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if the flow rate is sufficiently high, making practically equal the concentration inside the cell and the tanks [13]. However, for control strategies that require a variable flow rate, this condition is not always ensured, so a model that differentiates between the concentration in the tank and the cell is necessary. Moreover, in order to obtain a realistic estimation of the terminal voltage, the effect of different over-potentials must be considered. The activation over-potential, which is calculated using the Butler-Volmer equation, can be simplified to the Tafel equation [14]. Nevertheless, this simplification can lead to significant errors [15] depending on the current magnitude. Hence, a simple expression with less error is presented to take into account the effect of this over-potential in the model. Consequently to the different aspects presented, this work proposes a control strategy capable to ensure a desired output voltage, regardless of the current imposed by the grid in the charging process, or by the load in the discharge ones. One of the novelties yields on the use of a nonlinear dynamic model that differentiates between species concentrations in different parts of the system, and presents a more exact expression for the output voltage. To guarantee the correct operation of the controller throughout the total operational range, the H∞ technique is used, taking into account all possible operating points of the linearized model. A comparison between this technique and a PID tuned by means of a Particle Swarm Optimizer (PSO) is presented, to highlight the advantages of the proposed method, with respect to a classic controller highly used in the industry. The work is organized as follows: Section II presents the dynamic nonlinear model with the expression of the output voltage. Starting from the model, the design of the H∞ controller is presented in Section III. A PSO method is explained in Section IV that allows to obtain a PID controller based in the desired criteria of performance. Section V analyze the main results of both H∞ and classic PID controllers, while Section VI presents a comparative evaluation. Finally, the fruitful conclusions appear summarized in Section VII. II. MODEL FORMULATION There exist different models in the literature that define the behaviour of a VRFB. Among all existing ones, the dynamic electrochemical model of Skyllas Kazacos [16], who was pioneer in the use of vanadium for RFB, is the ones most used, presenting a compromise in terms of reality fitting and simplicity. Based on its model, it is possible to express the evolution of each vanadium species (V2+,V3+,VO2+ and VO+2 ) inside the cell and tanks, and use them to define the variables of interest of the system, which are the output voltage (which only depends on the cell concentration) and the SOC (that depends on tanks concentration). A. Dynamic concentration model The evolution of the concentration of each vanadium species can be represented in the state-space as: x ̇ =Ax+Bx·q+bj (1) being x=[cc2, cc3, cc4, cc5, ct2, ct3, ct4, ct5]T the state vector of species concentration, where the sub-index expresses the vana- dium species (2 for V2+, 3 for V3+, 4 for VO2+ and 5 for VO+2 ) and the super-index indicates where is it located (c for the cell and t for the tank). The inputs of the system are the the current density j and the flow rate q. All system parameters are defined by matrices A, B ∈ R8×8 that are related with the diffusion and flow rate, respectively, and the vector b ∈ R8 that contains the system parameters of the charging/discharging current effect. All model arrays appear summarized in the Appendix section. B. SOC computation The SOC of a VRFB gives information about the percentage of energy stored in the system, and is directly related with the concentration of vanadium species inside the tanks. Consider- ing that the evolution of species concentration in the anolyte and catholyte can be different, there exist a distinction: x5 SOC− = x+x (2) 56 SOC+=x8. (3) where SOC− and SOC+ compute the SOC in the anolyte and catholyte reservoirs of the system, respectively. In practice, the real SOC of the system would correspond to the minimum of them, determining the maximum energy that can be stored. C. Output voltage computation As with other types of batteries or fuel cells, the VRFB voltage can be computed by means of the Nersnt equation (V nernst) and considering the different over-potentials (η) and the formal electrode potential (V θ ). In this way, the following expression can be formulated: V = V θ + V nernst + η (4) where experimentally it has been seen that the formal potential V θ has a value of 1.4V [17]. The V nernst is computed taking into account the concentration of vanadium species and the hydrogen protons (cH+ ) formed in the catholyte: 2 Vnernst = RT ·ln x4 ·cH+ x1 F x3 catholyte x2 anolyte (5) where R is the gas constant, F is the Faraday constant and T is the temperature of the cell/stack. The formation of protons in the catholyte can be expressed as follows: cH+ = cH+(0) + x4 (6) where cH+(0) represents the initial concentration of hydrogen protons, that exist due to the presence of sulphuric acid in the composition of the electrolytes. An important factor to consider in the computation of the cell/stack voltage is the effect of the different over-potentials. Like other similar batteries, there are different types of losses, x7 + x8PDF Image | H control of a redox flow battery overpotentials
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