Milad Bahrami

Jean-Philippe Martin

Gaël Maranzana

Serge Pierfederici

Mathieu Weber

Farid Meibody-Tabar

Majid Zandi

Design and modeling of an equalizer for fuel cell energy management systems

IEEE TRANSACTIONS ON POWER ELECTRONICS



Abstract-During the lifespan of a Polymer Electrolyte Membrane

Fuel Cell (PEMFC) system, some heterogeneities between the cells constituting the stack can appear. The voltage of one particular cell in a stack may decrease because of specific aging or local malfunctioning such as drying. As a result, more heat is generated in this cell leading to an increase in its temperature and thus an additional voltage loss. This snowball effect can result in the failure of the cell. Therefore, the lifetime of a PEMFC stack can be increased by applying energy management to its cells. Notable that the output voltage of a cell is lower than a stack. Hence, a high conversion ratio converter is necessary to implement such energy management. An efficient way to increase the output voltage is to connect the output capacitors of the converters such as the boosts in series. Ensuring the converters' controllability is a key point to implement energy management. In this paper, an equalizer system is proposed to ensure the controllability of the boost converters. The balancing speed and the low number of switches are the main advantages of this system. The validity of the proposed system is verified through simulation and experiments.

Index

Terms-Controllability, converter, energy management, equalizer, Polymer Electrolyte Membrane Fuel Cell (PEMFC).

I. INTRODUCTION

URABILITY enhancement of Polymer Electrolyte Membrane Fuel Cells (PEMFCs) is one of the major challenges to enable the diffusion of this technology in the mass market. In a stack, different cells are electrically connected in series to increase the output voltage, whereas they are supplied with gas in parallel. The whole system lifetime strongly depends on each cell in such a connection. Therefore, energy management can have a key role to compensate for the faults that occurred in each cell. As an example, if the current production by one particular cell could be decreased to mitigate local flooding or could be increased to mitigate a local drying, the electrochemical stability of the stack would be improved. In the same way, if a faulty cell could be short-circuited, the stack may continue to operate in degraded mode and ensure the network reliability. Based on the new patent [START_REF] Maranzana | IMPROVED FUEL CELL[END_REF], the current of any number of cells in a single stack is accessible. As a result, the behavior of the cells in a stack can be checked and their fault can be compensated. However, a high voltage conversion ratio DC-DC converter is required in this case, because the number of cells in series is decreased. Isolated or high voltage ratio stepup DC-DC converters can be used but the basic challenge is efficiency improvement [START_REF] Arunkumari | An overview of high voltage conversion ratio DC-DC converter configurations used in DC microgrid architectures[END_REF], [START_REF] Shahin | High Voltage Ratio DC -DC Converter for Fuel-Cell Applications[END_REF].

An efficient way to solve these issues is the series connection of the DC-DC converters' output capacitors. This solution leads to an increase the freedom degree in management and control of the PEMFCs. The basic drawback of this structure is the possibility of the controllability loss. For instance, if the input power of one cell becomes lower than the required amount, the voltage of the corresponding capacitor will decrease. In this case and assuming that the DC-DC converter is a boost converter, the controllability will be lost if the output voltage becomes lower than the input voltage. In such conditions, a voltage equalizer or a balancing system can ensure the controllability.

The equalizer in the literature can be categorized into two basic groups: with an auxiliary source and without auxiliary source [START_REF] Imtiaz | Time Shared Flyback Converter ' Based Regenerative Cell Balancing Technique for Series Connected Li-Ion Battery Strings[END_REF]. The equalizers with auxiliary source use another source to compensate the lower voltage cells. These equalizers are not efficient, especially for embedded applications. As seen in Fig. 1, the equalizers without auxiliary sources, are categorized into two basic groups: dissipative and nondissipative equalizers [START_REF] Imtiaz | Time Shared Flyback Converter ' Based Regenerative Cell Balancing Technique for Series Connected Li-Ion Battery Strings[END_REF]. Dissipative equalizers use resistance to decrease the voltage of the higher voltage cells. These equalizers waste much energy. Therefore, the efficiency is very low. However, the low cost, simple implementation, and small size are their advantages [START_REF] Hijazi | Balancing Circuit New Control for Supercapacitor Storage System Lifetime Maximization[END_REF], [START_REF] Shang | A delta-Structured switched-Capacitor equalizer for series-Connected battery strings[END_REF]. The non-dissipative equalizers can be divided into three groups: capacitor based [6]- [START_REF] Ye | Analysis and Design of Zero-Current Switching Switched-Capacitor Cell Balancing Circuit for Series-Connected Battery/Supercapacitor[END_REF], converter based [START_REF] Imtiaz | Time Shared Flyback Converter ' Based Regenerative Cell Balancing Technique for Series Connected Li-Ion Battery Strings[END_REF], [START_REF] Uno | Single-Switch Single-Transformer Cell Voltage Equalizer Based on Forward -Flyback Resonant Inverter and Voltage Multiplier for[END_REF]- [START_REF] Anno | Double-input bidirectional DC/DC converter using cell-voltage equalizer with flyback transformer[END_REF] and other type equalizers [START_REF] Liu | Voltage Balance Control Based on Dual Active Bridge DC/DC Converters in a Power Electronic Traction Transformer[END_REF], [START_REF] Kang | A New Nonisolated High-Voltage-Gain Boost Converter With Inherent Output Voltage Balancing[END_REF]. A control approach is usually adopted for switches of a specific structure to balance the voltages in the third group of equalizers.

A number of capacitors are connected to cells by a number of switches in the capacitor-based topologies. The cells are periodically connected to these capacitors, and this operation continues to the point where all the cells and capacitors reach the same voltage. These equalizers require a numerous amount of switches and they are not cost-effective for a large number of cells or high power applications. In addition, these equalizers can balance the voltages through several repeated operations. As a result, they need much time to balance the voltages. The maximum of the balancing current can be controlled by sizing the capacitor capacitances. Therefore, the high efficiency in balancing can be obtained independently of imbalance states Design and modeling of an equalizer for fuel cell energy management systems [START_REF] Shang | A delta-Structured switched-Capacitor equalizer for series-Connected battery strings[END_REF]- [START_REF] Liu | A Novel Diode-Clamped Modular Multilevel Converter With Simplified Capacitor Voltage-Balancing Control[END_REF]. The capacitors of each level of a multi-level converter were frequently connected to next or previous level capacitors by using the numerous amount of switch in [START_REF] Liu | A Novel Diode-Clamped Modular Multilevel Converter With Simplified Capacitor Voltage-Balancing Control[END_REF]. In this topology, the switches and capacitors in the multi-level converter were used to balance the voltage of capacitors in each level without adding another device. To overcome the problem of the balancing time, the resonance phenomenon can be used. Using the resonance in such topologies, the balancing speed can be increased simultaneously with achieving the benefits of soft switching [START_REF] Ye | Analysis and Design of Zero-Current Switching Switched-Capacitor Cell Balancing Circuit for Series-Connected Battery/Supercapacitor[END_REF].

The converter based equalizer can be categorized into three groups: cell to stack equalizers [START_REF] Imtiaz | Time Shared Flyback Converter ' Based Regenerative Cell Balancing Technique for Series Connected Li-Ion Battery Strings[END_REF], stack to cell equalizers [START_REF] Uno | Single-Switch Single-Transformer Cell Voltage Equalizer Based on Forward -Flyback Resonant Inverter and Voltage Multiplier for[END_REF], [START_REF] Li | A high-efficiency voltage equalization scheme for supercapacitor energy storage system in renewable generation applications[END_REF]- [START_REF] Mizuno | A Voltage Equalizer Using Flyback Converter With Active Clamp[END_REF], stack to stack equalizers [START_REF] Anno | Double-input bidirectional DC/DC converter using cell-voltage equalizer with flyback transformer[END_REF], and cell to cell equalizers [START_REF] Chen | A MultiWinding Transformer Cell-to-Cell Active Equalization Method for Lithium-Ion Batteries With Reduced Number of Driving Circuits[END_REF]- [START_REF] Lee | Quasi-Resonant Zero-Current-Switching Bidirectional Converter for Battery Equalization Applications[END_REF].

A stack to cell equalizer with a flyback converter was used to equalize the voltage of series lithium-Ion batteries in [START_REF] Zhang | A Hierarchical Active Balancing Architecture for Lithium-Ion Batteries[END_REF]. A stack to stack equalizer was used in this topology to realize the modular approach.

A forward-flyback resonant inverter was used to send energy from the series cell to cells in [START_REF] Uno | Single-Switch Single-Transformer Cell Voltage Equalizer Based on Forward -Flyback Resonant Inverter and Voltage Multiplier for[END_REF]. A voltage multiplier circuit was connected to cells. This circuit is used two diodes for each cell to automatically select the lower voltage cells and send energy to them. The required amount of switches was minimized in this topology at a cost of lower efficiency due to the use of two diodes for each cell. In this topology, the number of switches is independent of the number of cells. Similar topology with a buck converter on the primary side of the transformer was used in [START_REF] Uno | Single-Switch Single-Magnetic PWM Converter Integrating Voltage Equalizer for Partially Shaded Photovoltaic Modules in Standalone Applications[END_REF] for photovoltaic application. The forward-flyback converter in [START_REF] Uno | Single-Switch Single-Transformer Cell Voltage Equalizer Based on Forward -Flyback Resonant Inverter and Voltage Multiplier for[END_REF] the proposed was divided by a bidirectional converter in [START_REF] Yashiro | Transformer-Less Bidirectional PWM Converter Integrating Voltage Multiplier-Based Cell Voltage Equalizer for Series-Connected Electric Double-Layer Capacitors[END_REF]. Therefore, the design became more flexible and circuit implementation was simplified. In [START_REF] Uno | String-to-Battery Voltage Equalizer Based on Half-Bridge Converter with Multi-Stacked Current Doublers for Series-Connected Batteries[END_REF], a half-bridge converter was used on the primary side of a transformer to send energy from the series connected cell to lower voltage cells. A multi-stacked current doubler was connected to the secondary side of the transformer in connecting with cells. The number of diodes and inductances must be increased by increasing the number of cells. Since this topology can be implemented with two switches independent of cells number without a multi-winding transformer, the circuit design and implementation can be simplified. In [START_REF] Li | A high-efficiency voltage equalization scheme for supercapacitor energy storage system in renewable generation applications[END_REF], [START_REF] Li | A rapid cell voltage balancing scheme for supercapacitor based energy storage systems for urban rail vehicles[END_REF], an improved push-pull converter with clamping capacitor was used to send energy from the series cells to lower voltage cells. A multi-winding transformer was used in this topology and Hbridge inverters were used in the secondary windings of the transformer. This topology needs to measure the voltage to drive the required H-bridge inverters. The balancing current can be very high in this topology and as a result, the balancing time can be reduced.

For the battery applications, it is not necessary to have a large balancing current, because the voltage difference among cells slowly increases. Therefore, there is much time to balance the cells with the lower current to decrease the losses [START_REF] Shang | A delta-Structured switched-Capacitor equalizer for series-Connected battery strings[END_REF]. However, in such application of which the voltage balancing between output capacitors of DC-DC converters is required to ensure the controllability of the converters, the balancing time is the most important parameter. On the other hand, when the difference between cell voltages is very high, using the resonance can be dangerous for power electronic devices due to the high current peak. Regarding the possible inability of a cell or a set of cells in injecting the nominal power in a stack, a large difference between the output capacitor voltages is possible in the fuel cell applications. As a result, a new equalizer is proposed in this paper. The proposed equalizer can send energy to the lower voltage cells. This topology can quickly reach the steady-state condition by using a small number of switches and ensure the controllability in the worst conditions. The number of controlled switches in some converter-based equalizers depends on the cell numbers [START_REF] Imtiaz | Time Shared Flyback Converter ' Based Regenerative Cell Balancing Technique for Series Connected Li-Ion Battery Strings[END_REF], [START_REF] Lee | Active Balancing of Li-Ion Battery Cells Using Transformer as Energy Carrier[END_REF]- [START_REF] Li | A high-efficiency voltage equalization scheme for supercapacitor energy storage system in renewable generation applications[END_REF], [START_REF] Li | A rapid cell voltage balancing scheme for supercapacitor based energy storage systems for urban rail vehicles[END_REF], [START_REF] Zhang | A Hierarchical Active Balancing Architecture for Lithium-Ion Batteries[END_REF], [START_REF] Anno | Double-input bidirectional DC/DC converter using cell-voltage equalizer with flyback transformer[END_REF]. These equalizers are not effective in term of cost and volume. In the proposed equalizer, the number of controlled switches is independent of the cells number. In this paper, the performance of this equalizer is validated by the simulation and experimental results. A state space based model is also proposed that can be used for design purposes.

The rest of this paper is organized as follows: The principals of the proposed equalizer and the operation modes are described in Section II. The most important design considerations are also summarized in this section. In Section III, the simulation results are presented and compared with the theoretical results. The experimental results are provided in Section IV to validate the simulation results. Finally, conclusions are listed in Section V.

II. PROPOSED EQUALIZER

As Mentioned before, a power electronic structure is required to manage the current of cells separately to increase the lifetime of the stack. Since the voltage of one cell or a small number of cells is very low, using a step-up converter is inevitable. In this case, the conventional boost converter is used for the purpose of simplicity. To further increase the output voltage, the output capacitors of boost converters are connected in series. Loss of controllability is possible in this case. To prevent this possibility, an equalizer is proposed. The proposed power electronic structure with the equalizer is shown in Fig. 2 where the classic boost converters are used to connect the cells to the DC bus. As seen in this figure, the energy can be transferred from the series capacitors to the lower voltage capacitors. The H-bridge inverter can send the energy to even or odd numbered cells through the transformer. CH is a film capacitor at the input of the H-bridge inverter to stabilize the input voltage of this converter. The DC link voltage (Vdc) is determined by the load requirements. Regarding the embedded application, the standard voltage of 48 v is adopted for DC link voltage. It is assumed that the cells inside one stack are broken down into some group of cells with an identical number of cells. Furthermore, it is assumed that only one group of cells has a problem in one moment. In the worst condition, one group of Converter based [START_REF] Imtiaz | Time Shared Flyback Converter ' Based Regenerative Cell Balancing Technique for Series Connected Li-Ion Battery Strings[END_REF], [START_REF] Uno | Single-Switch Single-Transformer Cell Voltage Equalizer Based on Forward -Flyback Resonant Inverter and Voltage Multiplier for[END_REF]- [START_REF] Anno | Double-input bidirectional DC/DC converter using cell-voltage equalizer with flyback transformer[END_REF] Other [START_REF] Liu | Voltage Balance Control Based on Dual Active Bridge DC/DC Converters in a Power Electronic Traction Transformer[END_REF], [START_REF] Kang | A New Nonisolated High-Voltage-Gain Boost Converter With Inherent Output Voltage Balancing[END_REF] cells can inject no power. As a result, the output capacitor voltage can be changed from zero to 16 v in serial connection of four output capacitors. The DC link voltage is converted to the AC voltage on the primary side of the high-frequency transformer by the H-bridge inverter. This AC voltage is shown by Vin.

A. Operation Modes

In this section, the following assumptions are considered in the operation analysis for the purpose of simplicity:

1. All the switches and diodes are considered as the ideal devices. 2. Turn ratio for all secondary windings are the same and equal to 𝑛 = 𝑁 2

𝑘 𝑁 1

and k is the coupling coefficient.

3. The coupling coefficient between the secondary windings is perfect. 4. The DC bus voltage is controlled to have a constant voltage of Vdc.

The theoretic waveforms of the proposed equalizer are shown in Fig. 3 in steady state when C1 and C2 are the lower voltage capacitors between the even and odd-numbered cells respectively. Other cells have the same voltage. The switching commands of the H-bridge inverter are generated in such a way that a symmetrical square wave is imposed to the primary side of the transformer with variable duty cycle. Due to the windings polarity and the direction of the diodes in the circuit, the odd and even-numbered diodes can be naturally turned on in the positive and negative sections of the square wave, respectively. All the even-numbered/odd-numbered diodes are biased negatively when the input voltage of the transformer Vin is positive/negative. The diode corresponding to the capacitor, which has the lower voltage between even/odd-numbered capacitors, conducts the current when the negative/positive voltage appears on the primary side of the transformer. All the secondary windings will have the same voltage equal to the lower voltage capacitor when one diode starts to conduct. As a result, all the other diodes are reverse biased. The different operation modes of the proposed equalizer are shown in Fig. 4 when C1 and C2 are the lower voltage capacitors. This assumption is considered in order to investigate all the modes.

where L is the inductance, r is the resistance, i is the current, m and f subscripts are used to indicate the magnetizing and leakage inductance or resistance, Vd is the diode drop voltages, Vin is the input voltage at the primary side of the transformer, V C j is the voltage of Cj, P FC j is the injected power of FCj, and iload is the load current.

The switching frequency is chosen in such a way that the diode D1 turns off and the leakage current is reached to the magnetizing current before the t2. The differential equations are changed when the diode D1 is turned off (after tf) as follows:

{ 𝐿 𝑚 𝑑𝑖 𝑚 𝑑𝑡 = 𝑟 𝑚 (𝑖 𝑓 -𝑖 𝑚 ) 𝐿 𝑓 𝑑𝑖 𝑓 𝑑𝑡 = -𝑟 𝑓 𝑖 𝑓 -𝑟 𝑚 (𝑖 𝑓 -𝑖 𝑚 ) 𝐶 𝑗 𝑑𝑉 𝐶 𝑗 𝑑𝑡 = 𝑃 𝐹𝐶 𝑗 𝑉 𝐶 𝑗 -𝑖 𝑙𝑜𝑎𝑑 𝑗 = 1,2,3,4 (3)
4) Mode 4 [t3-t4: Fig. 4 (d)]: This mode is similar to mode 2 with this difference that S3 is turned off and S1 is turned on. Before the diode D2 is turned off, the differential equations of the system are as follows:

{ 𝐿 𝑚 𝑑𝑖 𝑚 𝑑𝑡 = - 𝑉 𝐶 2 +𝑉 𝑑 𝑛 𝐿 𝑓 𝑑𝑖 𝑓 𝑑𝑡 = 𝑉 𝐶 2 +𝑉 𝑑 𝑛 -𝑟 𝑓 𝑖 𝑓 𝐶 2 𝑑𝑉 𝐶 2 𝑑𝑡 = - (𝑖 𝑓 -𝑖 𝑚 + 𝑉 𝐶 2 +𝑉 𝑑 𝑛 𝑟 𝑚 ) 𝑛 -𝑖 𝑙𝑜𝑎𝑑 + 𝑃 𝐹𝐶 2 𝑉 𝐶 2 𝐶 𝑗 𝑑𝑉 𝐶 𝑗 𝑑𝑡 = 𝑃 𝐹𝐶 𝑗 𝑉 𝐶 𝑗 -𝑖 𝑙𝑜𝑎𝑑 𝑗 = 1,3,4 (5)
Then, the diode D2 is turned off and the differential equations of the system change as (4). The power is transferred through the transformer to increase the voltage of lower voltage capacitors. Any losses in the transformer reduce the power that is received by lower voltage cells. The transformer leakage inductance can also affect its transmitting power by affecting its input current. Therefore, the transformer design has a key role in this topology.

B. Design considerations

The magnetizing inductance of the transformer must be maximized in order to reduce the magnetizing current. For instance, Lm and Lf of a core with two windings can be calculated by the following equation:

{ 𝐿 𝑚 = 𝑘 2 𝐿 1 𝐿 𝑓 = (1 -𝑘 2 )𝐿 1 ( 6
)
Where k is the coupling coefficient and L1 can be calculated by:

𝐿 1 = 𝑁 2 ℜ (7)
N is the number of turns and ℜ is the reluctance of the core. As seen in this equation, increasing the number of turns and decreasing the reluctance lead to increase the magnetizing inductance. Increasing the number of turns increases the volume of the transformer and, therefore, increase the hysteresis losses. Furthermore, increasing the number of turns also increases the wire length that leads to an increase of the wire resistance and, consequently, the copper losses. The reluctance is inversely proportional to the permeability. Hence, selecting a core with high permeability can make it possible to increase the magnetizing inductance with a lower volume. This also increases the leakage inductance. However, the leakage inductance affects the maximum transmitting power through the transformer in each cycle. This power is also the most important parameter in reducing the equalizing time. To obtain this power in steady state by considering the first group of cells as the faulty cells, the diode current can be used to calculate the transferred energy. Therefore, the transferred energy can be calculated as follows: [START_REF] Shang | A Switched-Coupling-Capacitor Equalizer for Series-Connected Battery Strings[END_REF] Where We is the energy that is received by the faulted cell, 𝑉 𝐶 1 is the voltage of the first capacitor, and iD is the corresponding diode current that is on. Based on Fig. 3, this equation can be solved by dividing its time interval into two intervals:

𝑊 𝑒 = (∫ 𝑖 𝑓 -𝑖 𝑚 𝑛 𝑑𝑡 𝑡 1 𝑡 0 + ∫ 𝑖 𝑓 -𝑖 𝑚 𝑛 𝑑𝑡 𝑡 𝑓 𝑡 1 - 𝑉 𝐶 1 +𝑉 𝑑 𝑛 𝑟 m (𝑡 𝑓 -𝑡 0 )) (9)
Assuming that the time constant of Lf/rf is much larger than tf-t1, the magnetizing and leakage current functions can be obtained based on [START_REF] Maranzana | IMPROVED FUEL CELL[END_REF] in the interval [t0,t1] as follows:

{ 𝑖 𝑓 (𝑡) = -𝑖 𝑚0 (1 - 𝑡 𝐿 𝑓 𝑟 𝑓 ) + ∑ 𝑉 𝐶 𝑗 𝑗 - 𝑉 𝑑 +𝑉 𝐶 1 𝑛 𝐿 𝑓 𝑡 𝑖 𝑚 (𝑡) = -𝑖 𝑚0 + 𝑉 𝑑 +𝑉 𝐶 1 𝑛 𝐿 m 𝑡 (10)
Where -im0 is the magnetizing current at t0. The magnetizing current value at tf is equal to im0. keeping the same assumption like [START_REF] Shu | Voltage balancing approaches for diode-clamped multilevel converters using auxiliary capacitor-based circuits[END_REF] for the time constant of Lf/rf, the magnetizing and leakage current functions can be obtained based on (2) in the interval of [t1,tf] as follows: Integration of this equation in the interval of t0-t1 and using [START_REF] Shu | Voltage balancing approaches for diode-clamped multilevel converters using auxiliary capacitor-based circuits[END_REF], an equation is obtained for the maximum current value of the diode. Another equation for the maximum current of the diode can be obtained by integrating [START_REF] Ye | Analysis and Design of Zero-Current Switching Switched-Capacitor Cell Balancing Circuit for Series-Connected Battery/Supercapacitor[END_REF] in the interval of t1-tf and using [START_REF] Liu | A Novel Diode-Clamped Modular Multilevel Converter With Simplified Capacitor Voltage-Balancing Control[END_REF]. The time tf-t1 can be calculated by solving these two equations.

To eliminate im0 from the equations, the equation of magnetizing current over the whole interval [t0,tf] can be used as follows:

𝑖 𝑚0 = -𝑖 𝑚0 + 𝑉 𝑑 +𝑉 𝐶 1 𝑛 𝐿 m (𝑡 1 -𝑡 0 ) → 2𝑖 𝑚0 = 𝑉 𝑑 +𝑉 𝐶 1 𝑛 𝐿 m (𝑡 1 -𝑡 0 ) (13)
A function for the transferred energy is obtained by replacing [START_REF] Shu | Voltage balancing approaches for diode-clamped multilevel converters using auxiliary capacitor-based circuits[END_REF] and ( 11) in (9) and using ( 12) and ( 13). The transmitting power through the transformer can be calculated with respect to this energy using the following equation:

𝑃 𝑒 = 𝑑𝑉 𝐶 1
As seen in this equation, the transferred power in steady state depends on the leakage and magnetizing inductances, capacitor voltages, frequency, duty cycle, iron losses, copper losses, and transformer turn ratio.

With the hypothesis that only one diode between oddnumbered cells/ even-numbered cells can be turned on at the positive/negative part of the input voltage of the transformer, the voltage difference can be calculated. In this case, just the first cell is supposed to have a problem. For this purpose, the Kirchhoff current law in steady state for each capacitor is written as follows:

{ 𝑃 1 +𝑃 𝑒 𝑉 𝐶 1 -𝑖 𝑙𝑜𝑎𝑑 = 0 𝑃 𝑗 𝑉 𝐶 𝑗 -𝑖 𝑙𝑜𝑎𝑑 = 0 ∀ 𝑗 ∈ {2, 3 , … , 𝑛} (15)
where 𝑉 𝐶 𝑗 is the voltage of capacitor number j, Pj is the supplied power by the cell number j, and iload is the load current. The equation of load current is as follows:

𝑖 𝑙𝑜𝑎𝑑 = 𝑃 𝑖𝑛 𝑉 𝑑𝑐 + 𝑉 𝑑𝑐 -𝑉 𝑏𝑎𝑡 𝑟 𝑏𝑎𝑡 ( 16
)
where Vdc is the DC bus voltage that is controlled to have a constant value, Vbat and rbat is the voltage and the resistance of the power supply respectively that controls the DC bus voltage, and Pin is the power injected to the equalizer system as follows:

𝑃 𝑖𝑛 = 𝑃 𝑒 + 𝑉 ̅ 𝑝 2 𝑟 𝑚 + 𝑟 𝑓 (𝐼 ̅ 𝑓 ) 2 ( 17
)
where 𝐼 ̅ 𝑓 is the RMS value of leakage current that can be calculated using ( 10) and ( 11), and 𝑉 ̅ 𝑝 is the RMS voltage on the primary side of the transformer that is the RMS value of By substituting ( 16) and ( 14) in [START_REF] Shang | An Automatic Equalizer Based on Forward-Flyback Converter for Series-Connected Battery Strings[END_REF], the resulting equations form a system of equations. 𝑉 𝐶 𝑗 can be determined by solving this system numerically. Then, Pe can be calculated by [START_REF] Chen | A MultiWinding Transformer Cell-to-Cell Active Equalization Method for Lithium-Ion Batteries With Reduced Number of Driving Circuits[END_REF]. The results of these theoretical equations are shown in Fig. 5 by assuming the four capacitors in series, and that all PEMFCs can inject the rated power of 126 W except FC1 that injects no power. Other information is summarized in TABLE I. Therefore, the corresponding capacitor voltage (𝑉 𝐶 1 ) is decreased. To investigate the impact of the leakage inductance on the transferred power to the lower voltage capacitor, the duty

Fig. 5. The amount of transferred power through the transformer of the proposed equalizer in steady state based on: (a) Changes in the amount of leakage inductance when the duty cycle of H-bridge inverter is equal to 0.4. (b) Changes in the amount of supplied power by the faulted cell and changes in the duty cycle.

cycle of the H-bridge inverter is assumed to be 0.6 and the leakage inductance is changed. Using ( 14)-( 16), the transferred power is calculated for different value of the leakage inductance. The amount of transferred power as a function of the leakage inductance is shown in Fig. 5 (a). As seen in this figure, the amount of power is decreased by increasing the leakage inductance. Thus, the leakage inductance should be minimized in one hand. A large leakage inductance is required to limit the overshoot of the current in another hand. Therefore, there must be a trade-off between the peak of current and maximum transferred power through the transformer.

To check the effectiveness of the proposed equalizer, it is assumed that the first group of cells cannot inject the nominal power. The injected power by the first cell is changed from zero to nominal power. The other groups of cells inject the nominal power. In this condition, the transferred power is calculated by changing the duty cycle of the H-bridge inverter based on ( 14)-( 16). The transferred power by changing the duty cycle of the H-bridge inverter for different value of first cells injected power is shown in Fig. 5 (b). As seen in this figure, the further reduction in injected power by the first group of cells can be compensated by increasing the duty cycle of the H-bridge inverter. In other words, the transferred power through the transformer is increased by increasing the duty cycle of the Hbridge inverter when the injected power of faulted cell was decreased.

To select a magnetic core based on the constraints mentioned above, the worst condition is taken into account. In this condition, one group of cells should be short-circuited. Thus, one group can inject no power while the other groups inject the nominal power. In this condition, the output voltage of the boost converter connected to the faulted cell decreased and the controllability is lost. The equalizer should send the power to this capacitor to ensure the controllability. The Leakage inductance of the transformer has an impact on the primary and secondary transformer side currents. Considering the first group of cells as the faulted cells, the first capacitor voltage is equal to zero in this condition. As a result and based on (10) and ( 13), the leakage current in transient conditions can be calculated as follows:

𝑖 𝑓 = ∑ 𝑉 𝐶 𝑗 𝑗 𝐿 𝑓 𝑡 (19)
The DC link voltage is controlled to have a constant value of 48 V and based on Fig. 3, the maximum of the leakage current can be calculated as follows:

𝑖 𝑓 𝑚𝑎𝑥 = 48 𝐿 𝑓 𝑑𝑇 2 (20)
Using [START_REF] Shu | Voltage balancing approaches for diode-clamped multilevel converters using auxiliary capacitor-based circuits[END_REF], [START_REF] Ye | Analysis and Design of Zero-Current Switching Switched-Capacitor Cell Balancing Circuit for Series-Connected Battery/Supercapacitor[END_REF], and (13), the maximum current of the diode can be calculated as follows:

𝑖 𝐷 𝑚𝑎𝑥 = 48 𝐿 𝑓 𝑑𝑇 2𝑛 (21)
In the worst case, the duty cycle is equal to the maximum value of one. The maximum of the leakage current in the transient conditions for different value of the leakage inductance is shown in Fig. 6. As seen in this figure, the peak of the leakage current is very high. Indeed, a low-pass filter can cope with this problem and control the rate of duty-cycle change.

The leakage current in steady-state can be used to make a decision about the rating of the semiconductor devices. Considering a proportional gain to control the duty cycle of the H-bridge inverter in closed-loop, the maximum of the leakage and diode currents can be calculated by ( 10) and ( 12)-( 16). These currents are shown in Fig. 7 by changing the leakage inductance of the transformer when the first group of cells can inject no power and other groups can inject the nominal power. As seen in this figure, if the leakage inductance is equal to 5 µF, the current rating of the H-bridge inverter switches should be very higher than 15 A and the current rating of the diodes should be higher than 60 A. Therefore, there is a trade-off between the current rating of the semiconductor devices and the maximum power that can be transferred through the transformer.

The coupling coefficient of 0.99 is assumed based on previous experiences. Replacing [START_REF] Shang | A delta-Structured switched-Capacitor equalizer for series-Connected battery strings[END_REF] in [START_REF] Chen | A MultiWinding Transformer Cell-to-Cell Active Equalization Method for Lithium-Ion Batteries With Reduced Number of Driving Circuits[END_REF] and using ( 15) and ( 16), the value of L1 can be calculated to send the nominal power by the equalizer. Equation ( 7) can be used to select the magnetic core by knowing L1. It is essential to minimize the volume of the transformer to reduce its losses. Therefore, the desired core can be chosen among the magnetic core in the market by using [START_REF] Ye | Topology , Modeling , and Design of Switched-Capacitor-Based Cell Balancing Systems and Their Balancing Exploration[END_REF] and the classical equation of the minimum number of turns with the aim of minimizing the transformer volume. Considering these design constraints, the devices used to make simulation and obtain experimental results are shown in TABLE I.

III. SIMULATION RESULTS

A simulation is made by four capacitors in series in Matlab Simulink to preliminary verify the validity of the proposed equalizer theories. For a precise study of the balancing system, the H-bridge inverter is controlled in the open-loop mode and duty cycle is set to 0.4. It is assumed that the PEMFCs connected to the first and second capacitors can inject the half of nominal power and the other cells inject the nominal power (126 W). The voltage and current waveforms on the primary side of the transformer for d=0.4 are shown in Fig. 8. As seen in this figure, the current is linear and the power can be transmitted through the transformer. These results are also in agreement with the theoretical waveforms.

To control the duty cycle of the H-bridge inverter in closedloop, a simple proportional control is used. This closed-loop proportional control is implemented by comparing the average voltage of capacitors and their minimum voltage. As seen in Fig. 9, a low-pass filter is used in this control to limit the maximum current in transient conditions. The low pass filter is designed by considering the worst condition (short circuit of one group of cells) to ensure that the leakage current has no overshoot during transient conditions. The cut-off frequency of this filter is equal to 10 rad/s. The proportional gain (P) of this controller is set to 0.1 for the purpose of simulation and experimental results. The voltage changes of different capacitors with this closed-loop controller are shown in Fig. 10 when the cells inject nominal power except the first cell. The first cell is turned off and it can inject no power in this simulation.

IV. EXPERIMENTAL RESULTS

For further verify the proposed balancing system and validation of the theoretical analysis and simulation, the experiments are conducted on a laboratory prototype as demonstrated in Fig. 11. dSPACE 1005 with FPGA board is used to receive the information and send the commands. Four programmable power are used to emulate the PEMFCs. These power supplies are connected to four boost converters, the output capacitors of the boosts are connected in series. The component part number used in this prototype are listed in Table II. The proposed balancing system connects to these capacitors. It is notable that a planar transformer is built for this paper. Each winding turn of this transformer means a printed circuit board. On each printed circuit board, a copper line with the width of 21.5 mm and the thickness of 35 µm is rotated in such a way that the circuits can be placed in the window of the magnetic core. This design is used to minimize the leakage inductance. The turn number of primary and secondary windings are shown in Table I.

To check the validity of the theoretical analysis and simulation, several experiments have been accomplished. First experiment conditions are similar to the first simulation. The cells inject nominal power except first and second cells. These cells inject the half of nominal power. The voltage and current waveforms on the primary side of the transformer for d=0.4 are shown in Fig. 12. As seen in this figure, the results are in agreement with the simulation results and the power can be transferred to the first and second capacitors during the positive and negative part of the input voltage respectively.

The second experiment conditions are similar to the second simulation. The first power supply is turned off and as a result, it can inject no power. The other cells inject the nominal power. The voltage changes of different capacitors are shown in Fig. 13 when the duty-cycle of H-bridge inverter is controlled in closed-loop with the same parameter of simulation .

To validate the proposed model for the design purpose, the capacitor voltages are measured when the first cell can inject the half of nominal power. In this experiment, the duty cycle is changed in the open-loop from 0.3 to 0.9. These voltages are also numerically calculated by ( 14)-( 16). The output power of each boost converter is measured. These powers are used in these theoretical equations. The results of the experiment and theoretical equations are shown in Table III The main objective of the proposed equalizer system is to guarantee the controllability of DC-DC converters connected to fuel cells for applying energy management (Fig. 2). The drying and flooding conditions are the most critical conditions for the membrane of the PEMFC. In these conditions, the energy management system must function properly to compensate the faults. If one cell suffers from drying, it should be shortcircuited to produce more water by keeping constant gas flow rates. This leads to a decrease in gas stoichiometry for this particular cell and a consequent membrane humidification. On the contrary, no current should be generated by a cell in flooding conditions in order to let it dry. In other words, the power generated by the cell should be controlled near zero. In the following, it is assumed that the first cell experiences these conditions among four other healthy cells.

The capacitor voltages are shown in Fig. 14 (a) and the transformer voltage and current on the primary side are shown in Fig. 14 (b) in the flooding condition of the first PEMFC. As seen in these figures, the current of the first PEMFC cannot be set to zero before running the equalizer system. As seen in Fig. 14, when the square wave voltage is supplied on the primary side of the transformer by the H-bridge inverter, the current of the first PEMFC reaches to zero after about 40 ms with the used parameters for the controller. Indeed, the boost converter connected to the first PEMFC will be controllable after this time. Furthermore, the capacitor voltage connected to the first PEMFC is increased and other capacitor voltages are decreased.

The output capacitor voltages of the boost converters in the drying condition of the first PEMFC are shown in Fig. 15 (a). In this condition, the voltage and current on the primary side of the transformer are also shown in Fig. 15 (b). As seen in this figure, the first capacitor voltage is increased from zero to 8 volts but the short-circuit conditions are maintained for the first PEMFC. For this experiment, the maximum current of the power supply is limited to 20A.

V. CONCLUSION

In this paper, an equalizer based on the multi-winding transformer was proposed to ensure the controllability of boost converters in a series connection of their output capacitors.

This equalizer can be used to implement the PEMFC energy management system. The validation of the proposed system was investigated through the simulation and experimental results. The validity of the theoretical model was confirmed using the experimental results. Therefore, this model can be used to predict the behavior of the equalizer system. In other words, the theoretical equations can be used for sizing the system. The experimental results confirm the effectiveness of the equalizer system in maintaining the control ability of the boost converters when a cell is not able to supply the nominal power. The performance of the system was also evaluated in two critical conditions of the fuel cell operation. In both operation conditions (emulation of drying and flooding conditions), the controllability of the converter connected to the faulty cell is ensured by increasing its capacitor voltage.

Fig. 1 .

 1 Fig. 1. Different types of equalizers.

1 )

 1 Mode 1 [t0-t1: Fig.4(a)]: before t0 two switches S1 and S2 were on. At t0, S2 is turned off and S4 is turned on. Therefore, the positive voltage is imposed on the primary side of the transformer. This voltage is induced in the secondary windings. This voltage leads to the reverse biased of the even-numbered diodes. Based on the reason mentioned above, only diode D1, which connects to the lower voltage capacitor between odd-numbered capacitors, conducts current. As a result, the energy through the leakage inductance of the transformer is transferred to the capacitor C1. Therefore, the leakage current and voltage of C1 is increased. The derivative equations of different state variables are shown in[START_REF] Maranzana | IMPROVED FUEL CELL[END_REF].-𝑖 𝑙𝑜𝑎𝑑 𝑗 = 2,3,4

2 )

 2 Mode 2 [t1-t2: Fig.4(b)]: S4 is turned off and S2 is turned on as a synchronous rectifier in this mode. Therefore, the primary side of the transformer is short-circuited. The diode D1 continues to conduct but the current passing through it, which is proportional to the leakage current, is linearly decreased. The derivative equations in this mode are as follows:

Fig. 2 .

 2 Fig. 2. Proposed equalizer topology.

Fig. 3 .

 3 Fig. 3. Theoretical waveforms in steady-state operation of the proposed equalizer.

Fig. 4 .

 4 Fig. 4. Different operation modes of the proposed equalizer. (a) Mode 1: t0<t<t1. (b) Mode 2: t1<t<t2. (c) Mode3: t2<t<t3. (d) Mode 4: t3<t<t4.

  solving (9), a function for the transferred energy is obtained based on Lf, Lm, m, Vd, tf-t1, t1-t0, V C 1 , and the sum of the capacitor voltages. The interval t1-t0 is equal to dT/2 where T=1/F and F is the switching frequency of the H-bridge inverter, and d is its duty cycle. The value of tf-t1 can be calculated by writing the diode current equation over this interval [t1,tf]. The diode current can be obtained from integrating the following equation:

𝑉 𝑑 + 𝑉 𝐶 1 n

 1 over the interval of tf-t0 in the switching period as follows:

Fig. 6 .

 6 Fig. 6. Impact of the leakage inductance on the maximum of leakage current in transient conditions.

Fig. 7 .

 7 Fig. 7. Impact of the leakage inductance on a) the maximum of leakage current b) the maximum of the diode current in steady state.

  𝑉 𝐶 𝑗 𝑡ℎand 𝑉 𝐶 𝑗 𝑒𝑥𝑝 are respectively the theoretical and experimental voltage of the j th capacitor.

Fig. 8 .

 8 Fig. 8. Simulation results of voltage and current waveforms on the primary side of the transformer for d=0.4.

Fig. 9 .

 9 Fig. 9. Schematic diagram of the control method used to determine the duty cycle (d) of the H-bridge inverter.

Fig. 10 .

 10 Fig. 10. Simulation results of the Output capacitor voltages in closedloop when the first group of cells injects no power.

Fig. 11 .

 11 Fig. 11. Proposed equalizer test bench.

Fig. 12 .

 12 Fig. 12. Experimental waveforms of the current and voltage on the primary side of the transformer when first and second fuel cells inject the half of nominal power and duty cycle of H-bridge inverter is equal to 0.4.

Fig. 13 .

 13 Fig. 13. Experimental results of the Output capacitor voltages when the first group of cells injects no power.

Fig. 14 .

 14 Fig. 14. Experimental results of the proposed equalizer during the flooding condition of the first cell. (a) Output capacitor voltages. (b) Voltage and current on the primary side of the transformer and current of the first cell.

Fig. 15 .

 15 Fig. 15. Experimental results of the proposed equalizer during the drying condition of the first cell. (a) Output capacitor voltages. (b) Voltage and current on the primary side of the transformer.

  This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TPEL.2019.2899150, IEEE Transactions on Power Electronics IEEE TRANSACTIONS ON POWER ELECTRONICS 𝑊 𝑒 = 𝑉 𝐶 1 ∫ 𝑖 𝐷 (𝑡)𝑑𝑡

	𝑡 𝑓 𝑡 0	= 𝑉 𝐶 1 ∫ 𝑡 0 𝑡 𝑓	𝑖 𝑓 -𝑖 𝑚 -𝑛 𝑉 𝐶 1 𝑛 𝑟 m +𝑉 𝑑	𝑑𝑡

0885-8993 (c) 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

  3 𝐿 𝑚 (𝑛𝑟 𝑚 𝑉 𝑑𝑐 -2𝐿 𝑓 (𝑉 𝐶 1 +𝑉 𝑑 )) 𝑟 𝑚 (-4𝐹𝐿 𝑓 𝐿 𝑚 +4𝐹(𝐿 𝑓 +𝐿 𝑚 )𝑟 𝑚 -𝑑𝑟 𝑓 𝑟 𝑚 ) + 𝑑 2 𝐹𝑟 𝑓 (-𝑛𝑟 𝑚 𝑉 𝑑𝑐 +2𝐿 𝑓 (𝑉 𝐶 1 +𝑉 𝑑 )) 𝐿 𝑓 (4𝐹𝐿 𝑓 𝐿 𝑚 -4𝐹(𝐿 𝑓 +𝐿 𝑚 )𝑟 𝑚 +𝑑𝑟 𝑓 𝑟 𝑚 ) -𝑑(-4𝐹(𝐿 𝑓 +𝐿 𝑚 )+𝑑𝑟 𝑓 +𝑑𝑟 𝑓 ) 4𝐿 𝑓 𝐿 𝑚 (4𝐹𝐿 𝑓 𝐿 𝑚 -4𝐹(𝐿 𝑓 +𝐿 𝑚 )𝑟 𝑚 +𝑑𝑟 𝑓 𝑟 𝑚 )

2 (𝑉 𝐶 1 +𝑉 𝑑 ) (4𝐹𝐿 𝑚 𝑟 𝑚 (𝑛𝑉 𝑑𝑐 -(𝑉 𝐶 1 + 𝑉 𝑑 )) + 𝑑𝑟 𝑓 𝑟 𝑚 (𝑉 𝐶 1 + 𝑉 𝑑 ) -4𝐹𝐿 𝑓 (𝐿 𝑚 + 𝑟 𝑚 )(𝑉 𝐶 1 + 𝑉 𝑑 )) 2 )

  . In this table, the error 𝑉 𝐶 1 is the relative error of the first capacitor voltage. In this table, the error is the root mean square of the relative errors as follows:

	𝑉 𝐶 𝑗 𝑡ℎ 𝐸𝑟𝑟𝑜𝑟 = √∑ ( 𝑉 𝐶 𝑗 𝑡ℎ -𝑉 𝐶 𝑗 𝑒𝑥𝑝

TABLE II COMPONENT

 II PART NUMBER FOR PROTOTYPE.

		Component	Part number
	FC	Power supply	TDK GENH 750W
	Boost converter	Inductance	(1mH)
		Switches	IGBT
		Capacitor	Electrochemical (4700𝜇𝐹)
	Equalizer	Diodes	DSS2x121-0045B
		Magnetic core	B66295G Material N87
		H-bridge	SiC MOSFET
		switches	CCS050M12CM2
		Capacitor	film (220 𝜇𝐹)

Milad Bahrami was born in Roodsar, Iran, in 1991. He received the B.Sc. degree in electrical engineering and M.Sc. degree in renewable energies engineering from Shahid Beheshti University, Tehran, Iran, in 2013 and 2016, respectively. He is currently working toward the Ph.D. degree with LEMTA, University of Lorraine, France. His current research interests include Multi-vector microgrids, power converters design, stability analysis, control of electric hybrid systems, and clean energy sources. His research activities deal with the stability study of distributed power systems, the control of DC and AC microgrids and the design of power electronic converters for specific applications like fuel cell and electrolyzer systems.

Jean-Philippe


Ali Djerioui
email: ali.djerioui@univ-msila.dz

Azeddine Houari
email: azeddine.houari@univ-nantes.fr

Mourad Saim Abdelhakim

Serge Pierfederici

Mohamed Fouad

Mohamed Machmoum

Malek Ghanes

Transactions on Industry Applications Flatness Based Grey Wolf Control for Load Voltage Unbalance Mitigation in Three-Phase Four-Leg Voltage Source Inverters

Keywords: Standalone Microgrids, Voltage unbalance, Four-Leg Voltage Source Inverter (FL-VSI), Differential flatness, Grey Wolf optimization

Standalone power-supply systems become a key solution to effectively address load demand in remote locations wherein the voltage asymmetry raises as a particular concern in view of the large number of single phase household and communities loads. Over these loading conditions, the use of Four-Leg Voltage Source Inverters (FL-VSIs) appears as a suitable topology to provide clean power with symmetrical voltage waveforms. The reported work employs the differential flatness theory associated with Grey Wolf optimization in order to guarantee proper operations of the FL-VSIs in different loading conditions. In fact, the differential flatness theory is applied to check the flatness of the FL-VSIs, which allows implementing a reduced control model. Then, a Grey Wolf algorithm which acts as a tracking controller (GWC) is developed. The GWC generates an optimal control signal used by the differentially flat model so as to ensure adequate control performance, even under severe load disturbances and model inaccuracies. A comprehensive experimental test is carried out to validate the effectiveness of the proposed flatness based grey wolf controller (FB-GWC).

I. INTRODUCTION

The output voltage quality in standalone power-supply systems is gaining more attention with regards to the increasing penetration of distributed renewable energy resources [START_REF] Kabiri | Control of active and reactive power ripple to mitigate unbalanced grid voltages[END_REF] [START_REF] Iyer | A generalized computational method to determine stability of a multi-inverter microgrid[END_REF]. However, the integration of such power sources as distributed generation entities must to comply with many technical and performance requirements [START_REF] Blaabjerg | Distributed Power Generation Systems and Protection[END_REF][START_REF] Iyer | Analysis and mitigation of voltage offsets in multi-inverter microgrids[END_REF][START_REF] Wu | Damping Methods For Resonances Caused By Lcl-Filter-Based Current-Controlled Grid-Tied Power Inverters: An Overview[END_REF]. Accordingly, a number of regulatory and grid considerations refer to power quality in standalone power-supply applications. Among them, European standard EN 50160 states that the allowable variation of supplying voltages cannot exceed 10% of their nominal values. The voltage harmonic content quantified by the total harmonic distortion index (THD) must, according to IEEE standard 519-1992 and recently 1547-2014, not exceed 5%. Furthermore; the voltage unbalance factor (VUF) should be in compliance with the IEC standards and kept below 2% as long as the consumers' current does not include which essentially characterizes the prevalence of single-phase loads in three-phase four-wire electrical networks [START_REF]IEEE Recommended Practice and Requirement for Harmonic Control in Electric Power Systems[END_REF][START_REF]IEEE standard for interconnecting distributed resources with electric power systems[END_REF][START_REF] Zhang | A near-state three-dimensional space vector modulation for a three-phase four-leg voltage source inverter[END_REF][START_REF] Pichan | Deadbeat control of the stand-alone four-leg inverter considering the effect of the neutral line inductor[END_REF][START_REF] Bifaretti | Modulation with Sinusoidal Third-Harmonic Injection for Active Split DC-Bus Four-Leg Inverters[END_REF][START_REF] Lidozzi | Load-Adaptive Zero-Phase-Shift Direct Repetitive Control for Stand-Alone Four-Leg VSI[END_REF][START_REF] Vechiu | Transient operation of a fourleg inverter for autonomous applications with unbalanced load[END_REF]. Indeed, the unbalanced loading can represent a large extent of the total load scenario in standalone microgrid configurations wherein the voltage asymmetry raises as a particular concern in view of the large number of single phase household and communities loads [START_REF] Zhou | Four-Leg Converters with Improved Common Current Sharing and Selective Voltage-Quality Enhancement for Islanded Microgrids[END_REF] [START_REF] Houari | Improved Control Strategy for Power Quality Enhancement in Standalone Systems Based on Four-Leg Voltage Source Inverters[END_REF].

In this context, the use of FL-VSIs as power source conversion units or even as power quality conditioners appears as a suitable topology to maintain balanced sinusoidal output voltage waveforms over all loading conditions in transformer-less applications [START_REF] Ufnalski | Particle swarm optimization of the multioscillatory LQR for a three-phase four-wire voltagesource inverter with an LC output filter[END_REF]. Compared to three phase three leg inverters with split DC-links, the addition of the fourth leg provides numerous extra advantages in terms of voltage rating. Indeed, reduced DC bus voltage is involved and larger AC voltage may be supported (over 15%) since it requires a reduced DC bus voltage and can support larger AC voltages (over 15%) [START_REF] Bifaretti | Modulation with Sinusoidal Third-Harmonic Injection for Active Split DC-Bus Four-Leg Inverters[END_REF]. Moreover, the additional fourth leg provides a path to draw zero current sequences, which sustains the converter capability to handle critical load conditions, such as the presence of single-phase or unbalanced phase loading [START_REF] Vyawahare | Design and analysis of standalone 4-wire, 4-leg inverter microgrid with unbalanced loads[END_REF]. FL-VSIs are widely used in different standalone and grid connected applications, namely distributed generation [START_REF] Burgos-Mellado | Experimental Evaluation of a CPT-Based Four-Leg Active Power Compensator for Distributed Generation[END_REF] with grid connected [START_REF] Golwala | New Three Dimensional Space Vector based Switching Signal Generation Technique without Null Vectors and with Reduced Switching Losses for a Grid-Connected Four-leg Inverter[END_REF], standalone [START_REF] Zhou | Four-Leg Converters with Improved Common Current Sharing and Selective Voltage-Quality Enhancement for Islanded Microgrids[END_REF] and uninterruptible power supply mode of operation [START_REF] Pichan | Sliding-Mode Control of Four-Leg Inverter With Fixed Switching Frequency for Uninterruptible Power Supply Applications[END_REF]: active power filtering [START_REF] Chebabhi | A new balancing three level three dimensional space vector modulation strategy for three level neutral point clamped four leg inverter based shunt active power filter controlling by nonlinear back stepping controllers[END_REF][START_REF] Bouzidi | Hybrid direct power/current control using feedback linearization of three-level four-leg voltage source shunt active power filter[END_REF][START_REF] Sawant | A multifunctional fourleg grid-connected compensator[END_REF], including distributed static compensators [START_REF] Lliuyacca | Grid-forming VSC control in four-wire systems with unbalanced nonlinear loads[END_REF] and dynamic voltage restorers [START_REF] Naidu | Dynamic voltage restorer based on a four-leg voltage source converter[END_REF].

Nevertheless, the control of such converter in standalone power-supply applications represents a quite complicated task and, therefore, an advanced control strategy has to be designed. The control strategy synthesis must have three main objectives, tracking performances, disturbance rejection and load voltage unbalance compensation. For this purpose, many control schemes have been previously analyzed in different reference frames. They focus on control of load voltage and must meet power sharing and quality requirements in power applications based FL-VSIs under unbalanced and various load conditions. In this way, a number of research contributions proposes classical control strategies, with proportional integral (PI) and proportional resonant linear regulators [START_REF] Lliuyacca | Grid-forming VSC control in four-wire systems with unbalanced nonlinear loads[END_REF] [START_REF] Demirkutlu | A Scalar Resonant-Filter-Bank-Based Output-Voltage Control Method and a Scalar Minimum-Switching-Loss Discontinuous PWM Method for the Four-Leg-Inverter-Based Three-Phase Four-Wire Power Supply[END_REF]. The obtained test results demonstrates the effectiveness and the simplicity of these controllers. However, their dynamic responses show sluggish performances in transients modes even more when cascaded control schemes are employed [START_REF] Patel | Threedimensional flux vector modulation of four-leg sine-wave output inverters[END_REF][START_REF] Loh | A comparative analysis of multiloop voltage regulation strategies for single and three-phase UPS systems[END_REF][START_REF] Loh | Analysis of multiloop control strategies for LC/CL/LCL-filtered voltage-source and current-source inverters[END_REF]]. An undesirable feature of these control schemes is the introduction of a decoupled control, which increases the control complexity, and correspondingly the computational load [START_REF] Ufnalski | Particle swarm optimization of the multioscillatory LQR for a three-phase four-wire voltagesource inverter with an LC output filter[END_REF] [START_REF] Demirkutlu | A Scalar Resonant-Filter-Bank-Based Output-Voltage Control Method and a Scalar Minimum-Switching-Loss Discontinuous PWM Method for the Four-Leg-Inverter-Based Three-Phase Four-Wire Power Supply[END_REF]. To deal with linear regulators limitations, several nonlinear control methods are proposed in the literature including feedback linearization control [21][29], backstepping [START_REF] Chebabhi | A new balancing three level three dimensional space vector modulation strategy for three level neutral point clamped four leg inverter based shunt active power filter controlling by nonlinear back stepping controllers[END_REF][30] and sliding mode control [START_REF] Pichan | Sliding-Mode Control of Four-Leg Inverter With Fixed Switching Frequency for Uninterruptible Power Supply Applications[END_REF]. These control methods use differentiation to achieve the control law, which results in high order derivative control terms. These derivatives can affect the control law capabilities under severe load disturbances and model inaccuracies.

Among the reported control strategies, the flatness-based control (FBC) appears as an interesting nonlinear control approach due to its useful performances when explicit trajectory planning is required under different operating conditions [START_REF] Rivera | Digital predictive current control of a three-phase four-leg inverter[END_REF]. Indeed, the flatness theory allows an accurate description of the transient and the steady state dynamic of the overall system state variables based on differentially flat outputs. In addition, and as reported in various works [START_REF] Yaramasu | Model predictive approach for a simple and effective load voltage control of four-leg inverter with an output LC filter[END_REF][START_REF] Bayhan | Finite Control Set Model Predictive Control for a Quasi-Z-Source Four-Leg Inverter Under Unbalanced Load Condition[END_REF][START_REF] Houari | Flatness-Based Control of Three-Phase Inverter With OutputLC Filter[END_REF][START_REF] Shahin | Reliability Improvement Approach Based on Flatness Control of Parallel-Connected Inverters[END_REF][START_REF] Soheil-Hamedani | Flatness-based control method: A review of its applications to power systems[END_REF], the FBC demonstrates stable performances even under large operating point and system parameters variation. For instance, the use of FBC has been proposed for three phase three leg VSIs and its effectiveness has been highlighted through comparisons with both proportional integral and feedback linearization controllers [START_REF] Houari | Flatness-Based Control of Three-Phase Inverter With OutputLC Filter[END_REF]. Also, the control of parallel three-phase VSIs is proposed in [START_REF] Shahin | Reliability Improvement Approach Based on Flatness Control of Parallel-Connected Inverters[END_REF], where the flatness theory is used with the aim to minimize the effect of circulating currents and reduce voltage distortion at the Point of Common Coupling (PCC). A literature review of the main properties and applications of the FBC in power systems is proposed in [START_REF] Soheil-Hamedani | Flatness-based control method: A review of its applications to power systems[END_REF].

More recently, optimization based control methods like model predictive controllers (MPC) and optimal linear quadratic regulators (LQR) emerge as effective tools [START_REF] Rivera | Digital predictive current control of a three-phase four-leg inverter[END_REF] [START_REF] Yaramasu | Model predictive approach for a simple and effective load voltage control of four-leg inverter with an output LC filter[END_REF]. For instance in [START_REF] Rivera | Digital predictive current control of a three-phase four-leg inverter[END_REF], a MPC strategy is proposed for FL-VSI, wherein the minimization of a cost function yields the inverter optimal switching state to apply at the next sampling instant. This strategy presents relevant performances, but suffers from a variable switching frequency. Moreover, the fourth leg operates at higher switching frequency than the phase legs, which complicates the output filter design as well as the neutral inductor sizing. In [START_REF] Yaramasu | Model predictive approach for a simple and effective load voltage control of four-leg inverter with an output LC filter[END_REF], a Particle Swarm Optimization (PSO) algorithm is proposed to on-line adapt the weighting matrices (Q and R) of an LQR controller. Besides the interesting performance of the proposed strategy, it can be mentioned that under severe nonlinear condition, the PSO technique can fall into local optimum, which can alter the control performances. In this paper, a novel control strategy is proposed for FL-VSI based standalone power supply systems. The proposed strategy combines the advantages of flatness control and optimization. Indeed, a FBC controller is designed and extended with an evolutionary search algorithm that acts as a tracking controller to achieve the desired power quality requirements. The use of a Grey Wolf (GW) algorithm is preconized since it presents several advantages for practical implementation in terms of low computing complexity, and convergence accuracy [START_REF] Djerioui | Grey Wolf based control for speed ripple reduction at low speed operation of PMSM drives[END_REF]. The GW algorithm is used to search for an optimum correction of the flat output references in such a way to secure the desired trajectory tracking performances especially in the presence of substantial nonlinear and/or unbalanced loads. The main contributions of this work are summarized as follows.

1)

A FBC is extended with a GW optimization algorithm to enhance the control performance of an LC-L interfaced FL-VSI in presence of disturbing loads such as single-/three-phase, unbalanced, and nonlinear loads.

2) The GW algorithm depends on few parameters (weighting factors) that are tuned in order to minimize the voltage THD and VUF. The GW algorithm is used to minimize a cost function and generate the convenient control correction signal.

3)

The robustness of the proposed controller is studied for large load disturbances and model parameters variations.

This paper is organized as follows. In Section II, the model of the studied system is presented, and, in Section III, the design methodology of the proposed FB-GWC is detailed. Section IV present the experimental results and discuss the effectiveness of the proposed approach. Finally, section V underlines the contributions of this work.

II. SYSTEM DESCRIPTION AND MODELING

The typical circuit configuration of FL-VSIs with an output LC filter stage is depicted in Fig. 1. The DC-link is assumed as a constant voltage source. A neutral line inductor L n is used to connect the midpoint 'o' of the fourth leg to the neutral point 'n' of the load bus with the aim to the neutral line current ripples [START_REF] Houari | Improved Control Strategy for Power Quality Enhancement in Standalone Systems Based on Four-Leg Voltage Source Inverters[END_REF]. For the sake of simplicity, the following voltage and current vectors are defined: Inverter voltages vector 𝒗 𝒊 = [𝑣 𝑢𝑜 , 𝑣 𝑣𝑜 , 𝑣 𝑤𝑜 ] 𝑇 , output voltages vector 𝒗 𝒇 = [𝑣 𝑎𝑛 , 𝑣 𝑏𝑛 , 𝑣 𝑐𝑛 ] 𝑇 , line currents vector 𝒊 𝒇 = [𝑖 𝑎 , 𝑖 𝑏 , 𝑖 𝑐 ] 𝑇 ,neutral current vector 𝐢 𝐧 = [𝑖 𝑛 , 𝑖 𝑛 , 𝑖 𝑛 ] 𝑇 and load currents vector 𝒊 𝑳_𝒂𝒃𝒄 = [𝑖 𝐿𝑎 , 𝑖 𝐿𝑏 , 𝑖 𝐿𝑐 ] 𝑇 . Note: Matrices and vectors are denoted with bold letters. By applying Kirchhoff's laws for voltage and current, to the nodes given in Fig. 1

After straightforward Park's transformation P(θ) the system state equations are obtained in the synchronous reference frame:

𝒙̇= 𝑨 𝒙 + 𝑩 𝒖 + 𝑪 𝒅 (3) with: 𝒙 = [𝑖 0 , 𝑖 𝑑 , 𝑖 𝑞 , 𝑣 𝑐0 , 𝑣 𝑐𝑑 , 𝑣 𝑐𝑞 ] 𝑇 the state variables vector. 𝒖 = [𝑢 0 , 𝑢 𝑑 , 𝑢 𝑞 ] 𝑇 is the 0dq-components of the input vector.

The 0dq-components of the load currents is defined by 𝒅 = 𝒊 𝑳 = [𝑖 𝐿0 , 𝑖 𝐿𝑑 , 𝑖 𝐿𝑞 ] 𝑇 , and

𝑨 = [ - 𝑅 𝑓 +3𝑅 𝑛 𝐿 𝑓 +3𝐿 𝑛 0 0 -1 𝐿 𝑓 +3𝐿 𝑛 0 0 0 -𝑅 𝑓 𝐿 𝑓 𝜔 0 -1 𝐿 𝑓 0 0 -𝜔 -𝑅 𝑓 𝐿 𝑓 0 0 -1 𝐿 𝑓 1 𝐶 𝑓 0 0 0 0 0 0 1 𝐶 𝑓 0 0 0 𝜔 0 0 1 𝐶 𝑓 0 -𝜔 0 ] , 𝑩 = [ 1 𝐿 𝑓 +3𝐿 𝑛 0 0 0 1 𝐿 𝑓 0 0 0 1 𝐿 𝑓 0 0 0 0 0 0 0 0 0 ] , 𝑪 = [ 0 0 0 0 0 0 0 0 0 -1 𝐶 𝑓 0 0 0 -1 𝐶 𝑓 0 0 0 -1 𝐶 𝑓 ]
A carrier based PWM technique presented in [START_REF] Kim | A carrier-based PWM method for three-phase fourleg voltage source converters[END_REF], where the block diagram is shown in Fig . 2, has been considered in this work due to its performance and ease of implementation. The input voltage references of the three principal phases with respect to the fourth-leg are denoted respectively:

𝑣 𝑛 𝑟𝑒𝑓 = 𝑚𝑖𝑑(- 𝑣 𝑚𝑎𝑥 2 , - 𝑣 𝑚𝑖𝑛 2 , - 𝑣 𝑚𝑎𝑥 + 𝑣 𝑚𝑖𝑛 2 ) (5) iɑ V DC n i Lɑ L ƒ , R ƒ C ƒ L n , R n i n o u v w a b c

DC bus

Four Leg Inverter LC Filter Unknown Loads

III. PROPOSED CONTROL APPROACH

This section presents the design methodology of FB-GWC. As shown in Fig. 3, the proposed voltage mode control strategy is subdivided into two main parts. The first part in which the differential flat model of the FL-VSI is used to derive the control inputs as function of the candidate flat outputs references and their successive derivatives. Meanwhile, the second part considers the use of a GW based controller to search for an optimum correction of the flat output references in such a way to ensure desired trajectory tracking performances especially in the presence of substantial nonlinear and/or unbalanced loads.

In the following, the flatness parameterization principle is first given, and then it is applied to check the flatness of the studied system. Afterward, the principle of the GW based tracking controller is provided. The end of this section summarizes the control flow implementation. ..

Fig. 3. Block diagram of the proposed FB-GWC.

A. Flatness description

The differential flatness is a structural property of some controllable dynamic systems. The property allows an explicit parameterization of both system inputs and states variables as function of a set of specified variables, commonly known as flat outputs, and a finite number of their time derivatives without requiring the resolution of differential equations [START_REF] Houari | Flatness-Based Control of Three-Phase Inverter With OutputLC Filter[END_REF]. In other words, the general system model 𝒙̇= 𝑓(𝒙, 𝒖) is considered as a flat system if its corresponding states vector 𝒙 ∊ ℝ 𝑛 and input vector 𝒖 ∊ ℝ 𝑚 can be basically expressed as function of the candidate flat outputs given by: 𝒚 = Λ(𝒙, 𝒖, 𝒖̇, … , 𝒖 (𝑝) ) , 𝒚 ∊ ℝ 𝑚 (6) Accordingly, the system variables should take the form:

𝒙 = Υ(𝒚̇, 𝒚̈, … , 𝒚 (𝑟) ) (7)
𝒖 = 𝜳(𝒚, 𝒚̇, 𝒚̈, … , 𝒚 (𝑟+1) ) (8) The flatness property is more useful if the candidate flat outputs correspond to the physical outputs of the system. In this way, the flatness-based description of most of power applications has gained in importance due to the existence of numerous output candidates having with straight mathematical and physical significance. The expressions given by ( 7) and ( 8) concord with the flatness property and allow a simple calculation of feed forward control law according to the desired flat outputs trajectories. The control law is finally designed to satisfy the control objectives and guarantee that the trajectories of the inputs and outputs are smooth.

B. Flatness of the studied system

Following the control objectives, the dq-axes energies presenting the electrostatic energy stored in the capacitor filters 𝐶 𝑓 and the homopolar output voltage 𝑣 𝑐0 are proposed to be the flat output candidates. Then, the flat output vector 𝐲 can be expressed as follow:

𝒚 = [ 𝑦 0 𝑦 𝑑 𝑦 𝑞 ] = [ 𝑣 𝑐0 1 2 𝐶 𝑓 𝑣 𝑐𝑑 2 1 2 𝐶 𝑓 𝑣 𝑐𝑞 2 ] (9)
According to part A, the parameterization (7) consists in the formulation of all system state variables as a function of the candidate flat output components and their respective derivatives. Regarding the proposed flat output vector (9), the 0dq-components of the output voltage can be expressed as follow:

𝒗 𝒄 = [𝑣 𝑐0 , 𝑣 𝑐𝑑 , 𝑣 𝑐𝑞 ] 𝑇 = [𝑦 0 , √ 2𝑦 𝑑 𝐶 𝑓 , √ 2𝑦 𝑞 𝐶 𝑓 ] 𝑇 (10)
After a derivation of the flat output vector (9), the 0dqcomponents of the line current can be expressed as:

𝒊 = 𝐶 𝑓 𝑑 𝑑𝑡 𝒗 𝒄 + [ 0 0 0 0 0 -𝜔 0 𝜔 0 ] 𝒗 𝒄 + 𝒊 𝑳 ( 11
)
where

𝑑 𝑑𝑡 𝒗 𝒄 = [𝑦̇0, 𝑦̇𝑑 √ 2𝑦 𝑑 𝐶 𝑓 , 𝑦̇𝑞 √ 2𝑦 𝑞 𝐶 𝑓 ] 𝑻
Notice that 𝑖 𝐿 is an external disturbance.

The last step to rule on the flatness of the studied system is to verify that the input vector can be parameterized as a function of the candidate flat output components and their successive derivatives. Regarding the differential equation of the system given in (3) and the expressions of the state variables defined in [START_REF] Bifaretti | Modulation with Sinusoidal Third-Harmonic Injection for Active Split DC-Bus Four-Leg Inverters[END_REF] and [START_REF] Lidozzi | Load-Adaptive Zero-Phase-Shift Direct Repetitive Control for Stand-Alone Four-Leg VSI[END_REF], the input vector 𝒖 = [𝑢 0 , 𝑢 𝑑 , 𝑢 𝑞 ] 𝑇 can be expressed as follow:

𝒖 = 𝒗 𝒄 + [ 𝑅 𝑓 + 3𝑅 𝑛 0 0 0 𝑅 𝑓 𝐿 𝑓 𝜔 0 -𝐿 𝑓 𝜔 𝑅 𝑓 ] 𝒊 + [ 𝐿 𝑓 + 3𝐿 𝑛 0 0 0 𝐿 𝑓 0 0 0 𝐿 𝑓 ] 𝑑 𝑑𝑡 𝒊 (12)
where the expression of the theoretical line current derivation term is given in relation [START_REF] Vechiu | Transient operation of a fourleg inverter for autonomous applications with unbalanced load[END_REF].

𝑑 𝑑𝑡 𝒊 = 𝑑 2 𝑑𝑡 2 𝒗 𝒄 + [ 0 0 0 0 0 - 𝜔 𝐶 𝑓 0 𝜔 𝐶 𝑓 0 ] 𝑑 𝑑𝑡 𝒗 𝒄 + 𝑑 𝑑𝑡 𝒊 𝑳 ( 13
)
with :

𝑑 2 𝑑𝑡 2 𝒗 𝒄 = [ 𝑦̈0 𝑦̈𝑑 √ 2𝑦 𝑑 𝐶 𝑓 - 𝑦̇𝑑 2 2𝑦 𝑑√ 2𝑦 𝑑 𝐶 𝑓 𝑦̈𝑞 √ 2𝑦 𝑞 𝐶 𝑓 - 𝑦̇𝑞 2
2𝑦 𝑞√ 2𝑦 𝑞 𝐶 𝑓 ] Regarding Eq. 10, 11, 12 and 13, the input vector can be parameterized as a function of the candidate flat output components and their successive derivatives.

𝒖 = 𝜳(𝒚, 𝒚̇, 𝒚̈) (14)

To conclude, the studied system fulfills the flatness conditions expressed in [START_REF]IEEE standard for interconnecting distributed resources with electric power systems[END_REF] and [START_REF] Zhang | A near-state three-dimensional space vector modulation for a three-phase four-leg voltage source inverter[END_REF] with 𝐲 as the flat output vector and 𝒖 as the input vector. The control goal is satisfied only if the parameters are exactly known and if there are no disturbances. Therefore, in the practice, the flat model should be extended with a tracking controller in order to follow the planned reference trajectories even in the presence of model error and load disturbance.

C. Grey Wolf Based Tracking Controller

In order to ensure that the measured flat outputs track appropriately the desired flat output trajectories, a GW based tracking controller is proposed. Grey wolf is a newly developed heuristic algorithm to handle nonlinear optimization problems [START_REF] Mirjalili | Grey Wolf Optimizer[END_REF][START_REF] Precup | Grey wolf optimizer algorithm-based tuning offuzzy control systems with reduced parametric sensitivity[END_REF][START_REF] Djerioui | Energy management strategy of Supercapacitor/Fuel Cell energy storage devices for vehicle applications[END_REF][START_REF] Sharma | Grey wolf optimisation for optimal sizing of battery energy storage device to minimise operation cost of microgrid[END_REF][START_REF] Bella | Model-Free Controller for Suppressing Circulating Currents in Parallel-Connected Inverters[END_REF]. It simulates the social hierarchy and hunting behavior of grey wolves in nature like searching, encircling and attacking the prey. It moves the wolves (agents) toward prey by updating location vector, which is an average of best locations of the pack. As reported in the literature, this algorithm presents advantages in terms of low computing complexity, solution accuracy, convergence independence of initial conditions and its ability to deal with local minima. In this subject, a comparative study with twenty nine tests [START_REF] Mirjalili | Grey Wolf Optimizer[END_REF] with Particle Swarm Optimization (PSO), Gravitational Search Algorithm (GSA), Differential Evolution (DE), Evolutionary Programming (EP), and Evolution strategy (ES) has shown that GW algorithm is highly competitive. These proprieties are particularly important from the point of view of real time implementation.

The construction mechanism of this algorithm involves three steps: objective formulation and evaluation, optimal hunting distance calculation, control terms selection.

Step 1) Objective function

The proposed optimization based tracking controller acts to find the minimal correction control terms (𝐲) which is then used by the differentially flat model [START_REF] Houari | Improved Control Strategy for Power Quality Enhancement in Standalone Systems Based on Four-Leg Voltage Source Inverters[END_REF] in such a way to ensure an adequate control performance under large load disturbances and model inaccuracy. For this aim, a cost function G on the flat output error is proposed.

𝐆 = ( k 0 0 0 0 k 𝑑 0 0 0 k q ) 𝛆 ( 15
)
where 𝐆 = [G 1 , G 2 , G 3 ] T is the objective vector. Terms 𝐺 1 , 𝐺 2 and 𝐺 3 are respectively associated with the error in the 0dq flat outputs, i.e 𝛆 = [(y 0 ref -y 0 ), (y d ref -y d ), (y q ref -y q )] T . k 0 , k d and k q are constant gains whose the synthesis is presented in the following part.

Step 2) Optimal hunting distance calculation

The algorithm mechanism search three best solutions in regard of the global possible hunting solutions and calculate the candidate input control. Regarding the chosen cost function each dqo-input control axis is decoupled, the first three best intermediate input control variables can be formulated as follows Where 𝑦̈𝑥 1 , 𝑦̈𝑥

𝑦̈𝑥[𝑘 + 1] = 𝑦̈𝑥 1 +𝑦̈𝑥 2 +𝑦̈𝑥 3 3 𝑥 = {𝑑, 𝑞, 𝑜} (17)
Step 3) Control terms selection As presented in the previous part, the final design step consists in the evaluation of the established control solutions in step 2. Herein, the established control solutions are parameterized by their specified distance 𝐴 𝑥 = 2𝑎 𝑥 𝑟 𝑥 𝑘 -𝑎 𝑥 .Therefore, the input control (d-axis) is selected as follow:

{ 𝑦̈𝑥 = 𝑦̈𝑥[𝑘 + 1] 𝑖𝑓(|𝐴 𝑥 | < 1) 𝑦̈𝑥 = 𝑦̈𝑥[𝑘] 𝑖𝑓(|𝐴 𝑥 | > 1) (18)
The established candidate input control, in step 2, with its specified distance 𝐴 𝑥 is evaluated (𝐴 𝑥 compared to 1) in order to predict the future input control

D. Reference Trajectory Generation

The reference trajectory used by the GW algorithm and the flat model is obtained through a second order low-pass filter. Note that this choice allows limiting the derivative terms and does not influence the required computation time. Therefore, desired trajectories of the flat outputs and its first derivative are defined by:

{ 𝑦 𝑥(𝑓) 𝑅𝑒𝑓 = (1 -𝑒 𝑡-𝑡 0 𝜏 - 𝑡 -𝑡 0 𝜏 𝑒 𝑡-𝑡 0 𝜏 ) (

E. Implementation Details

The step-by-step operation of the proposed control strategy at each sample time (𝑇 𝑠 ) is listed below.

1. The measured current and voltage load at time k are transformed to their dq0-components. 2. The actual 0dq-component of the output voltage are used by the GW controller to predict the optimal value of the correction control term 𝒚̈. The algorithm process is depicted in Fig. 4 and described below:  The cost function is computed for each agent, which allows to select the three best solutions.  Each pack element updates its position following Eqs ( 18), ( 19) and [START_REF] Chebabhi | A new balancing three level three dimensional space vector modulation strategy for three level neutral point clamped four leg inverter based shunt active power filter controlling by nonlinear back stepping controllers[END_REF].  The updated agents (𝑦̈𝑥 𝛼𝛽𝛿 (𝑘 + 1) with j=1,…, N) are classified to obtain the best one.  The process is repeated until the iteration number 𝑘 𝑚𝑎𝑥 .

 In the last step, the best solution 𝑦̈𝑥 𝛼 (𝑘 + 1) is selected. 3. The resulted optimal solution of the GW algorithm present the correction control term 𝒚̈ of the flat model, which is used to compute the inverter input controls thank's to equation ( 8). 4. For efficient working of the proposed strategy, the consumed time by the whole control (GW controller, park transformations and the flat model of the system) should be set sufficiently bellow the sampling periode (𝑇 𝑠 ). Then, the parameters of the on-line optimization algorithm are to be selected carefully. For this consideration, the research agents number (N) is limited to 4, while the maximum iteration number (𝑘 𝑚𝑎𝑥 ) setting is provided in the next section.

IV-SIMULATION RESULTS AND DISCUSSIONS

In this part, the GW tracking controller parameters selection is derived and the effectiveness of the control in terms of voltage harmonic contents and unbalance reduction is discussed. The main parameters taken into account for the simulation study are given in table I. The total harmonic distortion(THD) of the output voltage The voltage unbalance factor(VUF) The needed calculation time of the GW algorithm. For the above consideration, two load scenarios are considered. The THD criterion is evaluated under balanced per-phase nonlinear loads that consist of single-phase diode rectifiers with output RC circuits (R = 40 Ω, C =150 µF). Besides, the VUF, is based on an unbalanced linear resistive loading scenario where, in fact, the phase 'b' is opened while phases 'a' and 'c' are equally loaded (R = 25 Ω). The following calculation can determine the percentage of VUF unbalance defined by [START_REF] Houari | Flatness-Based Control of Three-Phase Inverter With OutputLC Filter[END_REF].

(a) Fig. 6. VU-Factor versus changes in the odq-axis objective functions parameters.

VUF (%) = max(𝑣 𝑎 , 𝑣 𝑏 , 𝑣 𝑐 ) -min(𝑣 𝑎 , 𝑣 𝑏 , 𝑣 𝑐 ) 𝑉 𝑎𝑣𝑔 × 100 [START_REF] Bouzidi | Hybrid direct power/current control using feedback linearization of three-level four-leg voltage source shunt active power filter[END_REF] For each load scenario, the cost function parameters are changed sequentially. In the left side of Fig. 6, the parameter k 0 is set to 7.5 and dq-axis gains (k d and k q ) are changed from 0.1 to 10. Herein, the THD rate is reduced when using dq-axis gains smaller than 5 and can be minimized for gains around 1. In the right side of Fig. 6, the gain k 0 is changed from 0.1 to 8 and dq-axis gains are taken equal and changed from 0.1 to 1. It can be seen that the THD evolution is more sensitive to the change dq-axis gains.

The evolution of VUF according to the cost function parameters is illustrated in Fig. 7. Thereafter, k d and k q are taken equal and are linearly changed from 0 to 1 while ko is changed from 0.1 to 8. The VUF rate, as it can be appreciated, is further reduced for grater gains of k 0 . This rate is minimized for k 0 to 7.5 which satisfy VUF <2.2 %.

(b)

implementation. The chosen value of N for our practical implementation is equal to 6. Fig. 7. Iteration number influence on the: THD (blue color), VUF (green color) and execution time (red color). In summary, dq-axis gains (kd and kq) values are related to the voltage quality and properly selected gain allows reducing the THD rate. Gain k0 acts on the voltage symmetry and allows reducing VUF rate. Notice that the iteration number is set to 6 in this test.

To illustrate the influence of the GW calculation burden on the effectiveness of the proposed tracking controller, the iteration number (N) is changed sequentially and several simulation tests under loading scenario are done. Fig. 7 groups the results obtained from the corresponding THD rate, the VUF rate and the calculation time. As it can be appreciated, the iteration number is a key parameter to satisfy the control effectiveness. Indeed, the obtained results show that iteration number 6 allows obtaining satisfactory control performance: i.e reduced THD and VUF rates with a minimized time calculation regarding real-time

A. Performance illustration

To underline the performance of the proposed controller, comparison tests with open-loop control based model and a conventional PI control method [START_REF] Bella | Model-Free Controller for Suppressing Circulating Currents in Parallel-Connected Inverters[END_REF] are proposed. The open loop control is based on the flatness model where ÿ is set to zero in Fig. 3. The scheme principle of the conventional PI control is shown in Fig. 8. The PI parameters are selected by the use of pole placement method where the outer loop parameters kpv=5.33x10 -3 kiv=1.42 and the inner loop kpi=120 et kii=316x10 3 .

The compared three control strategies are firstly tested under linear load based on balanced resistive loads where Ra = Rb = Rc = 15 Ω. The obtained RMS voltages are presented in Fig. 9). As it can be appreciated, the open-loop control can be sufficient under ideal working conditions (balanced linear load where all parameters are known). Notice that the THD are equal to 2.1 for the open-loop, PI and proposed FBC-GWC controllers respectively. The compared controllers robustness is evaluated under unbalanced nonlinear loading conditions. The loads consist of single-phase diode rectifiers that cause zero sequence distortion at the odd triplen harmonics. For instance, load current phase a cause 2% of 3rd harmonic and 5% of 9th harmonic. The open-loop model is no longer sufficient and the use of closed-loop controller is necessary to handle the unbalance and harmonic issues. In this way the use of a conventional PI controller and the proposed controller allow to significantly improve the power quality of the output voltage. The computed VUF values that correspond to open-loop, PI and proposed controller are 9.3%, 4.8% and 2.3% respectively, the correspondents VUF are reported in Fig 9c . The voltage THD of phase "a" are 4.6%, 2.6% and 1.3%, respectively; the correspondents THD rates are reported in Fig 9b.

The proposed control scheme achieves the lowest VUF and voltage THD values, which corresponds to better power quality.

To evaluate the effect of input voltage variation on the voltage regulation, the DC bus is changed from 400 to 600 V at t= 0.05s where the output RMS voltage reference is kept constant. As it can be appreciated, the AC voltage is not influenced by the DC bus voltage. The only condition is related to the controllability limit where the DC voltage level must be sufficient. The limit of DC bus voltage can be estimated as follow [START_REF] Rezkallah | Lyapunov Function and Sliding Mode Control Approach for the Solar-PV Grid Interface System[END_REF]: V dc ≥ V RMS 2√2/(√3 m). For a modulation index m = 0.5 , the minimal DC bus voltage to ensure controllability is around 360 V. Therefore, a DC bus voltage of 400 V is considered in this work. In case of single phase load (Ra= Rb =∞ Ω and Rc=15 Ω), Fig. 13 illustrates the load voltages and currents responses and their respective zooms before (open loop) and after activating the proposed control approach. The load voltages are balanced. The noted voltage THD with the proposed control is about 1 %.

Open loop case

Closed loop case 2. This test shows that the proposed control scheme allow keeping the load voltages balanced with a reasonable voltage THD.

Referring to these results, it can be concluded that in steady state with balanced or unbalanced linear or nonlinear loading conditions, the use of the proposed control approach allows to keep the load voltages balanced with a voltage THD less than 2%. The provided experimental results demonstrated that the proposed control approach allows keeping the load voltage balanced with a reasonable voltage THD rate when the system is subject to disturbances like nonlinear and unbalanced loads.

Fig. 1 .𝑚𝑖𝑛Fig . 2 .

 12 Fig. 1. Four-leg inverter topology with an output LC filter and a neutral line inductor.

{ 𝑦̈𝑥 1 =

 1 (1 -𝑀 𝑥 1 )𝑦̈𝑥 𝛼 -𝐺 𝑥 𝑦̈𝑥 2 = (1 -𝑀 𝑥 1 )𝑦̈𝑥 𝛽 -𝐺 𝑥 𝑦̈𝑥 3 = (1 -𝑀 𝑥 1 )𝑦̈𝑥 𝛿 -𝐺 𝑥 𝑥 = {𝑑, 𝑞, 𝑜}[START_REF] Vyawahare | Design and analysis of standalone 4-wire, 4-leg inverter microgrid with unbalanced loads[END_REF]

Fig. 4 .

 4 Fig. 4. Block diagram of the GW based tracking controller.

Fig. 5 .

 5 Fig. 5. Influence of the cost function odq-axis parameters on the voltage THD evolution: (a) ko gain influence when gains kd and kq are set equal, (b) kd and kq gains influence when ko is set to 10.

Fig. 8 .

 8 Fig. 8. Block diagram of a conventional PI based FL-VSI control[45].

Fig. 9 .

 9 Fig.9. Obtained output voltage RMS and THD rate with open-loop, PI controller and closed-loop under balanced linear load and unbalanced nonlinear loads: (a) output voltage RMS (b) output voltage THD (c) output voltage VUF

Fig. 10 .Fig. 11 .

 1011 Fig.10. Obtained output voltage RMS under input voltage variation IV. EXPERIMENTAL VERIFICATION In order to validate the proposed control strategy, experiments are carried out. The control algorithm is implemented in a dSPACE (Micro-Auto-Box rapid

Fig. 11 .

 11 Fig.11. shows the steady state response of the load voltages and currents and their respective zooms under balanced resistive load before (open loop) and after activating the proposed control approach. The balanced resistive loads are set to Ra= Rb= Rc=15 Ω.When the proposed control is enabled the load voltage waveforms become purely sinusoidal and the voltage THD decreased from 8% (open loop) to 0.8%. Fig.12. shows the steady state response of the load voltages and currents and their respective zooms under unbalanced resistive load before (open loop) and after activating the proposed control approach. The considered unbalanced resistive loads are set as Ra=∞ Ω and Rb= Rc=15 Ω. It can be noticed that the load voltage are balanced when the proposed control is applied and the measured voltage THD is decreased from 8.5% (open loop) to 1.1%.

Fig. 12 .Fig. 13 .Fig. 14 .Fig. 14 .Fig. 15 .Fig. 16 .

 121314141516 Fig. 12. Experimental results: load voltages and currents during steady state with unbalanced resistive loads (Ra=∞ Ω and Rb= Rc=15 Ω) before and after activating the proposed control.

  . The voltage THD values are reported in Table

Fig. 17 .

 17 Fig.17. Experimental results: load voltages and currents during steady state with unbalanced nonlinear loads before and after activating the proposed control.TABLE 2 LOAD VOLTAGE THD FOR A THREE-PHASE THD (%) Phase a

Fig. 18 .

 18 Fig.18. Load voltage THD versus value of Inductor (L f ) under balanced linear load (blue color) and unbalanced nonlinear loads (red color).

Fig. 19 .

 19 Fig.19. Load voltage THD versus value of capacitor (C f ) under balanced linear load (blue color) and unbalanced nonlinear loads (red color). In Fig 19, the evolution of the load voltage THD versus the variations in Cf with balanced ((Ra= Rb= Rc=15 Ω, blue curve) and unbalanced (Ra=∞ and Rb= Rc=15 Ω, red curve) loading conditions, when the parameter L is set to 1 and Cf is changed from 20 uF to 100 uF. It can be seen that the THD evolution is less sensitive to the Cf superior 40 uF. The analysis of the fig 18 and 19 shows that the variations in Cf have more impact on the load voltages than the Lf variations. V. Conclusion In this paper a voltage control strategy based on a Flatness theory with a Grey Wolf tracking control (FB-GWC) has been proposed for a four-leg inverter with an output LC filter for use in Three-Phase Four-Leg Voltage Source Inverters. The proposed FB-GWC control scheme can be split into two subsystems: the flat model of the system and the GW tracking controller. The differentially flat model of the FL-VSI allows calculating the control inputs as differential functions of the flat outputs and their derivatives. The GW tracking controller provide the correction control effort used by the differentially flat model in such a way to ensure the desired trajectory

, the following load dynamics equations (1) and (2) can be obtained, where the filter line currents 𝒊 𝒇 and the output voltage 𝒗 𝒇 are considered as the state variables.

	𝒗 𝒊 = 𝒗 𝒇 + 𝑅 𝑓 𝒊 𝒇 + 𝐿 𝑓	𝑑𝒊 𝒇 𝑑𝑡	+ (𝑅 𝑛 𝒊 𝒏 + 𝐿 𝑛	𝑑𝒊 𝒏 𝑑𝑡	)	(1)
	𝐶 𝑓	𝑑𝒗 𝒇 𝑑𝑡	= 𝒊 𝒇 -𝒊 𝑳_𝒂𝒃𝒄

  2 , 𝑦̈𝑥 3 are intermediate input control variables.

	𝑦̈𝑥 𝛼 , 𝑦̈𝑥 𝛽 , 𝑦̈𝑥 𝛿	are	the	first	three	best	optimal
	solutions to distance,		𝑀 𝑥 1 = (2 𝑎 𝑥 𝑟 𝑥 𝑘 ) 2 -2 𝑎 𝑥 2 𝑟 𝑥 𝑘 𝑥 ∈
	{𝑑, 𝑞, 𝑜}, 𝑘 ∈ {1,2,3} , 𝑟 𝑥 𝑘 are random vectors which values
	are within [0, 1], 𝑎 𝑥 are linearly decreased from 2 to 0 over.
	Finally, by the use of the predefined intermediate control
	variables, the candidate dq-input control axes are calculated
	by: