H∞ Control of Wrim Driven Flywheel Storage System to Ride-Through Grid Voltage Dips

Flywheel Energy Storage Systems (FESSs) are commonly integrated with wind farms to help them to provide many grid services, including frequency control, voltage control, and power smoothing. Although such systems are not concerned by the severe grid code requirements, their ability to ride-through voltage dips is important to ensure better stability of the power grid. In this paper, the authors propose a robust H∞ current controller for a Wound Rotor Induction Machine (WRIM) based FESS during grid voltage dips. The proposed H∞ controller decreases the negative effects of voltage dips in the WRIM system, such as the rotor over-currents and the active power oscillations. On the other hand, it also guarantees the robustness in the presence of parameter perturbation. The proposed controller is designed using a modified mixed-sensitivity H∞ technique to take into consideration grid disturbances and parameter perturbation. Finally, simulations are made in MATLAB/Simulink using SimPowerSystems to verify the effectiveness of the H∞ controller under grid voltage dips with WRIM parameter perturbation. The simulation results show that the proposed H∞ controller can improve the stability of the WRIM based FESS subject to grid voltage dips and guarantee the robustness with parameter perturbation.


Introduction
With regard to the progressive exhaustion of fossil energy, new alternative energies are considered key tools for reducing carbon dioxide emissions. Among these renewable energies, wind is the fast-growing source of electricity in the world. In 2018, nearly 51.3 GW of new capacity were installed worldwide, bringing the global total to around 591 GW [1].
Nevertheless, with rising penetration of wind energy and their variable nature, grid instability is a major concern. As these wind energy sources may lead to frequency and voltage instability, especially for a weak part of the grid [2]. In addition, the disconnection of wind generators from the power grid during grid faults may amplify grid instability issues [3].
In recent years, Flywheel Energy Storage System (FESS) has gained increasing attention because it can support wind energy sources to provide many grid services, including frequency control and power smoothing [4], and [5]. Considering their technical characteristics, fast response is the key to the FESS for providing the above-mentioned services and also for improving the ride-through capability of wind farms [6].
Among the existing electrical machines used in FESSs, Permanent Magnet Synchronous Machines (PMSM) and Induction Machines (IM) are perhaps the most commonly employed machines [7], [8], [9], and [10]. The Wound-Rotor Induction Machine (WRIM) is also of particular interest in the field of FESS, where it has the advantages of smaller rating converters, and decoupled control of active and reactive power [11], and [12]. Various advanced control strategies have been developed to control the FSS based WRIM for wind power smoothing such as; Fuzzy logic combined with sliding mode control [13], artificial neural network [14], adaptive neuro-fuzzy control [15], and adaptive wavelet fuzzy neural network combined with nonlinear predictive control [6].
Flywheel Energy Storage Systems (FESSs) are very susceptible to grid voltage dips. This is particularly true for FESSs based WRIMs that employ reduced converters in their design. Faults in the power system, even far away from the FESS, could cause grid voltage dips. This will result in an overcurrent on the rotor circuit of the WRIM and consequently may lead to a protective disconnection of the FESS from the power grid.
Due to their simple structure, PI controllers are the most common controllers used for WRIM control [11], and [16]. However, they have operating characteristics that result in excessive rotor currents during voltage dips, and its performance is not guaranteed under parameter perturbation [17]. This is due to the fact that PI controllers have limited bandwidth and gain margin. Therefore, different advanced control strategies such as neuro-fuzzy control [18], neural sliding mode control [19], input-output feedback linearization [20], and sliding mode combined with direct power control [21], have been used to enhance the fault ride-through ability of grid-connected WRIMs. Other methods based hardware implementation has also been proposed in the literature [22]. H∞ control has found numerous applications in electrical field control, such as wind power [23], Uninterruptible Power Supplies (UPS) [24], Dynamic Voltage Restorer (DVR) [25], and Power System Stabilizer (PSS) [26], etc. The mixed-sensitivity H∞ control is seen to be a powerful tool for robust controller design achieving Robust Stability (RS) and Robust Performance (RP) under system uncertainty [23]. In this approach, weights functions are used to shape the magnitudes of closed-loop transfer functions, such as sensitivity function S and the complementary sensitivity function T . An alternative to mixed-sensitivity H∞ control is the H∞ loop shaping synthesis [27]. This method combines the H∞ robust stabilization using normalized coprime factorization with the classical loop shaping. Such designs are formulated as an H∞ optimal control problem [28].
Considering the benefits of FESSs for wind turbines, their stability and robustness during grid voltages dips are very important. Although FESSs are not concerned by the severe grid code requirements, they must stay connected to the grid during voltage dips to ensure better stability of the electric grid. In this context, this work proposes a robust H∞ controller design of WRIM that can guarantee uninterrupted operation of the flywheel storage system during and after grid voltage dips. The main contribution of this paper is the design of a robust H∞ current controller which will decrease the negative effects of voltage dips in the WRIM system, such as rotor over-currents and active power oscillations, and guarantee the system RS and RP during grid voltage dips with parameter perturbation.
This work is organized as follows. First, the influence of grid voltage dip and WRIM parameter perturbation on the rotor currents is analyzed in Sec. 2.
Next, the H∞ current controller is designed, and the weights functions are selected to guarantee the system RS and RP in Sec. 3. Finally, simulations in Matlab/Simulink are presented in Sec. 4. to evaluate the performance of the proposed H∞ controller during grid voltage dips with parameter perturbation, followed by a conclusion in Sec. 4.

System Overview
A flywheel energy storage system incorporating a WRIM is connected to the point of common coupling of a wind farm with the power grid, as shown in Fig. 1. Adjusting the rotor speed of the WRIM makes the flywheel either release or absorb energy in order to smooth the wind power delivered into the grid.

Wind Farm Modelling
In this work, six wind turbines based on doubly fed induction generators that operate identically constitute a wind farm of 9 MW. Each wind turbine presents a rated power of 1.5 MW.
Due to various losses in a wind turbine, the power extracted from the wind is given by [29]: where: S is the surface swept by the blades of the turbine in (m 2 ), ρ is the density of the air in (kg·m −3 ), v w is the wind speed in (m·s −1 ), C P is the power coefficient (-), λ is the tip speed ratio, and β is the blade pitch angle in ( • ).
The tip speed ratio λ is defined by the following equation: The expression of C P is approached for a 1.5 MW turbine using the following nonlinear function [30]: The values of coefficients C 1 to C 5 are given in Tab Figure 2 demonstrates the efficiency coefficient curve C P against the tip speed ratio λ and the blade pitch angle β. We obtain a maximum value of C P equal to 0.4382 for a blade pitch angle β = 0 • and for an optimum value of the speed ratio (λ opt = 6.32). Maximum Power Point Tracking (MPPT) is used to capture the maximum power from each wind turbine and therefore increase the overall performance of the wind farm. Therefore, the pitch angle is kept at the optimal value β = 0 • . Tab. 1: The values of coefficients C 1 to C 5 .

WRIM Driven FESS Modelling
The flywheel energy storage system is modeled as a WRIM coupled with a rotating mass that stores the electrical power as kinetic energy. The kinetic energy stored in the flywheel energy storage system is proportional to its inertia J f and the rotational speed of the WRIM w rf as shown in Eq. (5): In order to smooth the wind power injected into the grid, the kinetic energy of the FESS is determined by [13]: where, P f is the reference active power exchanged between the FESS and the grid, P wf is the filtered wind power, and P g is the desired grid power.
When the reference active power P f is negative, it indicates that the WRIM driven FESS injects the power into the grid. While positive power indicates that the FESS stores the wind power.
Since the WRIM can be regarded as traditional IM with non-zero rotor voltages, its dynamic model in the dq-synchronous reference frame can be expressed as in Eq. (7), Eq. (8) and Eq. (9) [31]: where the subscripts "s" and "r" represent the stator and rotor R s and R r are resistances of the stator and rotor, L σs and L σr are the stator and rotor leakage inductances, L m is the magnetizing inductance, L s = L ls + L m and L r = L lr + L m are the stator and rotor winding total self-inductances, ϕ sf and ϕ rf are the stator and rotor magnetic flux linkages, v sf and v rf are the stator and rotor voltages, i sf and i rf are the stator and rotor currents, w sf is the electrical angular speed, and w rf is the rotor angular speed.

Uncertain Model of WRIM Driven FESS
Since our main objective in this work is protecting the converters and the machine under grid voltage dips against rotor over-current, the FESS model will focus on the rotor current loop control. In this work, the system uncertainty is described as WRIM parameter perturbation and grid voltage dips. Based on Eq. (7) and Eq. (8), an uncertain WRIM model can be described as: The matrices A, B, B d , and C are given by Eq. (11), Eq. (12), Eq. (13), and Eq. (14), respectively, where δ A , δ B and δ B d present the uncertain part of the matrices A, B, and C, respectively. Here, ±50 % perturbation with respect to the nominal values of the parameters R r , L m , L σs and L σr has been considered. The grid voltage dip is presented in the frequency domain using a multiplicative factor δ d , satisfying Figure 4 illustrates the frequency characteristics of the uncertain model based on Eq. (10). The bode plots from u dsf to i drf and from u drf to i def are shown in Fig. 4(a) and Fig. 4(b), respectively. The curve marked with '+' represents the nominal model, while the uncertain models are shown by the other curves. It can be seen that the grid voltage disturbance and the WRIM parameter perturbation have an influence on the rotor current control. Figure 3 shows the control structure for the rotor side converter of the WRIM. The proposed control ensures the wind power smoothing by adjusting the WRIM speed when a voltage dip occurs. K∞ is the rotor current controller which is designed based on mixedsensitivity H∞ control. The standard S/KS mixed sensitivity design with an additional weight W 0 to describe the multiplicative output uncertainty is adopted in this work, as shown in Fig. 5.

H∞ Controller Structure Design
The grid voltage is considered as an external disturbance, and therefore, G d (s) = C(sI − A) −1 B d is also added as an input to the S/KS mixed sensitivity design. The parameter uncertainties are represented by unstructured multiplicative output uncertainty ∆ which satisfies ∆ ∞ ≤ 1, such as G p = (I + W 0 ∆)G, where W 0 is the weight function, G p is the uncertain model, and G is the nominal model. The input and output vector of ∆ is marked as y ∆ and u ∆ , respectively. The input of the H∞ controller is v = i * drf − i drf , i * qrf − i qrf . w = r, d and z = z 1 , z 2 present the external disturbance vector and the weighted outputs, respectively.
The weighting functions W 1 and W 2 are used to shape v and u, respectively.
The block including G, G d , W 0 , W 1 , and W 2 represents the shaped generalized plant P which can be derived as [23]: The block including P and K represents the closedloop system N which can be derived as: where, F l (P, K) is the lower linear fractional transformation, and S = (I + GK) −1 is the system sensitivity.
The design goal is to find a stabilizing controller K to minimize the H∞ norm of the closed-loop system N , which is the transfer function from external disturbances w to weighted outputs z: where γ min is obtained by solving two algebraic Riccati equations, and by reducing γ iteratively [28].

Weighting Functions Selection
The parameters uncertainties are represented in the frequency domain using multiplicative output uncertainty, and the relative errors of perturbed plants G p can be expressed as [28]: The design goal is to select a weight W 0 to cover the maximum magnitude of all relative error curves: This can be achieved by using the following firstorder filter: Where τ −1 is approximately the frequency whereat the relative uncertainty reaches 100 %, r 0 is the relative uncertainty at steady state, and r ∞ is the magnitude of the weighting function at high frequency (typically, r ∞ ≥ 2).
The corresponding relative errors and the weighting function W 0 are shown as functions of frequency in Fig. 6. It can be seen that the curve of W 0 lies at each frequency above all relative error curves. W 1 must be designed to guarantee the tracking performance to the low-frequency reference signal and avoid high-frequency noises: where 1 |W1| is equal to A 1 ≤ 1 at low frequencies, and equal to M 1 ≥ 1 at high frequencies.
In order to avoid numerical problems in the algorithm used to synthesize the H ∞ controller, the weight W 1 was given a gain of 60 dB at low frequency instead of including a pure integrator.
On the other hand, W 2 is designed as a high-pass filter in order to limit the closed-loop bandwidth. The low frequency gain of W 2 was set as −100 dB to ensure that the function in Eq. (17) is dominated by W 1 at low frequencies.
The parameters of the weighting functions W 0 , W 1 , and W 2 are listed in Tab. 2.

Robust Stability and Robust Performance Analysis
The Robust Stability (RS) and Robust Performance (RP) of the proposed H ∞ controller should be validated to guarantee the system stability and control performance of all perturbed models.
The condition for robust stability of a system with a multiplicative output uncertainty can be expressed as follows [28]: Figure 7 shows the singular value curves of the uncertain closed-loop system T and 1/W 0 . As can be seen, the maximum magnitude of all possible curves of T is below 1/W 0 for all frequencies satisfying the RS condition in Eq. (23).
Tab. 2: Parameters of the weights.

Weights
Parameter Value The requirement of Robust Performance (RP) in the case of a system with multiplicative uncertainty is given in Eq. (24) [28]: Fig. 7: Robust stability analysis. Figure 8 illustrates the singular value curves of W 1 S ∞ + W 0 T ∞ by considering all perturbed models. As can be seen, the maximum magnitude of all possible curves of W 1 S ∞ + W 0 T ∞ is below 1 (0 dB) satisfying the RP condition in Eq. (24).

Simulation Tests
In order to examine the performance of the proposed H ∞ current controller in the presence of voltage dips and WRIM parameter perturbation, the simulation model of 9 MW wind farm with 2 MW WRIM driven FESS has been constructed and simulated in Matlab/Simulink using SimPowerSystems. The parameters of the studied system are listed in App. A.
The FESS based WRIM is controlled to smooth the wind farm power output. The simulation results presented below were obtained for a desired active power at the grid of P g = −8 MW. In first, a three-phase voltage dip of 20 % occurred from 5 s to 6 s. It is assumed that the wind speed variation is between 8 m·s −1 and 12 m·s −1 as shown in Fig. 9. Therefore, the wind farm generates the maximum active power from the wind, as shown in Fig. 10. The rotor speed variation of the FESS is shown in Fig. 11.
When active power is required by the grid, the rotor speed is decreasing and the WRIM runs as a generator extracting power from the flywheel storage system and injecting it into the grid. When wind energy is required to be stored in the flywheel, the WRIM runs as a motor and the rotor speed is increasing to charge the flywheel storage system.  Here, the PI controller of the rotor current loop was also simulated to compare the performance with the proposed H ∞ current controller. The behavior of the WRIM rotor currents as controlled by the PI and H ∞ controllers in the presence of voltage dip are shown in Fig. 12 and Fig. 13, respectively. Figure 12 displays the WRIM rotor currents ( Fig. 12(a)) and their zoom ( Fig. 12(b)) in the case of the PI current controller. It is very clear to seed that the voltage dip causes the WRIM rotor currents to distort. While the same voltage dip has no impact on WRIM rotor currents in the case of the proposed H ∞ current controller, as shown in Fig. 13. The comparisons show that the over-currents in the rotor windings of the WRIM can be well limited with the proposed H ∞ controller, which can enhance the capability of the WRIM driven FESS to withstand voltage dips. Figure 14 presents the WRIM active power (Fig. 14(a)) and its zoom (Fig. 14(b)) controlled by PI and H ∞ controllers in the presence of voltage dip. We can observe that the active power presents oscillations under voltage dip in the case of PI controller. While the H ∞ controller reduces the oscillations produced by the voltage dip on the WRIM active power. The above-described simulations demonstrate the ca-pability of the FESS to inject more clean power to the grid.   Figure 15 shows the grid active power (Fig. 15(a)) and its zoom ( Fig. 15(b)) in the case of the PI and H ∞ current controllers. As can be seen, the voltage dip produces oscillations on the active power injected into the grid in the case of PI controller. While, the voltage dip has less impact on the active power injected into the grid in the case of the proposed H ∞ current controller, which helps to ameliorate the quality of the grid active power in the presence of voltage dip. Furthermore, parameter variations often occur in real-time control, which can degrade the WRIM driven FESS performance. In the aim to test the impact of the WRIM parameter perturbation on the system performance, simulations were made supposing that the values of L m , L s , L r and R r are increased by 30 %. 4 Figure 16 and Fig. 17 show the comparison between the PI controller and the proposed H ∞ controller under parameter perturbation and for the same voltage dip. In Fig. 16, we can observe that the WRIM parameter perturbation has less effect on the rotor currents using the proposed H ∞ controller. While in the case of PI controller, the rotor currents present severe distortion which may threaten the safe operation of the FESS.  Figure 17 displays the WRIM active power zoom under voltage dip with parameter perturbation. It is shown that the WRIM parameter perturbation has an important effect on the active power controlled by the conventional PI controller, while this impact is effectively reduced by the H ∞ controller.
The comparisons show that the proposed H ∞ controller has excellent performance under voltage dip and is robust to WRIM parameter perturbation.

Conclusion
This work proposes a robust H ∞ control scheme for a Wound Rotor Induction Machine (WRIM) driven Flywheel Energy Storage System (FESS) connected to the grid. The proposed H ∞ controller is used to limit the rotor over-current, and enhance its Robust Stability (RS) and Robust Performance (RP) under grid voltage dips with parameter perturbation. The H ∞ controller is designed using a modified mixedsensitivity synthesis based multiplicative output uncertainty. Weighting functions are selected in this design to decrease the negative effects of grid voltage dips and to guarantee the RS and RP of the proposed H ∞ controller under parameter perturbation.
The comparison between PI and H ∞ controller for the WRIM rotor currents and active power control is investigated, taking into consideration robustness to grid voltage dips and WRIM parameter perturbation. The simulation results illustrate that the proposed H ∞ controller has excellent performance compared with the PI one. Moreover, the proposed H ∞ control scheme leads to smooth the wind farm power output and ensures better quality of wind power generation under grid voltage dips.
Experiments will be conducted in the future work to validate the proposed control strategy.