Voltage H∞ Control of a Vanadium Redox Flow Battery

Redox flow batteries are one of the most relevant emerging large-scale energy storage technologies. Developing control methods for them is an open research topic; optimizing their operation is the main objective to be achieved. In this paper, a strategy that is based on regulating the output voltage is proposed. The proposed architecture reduces the number of required sensors. A rigorous design methodology that is based on linear H∞ synthesis is introduced. Finally, some simulations are presented in order to analyse the performance of the proposed control system. The results show that the obtained controller guaranties robust stability and performance, thus allowing the battery to operate over a wide range of operating conditions. Attending to the design specifications, the controlled voltage follows the reference with great accuracy and it quickly rejects the effect of sudden current changes.


Introduction
It is a real fact that the use of renewable energy is growing more and more, due to its environmental benefits [1]. Mainly, these renewable energy sources (RES) are based on the use of solar, wind, or marine energy, between others. Because of their nature, RES are intermittent and the delivered power fluctuates in time, depending on non-controllable variables, such as the weather conditions. This results in the impossibility to satisfy the constant demand of energy. For that reason, trying to solve this problem, the use of energy storage systems (ESS) is having a great impact within the energy field [2][3][4][5][6]. There are different types of ESS according to the methodology used to convert and store the energy, which are mechanical, electrical, or electrochemical systems.
Inside the group of electrochemical ESS, redox flow batteries (RFB) are being widely studied and used today. One of their main advantages, is the possibility to store large quantities of energy, which is why they are a great option for large-scale energy applications [7,8]. In addition, it is worth highlighting other characteristics, such as the modularity between power and energy, efficiencies in a range from 75% to 85% [9], and being a clean storage system without producing emissions of any kind into the atmosphere. For that reason, taking into account all of the advantages that RFB present, a lot of research is being carried out to improve the design of these systems.
RFB consist on an electrochemical cell, where the energy is generated from a redox reaction that takes place inside the cell with different chemical components that are dissolved in liquids that are pumped from external tanks. Thereby, the energy is stored inside the tanks and, for that reason, uncertainty. The uncertainty will represent the variation of the linearised models when the equilibrium point changes. A large set of linear systems has been used to widen the operational range of the controller. As a result, a voltage controller that guarantees the closed-loop stability and performance in all equilibrium points will be obtained. Robustness and performance are considered in the design and these characteristics can be tuned by the designer. Moreover, the developed control system is robust in front of possible uncertainty in the current or sudden changes in its value, allowing for delivering the power that is desired by the user in this possible scenario.
The main novelty of this work is introducing a constructive and formal methodology to design a voltage controller for VRFB. Differently from previous works, the proposed approach guarantees closed-loop performance in all operation space using only the voltage measurement.
This work has been organized, as follows: Section 2 describes the used VRFB model; Section 3 contains an analysis of the system equilibrium points; Section 4 describes the controller specification and design; Section 5 contains different simulation results; and finally, Section 6 contains some conclusions and future works. Figure 1 shows the conventional scheme of a RFB; it is composed by two tanks containing the electrolytes dissolved in solutions. These electrolytes are pumped by two coordinated pumps through a cell stack. It is in this cell where the oxidation reduction reaction (REDOX) and the electric current is absorbed or generated. The electrolytes return to the tanks with a different concentration of species from the initial one. The species concentration in the tanks is directly related with the stored energy, i.e., the state of charge (SOC); while, the species concentration in the cell is directly related with the output voltage. In a VRFB, the electrolytes contain salts of vanadium dissolved in solutions of sulphuric acid. In this case, all of the species are vanadium in different oxidation states (+II, +III, +IV, +V). The anolyte is composed by V 2+ and V 3+ vanadium species, while the catholyte is composed by V 4+ and V 5+ . It is important to notice that the vanadium species V 4+ and V 5+ exist as oxides, which are VO 2+ and VO + 2 , respectively.

Operation of a VRFB
During the charging process, V 4+ oxidizes to V 5+ , releasing an electron, which is transferred from the cathode to the anode, reducing V 3+ to V 2+ . The total reaction can be written as The process occurs in the reverse direction during the discharge. Subsequently, it is possible to conclude that, depending on the amount of electrons that appear on the cell in a discharging or charging process, the concentration of each specie will vary according to the reaction that is presented in (1).
The voltage in the cell can be computed while using the Nerst equation. This allows to know which is the electrode potential that appears in the extremes of a cell while taking the different concentrations of vanadium species contained on it into account. The electrode potential E has the following expression [31]: where R and F are, respectively, the gas and Faraday constant, T is the temperature, E θ is the standard electrode potential, and c cell i is the concentration of vanadium specie i in the cell. Experimentally, it has been found that the maximum value of the cell voltage during the charging process varies between 1.6 and 1.7 V, and then drops to 1.1 V during discharging [31,32]. As this value is low, in order to increase the voltage, different cells are connected in series, defining what is usually known as a stack.
The RFB state of charge (SOC), which corresponds the amount of energy that is stored in the tanks, can be computed as a function of the vanadium species contained in the tank as: where c tank i is the concentration of vanadium specie i in the tank. The index i = 2 corresponds to V 2+ , i = 3 to V 3+ , i = 4 to V 4+ (which exists as VO 2+ ) and i = 5 (which exists as VO + 2 ). This same convention is used in the cell concentration.

VRFB Electrochemical Model
One of the most popular models for describing the electrochemical behaviour was introduced by Maria Skyllas-Kazacos [18]. It allows for describing the evolution of the vanadium species concentrations, both in the tanks and the cells and it has been experimentally validated with excellent results [18,25].
The equation describing the evolution of the vanadium species concentrations in the cell is given by: where x cell = c cell 2 , c cell 3 , c cell 4 , c cell , V cell is the volume of the cell, q cell is the electrolyte flow rate, i cell is the current in the cell (positive and negative currents define the charging and discharging process, respectively), z is the number of electrons that are involved in the reaction (z = 1), S is the surface and d the thickness of the membrane, and finally, k i is the diffusion coefficient for vanadium specie i. Equation (4) can be rewritten in a more compact form as: The equation describing the tank concentration considering the number of cells that compose the stack is: where V tank is the volume of each electrolyte tank and N is the number of cells that compose the stack. Equations (5) and (6) define a dynamical system that describes the electrochemical behaviour in the RFB. In this model, x cell and x tank are the state variables and and q cell and i cell are two exogenous variables. From a control perspective, q cell is usually assumed as the control action, while i cell is considered to be a measurable disturbance.
In this work, a 3 kW and 15 kWh VRFB system has been considered with a total vanadium concentration of 2 M for each couple of species. The other parameters that are needed to obtain the model are summarized in Tables 1 and 2.
Number of cells of the stack 19

Equilibrium Points Analysis
Equilibrium points play a very important role in dynamical systems [33,34]. These points correspond to the configurations (values of the state variables) where the dynamic system can remain stationary, due to this they are also called operation points. The equilibrium points are the solutions that make the derivatives of the differential equations of a system equal to zero.
To compute the equilibrium points for the dynamic system defined by (5) and (6), it is necessary to forceẋ cell = 0 andẋ tank = 0 and isolate the value of x cell and x tank . Because (5) and (6) depend on i cell and q cell , there is no solution, but i cell = 0, q cell = 0, x cell = 0, and x tank = 0, which is clearly not of interest.
In RFB, the volume of the tanks, V tank , is much bigger than the volume of the cell, V cell , and, consequently, the variation of x tank is tiny in comparison to the variation of x cell . Therefore, a natural assumption is to assume thatẋ tank ≈ 0, which implies that x tank is constant, i.e., SOC is constant. Under this assumption, the equilibrium points of (5) can be found. These equilibrium points will depend on the value of x tank , i.e., the SOC, i cell and q cell .
From the analysis of these points, it is possible to determine which is the optimal q cell , once the SOC and i cell have been fixed. Figures 2 and 3 show the values of V 2+ and V 3+ in the cell for a 10% of SOC, a different values of i cell , from 10 to 140 A, and q cell from 75 mL/s to 800 mL/s [26,35,36].   Becasue the total vanadium concentration is 2000 mol/m 3 , the vanadium concentration in the cell is limited to this value. For a 10% of SOC, the concentration in the anolyte tank corresponds to 200 mol/m 3 for V 2+ and 1800 mol/m 3 for V 3+ .
Looking at Figures 2 and 3, it can be observed that the surface stay as an horizontal plane at the majority of current and flow points. This behaviour is observed, especially at flow rates that are greater than 0.2 L/s. This means that concentration in the cell is almost the same in the tank for high flow rates. In the case of V 2+ is 200 mol/m 3 , and for V 3+ is 1800 mol/m 3 .
However, at a low enough flow rate, the surface is heading upwards for the more negative specie, which is V 2+ , and downwards for the more positively charged V 3+ . This behaviour can be better appreciated as the current increases, reaching the corresponding maximum or minimum at the lowest flow with the highest current.
When the RFB is charging, introducing a certain positive current, the more positively charged ion, which is V 3+ , is reduced by the corresponding electrons of the current, which convert them to their counterpart, the more negatively charged ions V 2+ .
When the flow is lower, more time is necessary in order to take the ions in the cell to get out of it. In terms of current, as it increases, a faster reduction happens. When both circumstances happen, the reduced ions begin to build up on the cell itself, which is why that upwards tilting begins to happen with vanadium specie V 2+ . In counterpart, the vanadium concentration of ions V 3+ decreases with a behavior that is contrary to that of V 2+ . Figures 4 and 5 show the values of V 2+ and V 3+ in the cell for an 80% of SOC, which implies having 1600 mol/m 3 of V 2+ specie, and 400 mol/m 3 V 2+ inside the tank. In this new situation, the surface gets to his maximum possible concentration for V 2+ , which is 2000 mol/m 3 , and the minimum, which is 0 mol/m 3 for V 3+ specie. This maximum value can be observed in Figure 4, which is represented in yellow. On the other case, the minimum concentration of 0 mol/m 3 is observed in dark blue in Figure 5.  With these studies, it is possible to characterize the stationary behavior of the different concentrations and, therefore, of the voltage in the cell. The behavior has been analyzed for a loading and unloading scenario, but it is possible to do it for other SOC values. The behavior evolves smoothly between the two examples presented.

Introduction
Differently from other types of batteries, like Li-on, RFB are active elements, i.e., it is necessary to feed the cell with reactants. Because of this, it is necessary to design a control system that guarantees the correct closed-loop behavior. The RFB has two inputs, the current, i cell , and the flow rate, q cell , as discussed in Section 2. From a control point of view, i cell is treated as measurable disturbance, while q cell is used as control action. A very relevant measurable variable is the output voltage, E. The voltage combined with the current determine the input/output power.
In this section, a voltage controller will be designed. The methodology that is used to design the controller is based on the H ∞ control theory, which allows for synthesizing controllers that guarantee performance specifications and, at the same time, achieve the stabilization of the system.
Linear H ∞ methods [37] are used to shape the frequency response of the most relevant closed-loop transfer functions in a control system. This shaping is formulated as an optimization problem in terms of the infinity norm: || · || ∞ . The procedure to design a controller is based on two steps; in the first one, the specifications are defined through the use of weighting functions that can be understood as bounds over the frequency response. Once the specifications are made, the optimal controller is obtained, i.e., the controller is the one that better fulfils the specifications in terms of the || · || ∞ . One of the main advantages of the H ∞ design method is that it permits dealing with model and parametric uncertainties. In this way, we can design controllers that preserve closed-loop stability and desired performance in spite of plant uncertainties. These characteristics can be applied in order to design the control system of the redox battery. In this case, with the distinct equilibrium points of the system, we can create a structure for the plant composed of a nominal model of the battery that is associated to a norm-bounded uncertainty. On this base, the H ∞ design is used to accomplish closed-loop stability and performance in a wide range of operating conditions.

Model Simplification
Before designing the controller, the plant model (5) will be simplified. To do so, the following assumptions can be made: • Vanadium concentrations are the same on both sides of the half cell. Therefore, it is possible to have the following correspondence: The tanks concentration can be expressed in terms of SOC and total vanadium concentration c v : Taking into account these equalities, it is possible to express the electrochemical model, reducing Equations (4) and (2) to: respectively.

System Linearization
The system that is defined by (7) and (8) can be easily transformed in the standard nonlinear state space form:ẋ and u = i cell , q cell T . This model can be linearized around its equilibrium points (x,ū), using the Taylor series, as follows: where h.o.t denotes high order terms, which will be very close to zero if states (x, u) are sufficiently close to its equilibrium points (x,ū). It is possible to rewrite the linearized system in the state space form, as follows: where ∆x = x −x, ∆u = u −ū, and ∆y = y −ȳ, and A, b, c, d are the following jacobian matrices: For the system that is defined by (7) and (8), the Jacobian matrices become: The system defined by (A, b, c, d) allows for analyzing the nonlinear system behavior around the equilibrium point.

Uncertainty Modeling
The values (A, b, c, d) defined by (9)-(12) depend on the value of different parameters, the value of the SOC, and the concrete equilibrium point. Figure 6 shows the frequency response of different linear plants obtained sweeping the value of SOC between 10% to 90%, which corresponds to almost a full charged or discharged VRFB and sweeping the current value from 15 to 120 A (for both charging and discharging modes). Similarly, Figure 7 shows the step response of these linear models. As it can be seen, there is an important variability in both gain and time constant. It is important to take this variability in the controller design into account. A model with uncertainty will be developed in order to design a controller that can operate while taking into account all possible range of operation of SOC, current, and flow rate.  Step response for linear plants obtained around different equilibrium points. Figure 8 shows the block scheme of a plant with additive uncertainty, in this scheme G n (s) is the nominal plant, W a u (s) is a weighting function that provides bounds on how the real plant might change from the nominal one, and ∆(s) is an uncertainty function, such that ∆(s) < 1.
The nominal plant has been selected while using the nominal parameters (Table 2) Once the nominal plant has been defined, the value of W a u (s) will be obtained. Hence, the following procedure will be followed:  An almost perfect fitting is obtained, as it can be seen. This weighting function is calculated while using the cover function from the MATLAB Robust Control Toolbox [38].  Figure 10 shows the proposed feedback configuration for the control of the redox battery. The configuration is composed of the nominal plant G n (s), the controller C(s), the uncertainty term ∆(s), and the weighting functions W a u (s) to shape the plant uncertainty. Finally, a new term W e (s) is added in order to define the system performance specification. With this configuration, it will be possible to determine the conditions for performance and stability. Firstly, the performance specification will be defined. In this way, the nominal sensitivity function will be used to specify the performance; this transfer function relates the error and reference signals:

Controller Design
The sensitivity function also relates the output disturbance with the error. To define bounds on the sensitivity function, the following weigthing function is usually used [37]: where M s takes a value between 1 and 2 and it is directly related to the nominal robustness, i.e., gain and phase margins, is a bound on the steady-state error, and w b is used to fix the closed-loop bandwidth, which is related to the system settling time.
The steady-state error tolerance has been selected to be less than 1% to guarantee a precise tracking of the reference. A value of M s equal to 2 has been selected; this value guarantees a distance of 0.5 to the critical point (−1,0), a gain margin that is greater than 2 and phase margin greater of 30 • . In terms of bandwidth, it has been selected a frequency of 10 −4 rad/s. Figure 11 shows the frequency response of W −1 e (s), in blue, which defines the desired bounds over the nominal sensitivity function, S n (s). Therefore, to accomplish the performance specifications, we have: This equation corresponds to the Nominal Performance (NP) condition, which implies that the closed-loop system without uncertainty, i.e., W a u (s) = 0, fulfils the specified performance. Secondly, the design must guarantee closed-loop stability despite the plant uncertainty. This is known as Robust Stability Condition (RS). Based on the control system structure that is shown in Figure 10, the RS condition is [37]: An H ∞ problem will be defined in order to design a controller that guarantees NP and RS conditions. Figure 12 shows the required augmented plant [37]. It can be observed the appearance of In order to obtain a controller that guarantees (14) and (15)m the following Mixed sensitivity problem can be formulated: In case the obtained controller makes the H ∞ norm less than one, then the NP and RS conditions will be guaranteed . From Figure 12 and Equation (16), we seek a stabilizing controller C(s), with input Y(s) and output U(s), that minimizes the H ∞ norm from R(s) to E(s) U(s) T . With all previous definitions and configurations for performance and plant uncertainty, the optimization procedure yields a fifth order controller, C(s). This order is exactly the same as that of the augmented plant. Note that, the augmented plant, P(s) (see Figure 12), is composed by the nominal plant, G n (s), which is order 2 , the sensitivity function weighting function, W e (s), which is of order 1 and the uncertainty weighting function, W a u (s), which is order 2.
The resulting controller makes W e (s)S n (s) W a u (s)C(s)S n (s) = 0.86, which is less than 1, so both the NP and RS are obtained. Figure 11 shows the frequency responses of the obtained nominal sensitivity function, S n (s), and the inverse of W e (s). The NP is fulfilled, as the frequency response of S n (s) is below the performance specifications. Figure 13 shows a graphical representation of the the RS condition. Because the curve for C(s)S n (s) is below the frequency response of 1/W a u (s), the robust stability condition is met.
The resulting order of the controllers designed using the H ∞ method is usually high, which is a drawback for practical implementations. The order of the previous controller is five and it is desirable to obtain a lower order controller.
For that reason, a reduction of the order of the controller has been performed. The reduction is based on obtaining a balanced realization, which is a representation in the state space. The states are ordered by the so-called singular Hankel values, which give an idea of the observability and controllability of the different states by assigning them a certain value. The states that are below a certain tolerance of singular value are less significant to the system, having a negligible contribution to the I/O response. Eliminating these states, it is possible to reduce the order of the controller without altering their behaviour against the control of the system [37]. Using this approach, a reduced 2nd order controller,C(s), has been obtained. Figure 14 compares both the original and the reduced-order controller. As it can be seen, both controllers shows approximately the same frequency response. Only at high frequencies appears the difference. This is not a problem, since the controlled system works at low frequencies where it has been seen is the part where the uncertainty has more weight. Both NP and RS are preserved; consequently, the reduced order controller offers similar performance while facilitating the implementation and requiring less computational resources, as shown in Figures 11 and 13. Non-NP or RS guarantee that all possible plants fulfil the performance specifications; the proposed design procedure only guarantees performance over the nominal plant. Figure 15 shows the frequency response of the closed-loop sensitivity function of different plants, using the reduced order controller. As all drawn sensitivity frequency responses are below the specification curve, it can be assumed that, despite the uncertainty, all closed-loop systems fulfil the performance. This is called the Robust Performance (RP) condition. A more formal condition for RP, is [37]: |W e (s)S n (s)| + |W a u (s)C(s)S n (s)| ∞ < 1 This condition can not be expressed as a regular H ∞ problem, but it can be painted differentiating the NP and RS conditions, as it is shown in Figure 16. Figure 16 shows the formal robust performance analysis for the reduced order controller,C(s), as it can be seen, the black line corresponding to the bound defined in (17) is below 1 in almost all of the frequency range, but there is a small range of frequencies, around 10 −3 rad/s, where the bound is over 1; this means that the RP condition is not fulfilled in this frequency range. For that reason, it can not be stated that the closed-loop system, as shown in Figure 10, fulfills RP, but, in practice, this controller can be considered to be good enough.

Robust Performance
Frequency (rad/s) Figure 15. Aproximated robust performance analysis for the reduced order controller,C(s).

Integral Controller Design
It is well-known that integral action guarantees null steady-state error for constant or piecewise constant references. Unfortunately, H ∞ control does not generate controllers with integral action in general. In order to eliminate the error against step-type reference changes, an integral part can be applied to the designed controller.
To force the appearance of an integral action in the controller, while preserving the nice properties of the H ∞ framework, the weighting function for the nominal sensitivity function has been modified as: where must take the lowest possible value and K the highest one. Under these conditions, the dc gain of W e (s), will be K/ . Consequently, an effect that is similar to placing a zero at s = 0 in W e (s) will be achieved. This will force |W e (jω)| to be very high in the very low frequency range while preserving the frequency response of the original weighting function in the other frequencies.
Ideally, should be practically zero, but computationally it is not possible to do so. In this work, after some numerical testing, K = 10 3 and = 10 −10 have been selected. Figure 17 shows a comparison between the bounds imposed by the original, W e (s), and the new, W e (s), weighting functions. As it can be seen, they are almost the same, but in the very low frequency range where W e (s) imposes harder constrains.
Once the weighting function has been redefined, the optimal controller is obtained again. Previous manipulations will force that the controller has a pole close to the origin. This pole is latter manually replaced by an integrator. This will generate an order 7 controller, C i (s). Note that, as the weighting function has increased its order, the controller also increased the order. This controller makes that W e (s)S i,n (s) W a u (s)C i (s)S i,n (s) = 0.8605 which is small than 1, this implies that this controller guarantees both NP and RS. As in the previous section, the order of this controller can be reduced up to order two, C i (s). This controller can be rewritten as the Proportional, Integral, and Derivative (PID) one: with k p = −3.7053·10 −5 , k i = −6.7840 · 10 −8 , k d = 0.0679 and τ = 10 6 . The PID Controller will be easier to implement on most platforms.  Figure 18 shows an informal proof of the RP condition. As it can be seen, all selected plants fulfil the specifications, which point up that the RP might be fulfilled. Figure 19 contains a formal analysis of the RP condition. The RP condition is not formally fulfilled, but it holds in almost all frequencies, which can be considered good enough from a practical point of view, as it can be observed. Finally, Figure 20 contains the step response of a number of plants, as it can be seen in all cases that a smooth response is obtained with null steady-state error.

Robust Performance
Frequency (rad/s) Figure 18. Approximated robust performance check for the closed-loop system step response with C i (s) and W e (s).

Materials and Methods
The controller design procedure is one important result of this work. Figure 21 shows flowchart which summarizes the methodology introduced in previous subsections. Once the detailed non-linear model is defined, the design starts with the equilibrium point analysis. From this analysis, a set of representative operation points is selected, in each of these points the non-linear system is linearised; consequently, a set of linear models which cover the operation space is obtained. From this set of plants, a nominal plant, G n (s), with additive uncertainty, W a u (s), is defined. At this point, the original non-linear system has been transformed into a linear system with uncertainty; consequently, linear techniques can be applied. At this point, the closed-loop specifications are defined through a weighting function, W e (s), which define bounds over the closed-loop tracking error and the frequency response. Combining G n (s),W a u (s) and W e (s), an augmented plant is built. Based on the augmented plant, the controller is synthesized using numerical methods. Once the controller is obtained, NP, RS and RP needs to be checked. If the specifications are meet, then a simulation against the original non-linear model is performed in order to guarantee the closed-loop behaviour, even for big movements (small movements around operation pints are constructively guaranteed). In case a wider operational range is needed, a larger set of linearised plant should be defined. The method should be iterated until all requirements are satisfactorily accomplished.

Results and Discussion
This section will show the results that were obtained using the controller proposed in Section 4.6. The controller will be validated using the complete non-linear model presented in Section 2. This model has been previously experimentally validated in the literature and it precisely reproduces the reality [18,25]. Figure 22 shows the simulation block scheme. As the stack is composed by 19 cells, according to the Nerst equation, during the charging process, the voltage values might vary in a range between 26 to 31 V approximately (see Figure 23). Therefore, to analyse the behaviour of the controlled system, the voltage will be asked to follow a given voltage profile that is composed by different constant voltages. At the same time, it is important to study whether the controller is capable of operating under different current values. In this manner, uncertainty and disturbances are introduced in the system in terms of current. Hence, the current has been changed from 20 to 100 A in order to cover the full range. Figure 24 shows the profile of the current changes.  Figure 23 shows how the voltage tracks the voltage reference when using the low-order controller with integral part. As it can be seen, the stack voltage follows the reference with great precision. The steady-state error is 0 and the disturbances due to the current variations are successfully rejected. The transient response is satisfactory, even when voltage and current change simultaneously. This confirms that the robust designed controller behaves correctly against current and voltage variations. Figure 25 shows the evolution of the flow rate generated by the proposed controller. As the flow rate is the control action, it is important to keep it bounded. With the defined operating conditions, it can be observed that the flow rate remains under 0.12 mL/s. Larger peaks in flow rate can be seen when voltage and current change simultaneously; however, the maximum level of the control action (4.5 × 10 −5 m −3 /s) is not reached. It is important to note that neither current nor SOC are measured, only the voltage that is generated is used as the feedback signal. However, it is important to observe the evolution of the SOC. Figure 26 depicts the SOC against time for the given voltage and current profiles. Because current is always positive, the SOC is always increasing; this reflects the charging process of the VRFB. Finally, it can be seen that, with the designed robust controller, the VRFB works as expected, preserving the stability and performance. Furthermore, the specifications are kept under the operating conditions that are defined by current, flow rate, and SOC, which demonstrates that the controller successfully deals with uncertainty and disturbances.

Conclusions and Future Work
In this paper, a methodology to design robust feedback controllers to regulate the RFB voltage has been presented. The proposed methodology is based on linear H ∞ theory and it offers excellent performance in a wide range of operation points. The methodology offers a constructive and rigorous framework to design controllers for this type of system. Different from other controllers that were previously proposed in the literature, this controller only uses the output voltage and it is very simple. Furthermore, it is demonstrated that the low order controller has the structure of a PID compensator. The low order controller has the advantage of easier implementation that requires lower computational burden.
Currently, work is being done on the incorporation of current/SOC information in order to work at constant power or obtain a specific impedance. Another important line of work corresponds to the search for controllers that allow guaranteeing a long lifetime, by minimizing the system degradation, offering the desired performance at all operating points.