking into account the positive term, it is possible to formulate Fig. 1. 3 piecewise approximation for α=0.51. in a compact form the activation over-potential expression:    R · T 􏰅 j 􏰆  hsin−1 (1−α)F j0 if j/j0 < lb iflb ub  􏰅􏰆 ηact=R·Tln j 0 0.5·F j 0 α·F j0 where lb and ub are the lower and upper bounds, respec- tively, that define the extreme values of the linear approxi- mation. For the example shown in Fig 1 the lower and upper bounds are in -30 mV and 30 mV, which correspond to -2 and 2 for the term j/j0. Therefore, the expression of the over-potential (η) can be formulated considering (7) and (10) as: η = ηohm + ηact (11) By substituting (11) in (4) the global expression of the output voltage is achieved. III. H∞ CONTROL DESIGN This section presents the design of the H∞ controller that will be used to control the output voltage of a VRFB. The procedure follows a theory formulation that allow to specify the performance and stability conditions of the desired controller. This theory design is based on the use of the infinity norm || · ||∞ , that makes possible to define the desired specifications of the controller [21]. By means of different weighting func- tions is possible to specify the stability of the system and the desired performance. One of the advantages of the use of the || · ||∞ is that makes possible to formulate the design as an optimization problem, where the solution is the H∞ controller that best fulfills with the desired characteristics. Moreover, one of the most important advantages is that can deal with

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