CONTROL SYSTEM FOR AN ACTIVE BRIDGE CONVERTER WITH HYBRID POWER SUPPLY, METHOD AND DEVICE FOR IT

Description

FIELD OF INVENTION

[0001] The present invention relates to a control system for a hybrid-powered, active multi-bridge converter. The invention also relates to an associated control method and converter. TECHNOLOGICAL BACKGROUND OF THE INVENTION

[0002] The integration of renewable energy sources and energy storage systems into electronic power applications has generated interest in multiport converters.

[0003] A specific topology of such converters is that of MAB converters, the acronym MAB referring to the English name "Multi-Active Bridge" literally meaning "Multiple active bridges".

[0004] An example of the use of such a converter is shown on the figure 1 where the converter interacts with the grid, a battery, a load (resistor in this figure) and a solar panel.

[0005] This shows that such a structure has the advantage that the production, consumption and storage of electrical energy can be carried out in one place.

[0006] Furthermore, MAB converters exhibit intrinsic galvanic isolation because the transformer connects the ports via their respective windings. This is important for enabling the use of converters with power and load sources exhibiting significant differences.

[0007] It is known to use MAB converters with voltage ports.

[0008] However, for certain uses, particularly in the case of photovoltaic panels, it is desirable to use current-fed bridges, but this leads to a hybrid power supply converter where overvoltage control is tricky.

[0009] US patent 2008 / 212340 A1 states that, to convert a first DC voltage into a second DC voltage, a first bridge circuit within a power converter is controlled to convert the first DC voltage into a first AC voltage. The first AC voltage is then converted into at least one second AC voltage. Each second AC voltage is converted back into a DC voltage by its respective bridge circuits. To increase the efficiency of the power converter, the power converter switches are controlled to operate in low-loss switching mode. The duty cycle of each AC voltage is thus controlled. In one embodiment, the circuit is regulated so that the value of a time integral of the voltage over half a cycle of each AC voltage is substantially equal. SUMMARY OF THE INVENTION

[0010] Therefore, there is a need for a regulation system for a hybrid-powered active multi-bridge converter to ensure proper operation of the converter.

[0011] To this end, the description describes a regulation system for a hybrid-powered active multi-bridge converter, the active multi-bridge converter comprising: an input current port comprising an H-bridge switch for which a reference switch is defined, the reference switch being called the first reference switch and being controlled by a first control law, the first control law having a first duty cycle, at least one output voltage port comprising an H-bridge switch for which a reference switch is defined, each reference switch being called the second reference switch and being controlled by a respective second control law, each second control law having a predefined second duty cycle and a phase shift relative to the first control law, a transformer having windings, each winding being connected to a port by a respective isolation interface, the control system being suitable for regulating the active multi-bridge converter to a setpoint, the setpoint comprising a setpoint current value for the current port and at least one setpoint voltage value for a voltage port, the control system comprising: a unit of measurement for measured values, the measured values ​​including the current of the current port and the voltage of each voltage port, a first control unit for the first reference switch, the first control unit comprising: a first subunit for calculating a first desired initial value, the first desired initial value being a desired value for the first duty cycle equal to the sum of the result of a first calculation function applied to the setpoint values ​​and a safety margin, a first subunit for determining the difference between the measured current value and the setpoint current value, to obtain a determined current difference, a first correction subunit for converting the determined current difference into a first correction value for the first duty cycle by applying a first conversion function,a first subunit for adding the first desired initial value and the first correction value, to obtain a first candidate value, a first adjustment subunit for adjusting the first candidate value to obtain a first value to be applied between two first extreme values, the lowest first extreme value being the result of the first calculation function applied to the measured values, a first application subunit for the first control law having as its duty cycle the first value to be applied, and for each voltage port having a setpoint voltage value, a second control unit for the second reference switch of the voltage port considered, each second control unit comprising: a second calculation subunit for a second desired initial value,the second desired initial value being a desired value for the phase shift equal to the result of a second calculation function applied to the first desired initial value, a second subunit for determining the difference between the measured voltage value for the considered voltage port and the setpoint voltage value for the considered voltage port, to obtain a determined voltage difference, a second correction subunit for converting the determined voltage difference into a second correction value for the phase shift by applying a second conversion function, a second subunit for adding the second desired initial value and the second correction value, to obtain a second candidate value, a second adjustment subunit for adjusting the second candidate value to obtain a second value to be applied between two extreme second values,the second lowest extreme value being the result of the second calculation function applied to the first desired initial value, and a second subunit for applying the second control law having the phase shift of the second value to be applied.

[0012] In specific embodiments, the control system has one or more of the following characteristics, taken individually or in all technically possible combinations: Each adjustment subunit is designed to apply the unchanged candidate value when the candidate value is between the two extreme values, and otherwise the extreme value closest to the candidate value. The second highest extreme value is the result of a third calculation function applied to the first desired initial value. At least one of the calculation functions, the second and third, is a linear function. Each correction subunit is a proportional-integral controller. Each conversion function is a first-order function. Each H-bridge has two midpoints. Each isolation interface has two lines connecting a respective midpoint to one end of the associated winding, one of the two lines having a resistor in series with an inductor.Each winding has turns; the first conversion function has a gain dependent on the measured voltage values, the desired initial value, the number of turns in each winding, and the inductances of the insulation interfaces. Each winding has turns; the second conversion function has a gain dependent on the measured current value, the desired initial value, the desired initial value, the number of turns in each winding, and the inductances of the insulation interfaces. Each winding has turns; the first calculation function (also dependent on the number of turns in each winding and the inductances of the insulation interfaces). An output voltage port is a bidirectional port. Each subunit of determination is a subtractor. The safety margin is a predefined value.

[0013] The description also describes a hybrid-powered active multi-bridge converter, the active multi-bridge converter comprising: an input current port operating at a current and comprising an H-bridge switch for which a reference switch is defined and operating at a current, the reference switch being called the first reference switch and being controlled by a first control law, the first control law having a first duty cycle, at least one output voltage port operating at a voltage, each voltage port comprising an H-bridge switch for which a reference switch is defined, each reference switch being called the second reference switch and being controlled by a respective second control law, each second control law having a predefined second duty cycle and a phase shift relative to the first control law, a transformer having windings, each winding being connected to a port by a respective isolation interface,and a regulatory system as previously described.

[0014] The description also proposes a method for regulating a hybrid-powered active multi-bridge converter, the active multi-bridge converter comprising: an input current port comprising an H-bridge switch for which a reference switch is defined, the reference switch being called the first reference switch and being controlled by a first control law, the first control law having a first duty cycle, at least one output voltage port comprising an H-bridge switch for which a reference switch is defined, each reference switch being called the second reference switch and being controlled by a respective second control law, each second control law having a predefined second duty cycle and a phase shift relative to the first control law, a transformer having windings, each winding being connected to a port by a respective isolation interface, The method being suitable for regulating the active multi-bridge converter to a setpoint, the setpoint comprising a setpoint current value for the current port and at least one setpoint voltage value for a voltage port, the method comprising the steps of: measurement of measured values, the measured values ​​including the current of the current port and the voltage of each voltage port, control of the first reference switch, the control step comprising: a first calculation of a first desired initial value, the first desired initial value being a desired value for the first duty cycle equal to the sum of the result of a first calculation function applied to the setpoint values ​​and a safety margin, a first determination of the difference between the measured current value and the setpoint current value, to obtain a determined current difference, a first correction converting the determined current difference into a first correction value for the first duty cycle by applying a first conversion function, a first addition of the first desired initial value and the first correction value, to obtain a first candidate value,a first adjustment to fine-tune the first candidate value to obtain a first value to be applied between two first extreme values, the lowest first extreme value being the result of the first calculation function applied to the measured values, a first application of the first control law having as its duty cycle the first value to be applied, and for each voltage port having a setpoint voltage value, control of the second reference switch of the voltage port considered, each control step of the second switch comprising: a second calculation of a second desired initial value, the second desired initial value being a desired value for the phase shift equal to the result of a second calculation function applied to the first desired initial value,a second determination of the difference between the measured voltage value for the considered voltage port and the setpoint voltage value for the considered voltage port, to obtain a determined voltage difference, a second correction converting the determined voltage difference into a second correction value for the phase shift by applying a second conversion function, a second addition of the second desired initial value and the second correction value, to obtain a second candidate value, a second adjustment to fine-tune the second candidate value, to obtain a second value to be applied between two extreme values, the lowest second extreme value being the result of the second calculation function applied to the first desired initial value, and a second application of the second control law with the second value to be applied as its phase shift.

[0015] In this description, the expression "specific to" means interchangeably "suited for", "adapted to" or "configured for". BRIEF DESCRIPTION OF THE DRAWINGS

[0016] Features and advantages of the invention will become apparent from the following description, given solely by way of non-limiting example, and made with reference to the accompanying drawings, in which: there figure 1 is a schematic representation of an example of the use of a hybrid-powered, active multi-bridge converter, the figure 2 represents an electrical diagram of an example of a hybrid-fed, active multi-bridge converter, the converter including a regulation system, the figure 3 represents a diagram of part of the regulatory system, the figure 4 represents a diagram of another part of the regulatory system, the figure 5represents the evolution of the input and output signals of the converter figure 2 in ideal operation, the figure 6 schematically illustrates the star-delta conversion, and the Figures 7 to 10 present simulation results obtained by the applicant. DETAILED DESCRIPTION OF PREFERRED IMPLEMENTATION METHODS

[0017] According to the example described, the hybrid-powered active multi-bridge converter 10 has three ports 20, 22 and 24, a transformer 26 and a regulation system 28.

[0018] In this sense, converter 10 is a triple active bridge converter.

[0019] Such a converter 10 is sometimes called a TAB converter, the acronym "TAB" referring to the corresponding English name "Triple Active Bridge".

[0020] Following the example of the figure 2, the converter 10 has an input current port 20, a first voltage port 22 and a second output voltage port 24.

[0021] The first voltage port 22 is here both an input port and an output port (bidirectional character).

[0022] Such a converter 10 can, for example, be used to connect a photovoltaic module to the input current port 20, a battery to the first voltage port 22 and a DC load to the second output voltage port 24.

[0023] In such a case, the input current reference of input current port 20 would correspond to the maximum power point tracking of the photovoltaic module. This tracking is more often referred to by the English abbreviation MPPT, which stands for "Maximum Power Point Tracker".

[0024] The power port 20 includes a power source 30 and a first H-switch bridge 32.

[0025] The current source 30 is capable of delivering a current noted I 1 for the next part.

[0026] The current source 30 comprises a voltage generator 34, a resistance R f and an inductance L f in series.

[0027] In each of the following notations, for ease of reading, the reference sign of a component may correspond to its value. Typically, the resistance of the current source 30 is denoted by the reference sign Rf and the value of the resistance is denoted Rf.

[0028] The voltage generator 34 produces a voltage V 1.

[0029] The voltage generator 34 has a positive pole and a negative pole.

[0030] The current source 30 is designed to operate at a current that determines the operating current of the current port 20.

[0031] Power port 20 is a unidirectional port.

[0032] This means that the power port 20 is only suitable for sending power to the transformer 26.

[0033] To simplify the notation, each H-switch bridge will be referred to as an H-bridge.

[0034] The first H-bridge 32 has four branches 32B1 to 32B4, a first end 32E1 and a second end 32E2 and two midpoints 32M1 and 32M2.

[0035] The first branch 32B1 extends between the first endpoint 32E1 and the first midpoint 32M1, the second branch 32B2 extends between the first endpoint 32E1 and the second midpoint 32M2, the third branch 32B3 extends between the first midpoint 32M1 and the second endpoint 32E2 and the fourth branch 32B4 extends between the second midpoint 32M2 and the second endpoint 32E2.

[0036] The first end 32E1 is connected to the inductance L f of the current source 30 and the second end 32E2 to the negative pole of the voltage generator 34.

[0037] As its name "switch bridge" indicates, each branch 32B1 to 32B4 of the first H 32 bridge respectively includes a switch T 1 to T 4.

[0038] Each switch here is a transistor.

[0039] Furthermore, since each switch in an H-bridge in this example operates in an on / off manner, the term "switch" will be used throughout this description to refer to the transistors. However, what follows is more generally applicable to a switch in place of a transistor.

[0040] In addition, each branch 32B1 to 32B4 of the first H-bridge 32 has two diodes, a diode DA1 to DA4 in parallel with the switch T1 to T4 and a diode D1 to D4 in series with the switch T1 to T4.

[0041] In addition, the first switch T 1 is defined as the reference switch for current port 20 and will be referred to as the first reference switch T 1 hereafter.

[0042] The first reference switch T1 of the current port 20 is controlled by a first control law LC1. The first control law LC1 has a first duty cycle denoted D1.

[0043] The fourth switch T4 is controlled by the same first control law LC1, while the second switch T2 and the third switch T3 are controlled by the same law, phase-shifted by 180° with respect to the first control law LC1. The 180° phase shift corresponds to a time shift of half the switching period TS / 2.

[0044] In this respect, it would be possible to apply what will be described by choosing as the first reference switch one of the switches T 2 , T 3 or T 4 .

[0045] The first voltage port 22 includes a second voltage source 36 and a second H-bridge 38.

[0046] The first voltage port 22 operates at a voltage, called the second voltage V2. This second voltage V2 corresponds here to the voltage delivered by the second voltage source 36.

[0047] The first voltage port 22 is a bidirectional port.

[0048] This means that the first voltage port 22 receives power from transformer 26 and delivers power through transformer 26.

[0049] The second H-bridge 38 has a similar structure to the first H-bridge 30 (the one at the port of current 20) with four branches 38B1 to 38B4, ends 38E1 and 38E2 and two midpoints 38M1 and 38M2.

[0050] In the case shown, each branch 38B1 to 38B4 of the second H-bridge 38 also has a transistor and a diode in parallel similarly to the branches of the first H-bridge 30 but does not have a diode in series.

[0051] For the rest, the transistor of a 38Bi branch of the second bridge in H 38 is noted T 2i.

[0052] The first end 38E1 of the second H-bridge 38 is connected to the positive pole of the second voltage source 36 and the second end 38E2 of the second H-bridge 38 is connected to the negative pole of the second voltage source 36.

[0053] In addition, the first switch T 21 is defined as the reference switch of the first voltage port 22, which will in the following be the second reference switch T 21 to avoid any confusion with the first reference switch T 1.

[0054] The second reference switch T 21 of the first voltage port 22 is controlled by a second control law LC 2.

[0055] The second control law LC 2 presents a second duty cycle noted D 2 and a phase shift, denoted second φ 2, compared to the first control law LC 1.

[0056] Next, the second cyclical report noted D 2 is fixed to a predefined value, chosen here in a non-limiting way as equal to 0.5.

[0057] The second voltage port 24 has a load 39 and a third H-bridge 40.

[0058] Load 39 includes a capacitor C3 in parallel with a resistor R L3.

[0059] The second voltage port 24 operates at a voltage, called the third voltage V3. This third voltage V3 corresponds here to the voltage across the resistor R L3.

[0060] The second 24V voltage port is a unidirectional port.

[0061] This means that the second voltage port 24 receives power from the transformer 26.

[0062] The third H-bridge 40 has a similar structure to the second H-bridge 38 (that of the first tension port 22) with four branches 40B1 to 40B4, ends 40E1 and 40E2 and two midpoints 40M1 and 40M2.

[0063] In the case shown, each branch 40B1 to 40B4 has the same components as branches 38B1 to 38B4 of the third bridge in H 38, namely a transistor and a diode in parallel.

[0064] For the rest, the transistor of a branch 40B1 to 40B4 of the third bridge in H 40 is noted T 3i.

[0065] The first end 40E1 of the third H-bridge 40 is connected to one terminal of the capacitor C3 and one terminal of the resistor R L3 so that the second end 40E2 of the third H-bridge 40 is connected to another terminal of the capacitor C3 and another terminal of the resistor R L3.

[0066] In addition, the first switch T 31 is defined as the reference switch of the second voltage port 24, which will subsequently be the third reference switch T 31 to avoid any confusion with the other reference switches T 1 and T 21.

[0067] The third reference switch T 31 of the second voltage port 24 is controlled by a third control law LC 3.

[0068] The third control law LC 3 presents a third duty cycle noted D 3 and a phase shift, denoted third phase shift φ 3 , compared to the first control law LC 1.

[0069] Next, the third cyclical report noted D 3 is fixed to a predefined value, chosen here in a non-limiting way as equal to 0.5.

[0070] The transformer 26 has windings 42, 44 and 46 connected by a core 47

[0071] Each winding 42, 44 and 46 is connected to a port 20 to 24 by a respective isolation interface 48 to 52.

[0072] More specifically, the transformer 26 has a first winding 42 connected to the current port 20 by a first isolation interface 48, a second winding 44 connected to the first voltage port 22 by a second isolation interface 50 and a third winding 46 connected to the second voltage port 24 by a third isolation interface 52.

[0073] Each winding is a set of turns extending between a first end marked XE1 (where X is the reference sign of the winding considered) and a second end XE2.

[0074] The first isolation interface 48 includes a first inductance L1 in series with a first resistance R1.

[0075] The first midpoint 32M1 of the first H-bridge 32 is connected to the first resistor R1 and the second midpoint 32M2 of the first H-bridge 32 is connected to the second end 42E2 of the first winding 42.

[0076] The second isolation interface 50 includes a second inductance L2 in series with a second resistance R2.

[0077] The first midpoint 38M1 of the second H-bridge 38 is connected to the inductance L2 and the second midpoint 38M2 of the second H-bridge 38 is connected to the second end 44E2 of the second winding 44.

[0078] The third isolation interface 52 includes a third inductance L3 in series with a third resistance R3.

[0079] The first midpoint 40M1 of the third H-bridge 40 is connected to the third inductor L3 and the second midpoint 40M2 of the third H-bridge 40 is connected to the second end 46E2 of the third winding 46.

[0080] In each of the isolation interfaces 48, 50, or 52, the resistance R1, R2, or R3 and the inductance L1, L2, or L3 are those of the associated windings 42, 44, or 46. As such, the resistance R1, R2, or R3 corresponds to a parasitic resistance.

[0081] In some embodiments, these resistances or inductances come from external components.

[0082] The regulation system 28 is designed to regulate the converter 10 with multiple active bridges to a setpoint.

[0083] In this case, the setpoint is a setpoint current value for the current of current port 20 and a setpoint voltage value for the second voltage port 24.

[0084] In fact, in the example described, the first voltage port 22 has no setpoint. The voltage of the first voltage port 22 is imposed by the voltage value V2 delivered by the second voltage source 36 and its current is imposed by the fact that the algebraic sum of the input and output powers of converter 10 is zero.

[0085] The setpoint current value is noted I 1 ,ref and the setpoint voltage value for the second voltage port 24 is noted V 3 ,ref .

[0086] The control system 28 ensures that at all times, the operation of the converter 10 does not damage any component of the converter 10.

[0087] The various conditions are detailed in the Appendix section.

[0088] The control system 28 comprises a measuring unit 54, a first control unit 56, a second control unit 58 and a third control unit 60.

[0089] The unit of measurement 54 is suitable for measuring several values ​​of the converter 10.

[0090] According to the example described, the measured values ​​include the current I 1 of the current port 20 and the voltage V 2 and V 3 of each voltage port 22 and 24.

[0091] The measuring unit 54 is connected to the other control units 56 to 60 to communicate the measured values ​​to them.

[0092] The first control unit 56 is designed to control the first reference transistor T1 using a first adapted control law LC1.

[0093] As seen on the figures 2 And 3, the first control unit 56 comprises a first calculation subunit 62, a first determination subunit 64, a first correction subunit 66, a first addition subunit 68, a first adjustment subunit 70 and a first application subunit 72.

[0094] The first calculation subunit 62 is used to calculate a first desired initial value, denoted D 1 ,eq .

[0095] To do this, the first calculation subunit 62 applies a first calculation function FC1 to the setpoint values ​​and then adds a safety margin to the result ε .

[0096] The first calculation function FC1 depends, moreover, on the number of turns of each winding 42 to 46 and the inductance of the insulation interfaces 48 to 52.

[0097] The result of the first calculation function FC1 is noted D 1 , min _ref .

[0098] The first calculation function FC1 is such that: D 1 , min _ ref = 2 L 1 I 1 , ref L b . V 2 , ref . n 1 n 2 + L c . V 3 , ref . n 1 n 3 . T s + 1 2 Or : n 1 is the number of turns in the first winding, 42. n 2 is the number of turns in the second winding, 44. n 3 is the number of turns in the third winding, 46. L b = 1 1 L 1 + 1 L 3 ′ L 2 ′ + 1 1 L 1 + 1 L 3 ′ , with L k ′ = n 1 n k 2 L k for k = 1, 2 or 3 L c = 1 1 L 1 + 1 L 2 ′ L 3 ′ + 1 1 L 1 + 1 L 2 ′ , And T s is the switching period corresponding to the inverse of the chosen conversion frequency fs1.

[0099] Once this value D 1 min _ref is obtained, he comes: D 1 , eq = D 1 , min _ ref + ϵ

[0100] According to the example described, the safety margin ε is a predefined value, for example set to 0.01.

[0101] According to a more elaborate variant, the safety margin ε takes into account other elements such as variations in voltage sources or uncertainty in the value of inductance.

[0102] The first subunit of determination 64 is suitable for determining the difference between the current value I 1 measured and the setpoint current value I 1 ,ref .

[0103] The first subunit of determination 64 is, here, a subtractor.

[0104] The first subunit of determination 64 thus obtains the determined current deviation, denoted Δ I 1 and which therefore verifies: Δ I 1 = I 1 , ref − I 1

[0105] The first correction subunit 66 is suitable for converting the determined current deviation Δ I 1 in a first correction value for the first duty cycle.

[0106] The first correction value is denoted Δ D 1 .

[0107] To obtain the first correction value Δ D 1 , The first correction subunit 66 applies a first conversion function to the determined current difference Δ I 1.

[0108] The first conversion function is denoted G i 1st.

[0109] The first conversion function G i 1r is here a first-order function.

[0110] More specifically, the first conversion function G i 1r exhibits a gain dependent on the measured values ​​for the voltages, the second desired initial value, the number of turns of each winding 42 to 46 and the inductance of the insulation interfaces 48 to 52.

[0111] According to the example described, the first conversion function G i 1r is such that: G i 1 r s = I 1 s D 1 s φ ^ 2 = 0 φ ^ 3 = 0 V ^ 3 = 0 = a s + c

[0112] Or : s denotes the Laplace variable, a = 2 L b L f . 1 − L a . n 1 n 2 . V 2 + 2 L c L f . 1 − L a . n 1 n 3 . V 3 , ref , The notation ^ signifies small variation around the equilibrium point. L a = 1 1 L 2 ′ + 1 L 3 ′ L 1 + 1 1 L 2 ′ + 1 L 3 ′ , And c = R f L f + 4 L 1 L f . 1 − L a . T s .

[0113] The first correction subunit 66 thus forms a proportional integral corrector.

[0114] Alternatively, it is also possible that the first correction subunit 66 uses a first correction function G i 1r2 more elaborate as proposed in the appendix

[0115] The first addition subunit 68 is suitable for obtaining a first candidate value D 1 ,c by adding the first desired initial value D 1 ,eq with the first correction value Δ D 1 .

[0116] It comes like this: D 1 , c = D 1 , eq + Δ D 1

[0117] The first adjustment subunit 70 is used to obtain a first value to be applied D 1, app.

[0118] To do this, the first adjustment subunit 70 adjusts the first candidate value D 1 ,cto obtain the first value to apply D 1, a .

[0119] This adjustment aims to ensure that the first value to be applied D 1, a is between two initial extreme values.

[0120] The first two extreme values ​​are noted D 1, min And D 1, max .

[0121] The first extreme value D 1 ,min is the result of the first calculation function FC1 applied to the measured values ​​according to the formula described above, that is to say, it comes from: D 1 , min = 2 L 1 I 1 L b . V 2 . n 1 n 2 + L c . V 3 . n 1 n 3 . T s + 1 2

[0122] According to the example described, the first extreme value D 1, max is set at the maximum value for the first duty cycle, namely 1.

[0123] The first adjustment subunit 70 is designed to give as the value to be applied the unchanged candidate value when the candidate value is between the two extreme values ​​and otherwise the extreme value closest to the candidate value.

[0124] Put another way, it comes from:

[0125] This yields a first control law to be applied, this first control law having as its duty cycle the first value to be applied.

[0126] The first application subunit 72 is specific to applying the first control law to be applied.

[0127] The second control unit 58 and the third control unit 60 respectively control the second reference switch T 21 and the third reference switch T 31.

[0128] The third control unit 60 being the control unit with the most sub-units because a setpoint is fixed for the third reference switch T 31, it is described first before the second control unit 58.

[0129] As seen on the figure 4 , the third control unit 60 comprises a third calculation subunit 82, a third determination subunit 84, a third correction subunit 86, a third addition subunit, a third adjustment subunit 90 and a third application subunit 92.

[0130] The third calculation subunit 82 is used to calculate a third desired initial value, denoted φ 3, eq .

[0131] To do this, the third calculation subunit 82 applies a second calculation function FC2 to the first desired initial value D 1 ,eq .

[0132] The second calculation function FC2 is here a function calculated from the mathematical model presented in the following part, at a certain point of operation.

[0133] As will appear, the value φ 3 ,eq is obtained by solving a system of two equations with two unknowns once the value of the first duty cycle is fixed.

[0134] The third subunit of determination 84 is used to determine the difference between the measured voltage value V 3 for the second output voltage port 24 and the setpoint voltage value V 3 ,ref for the second output voltage port 24.

[0135] The third subunit of determination 84 is, here, a subtractor.

[0136] The third determination subunit 84 thus obtains the determined voltage difference, denoted Δ V 3 and which therefore verifies: Δ V 3 = V 3 , ref − V 3

[0137] The third correction subunit 86 is specifically designed to convert the determined voltage difference Δ V 3 in a third correction value for the third phase shift.

[0138] The third correction value is denoted Δ φ 3 .

[0139] To obtain the third correction value Δ φ 3 , The third correction subunit applies a second conversion function to the determined voltage difference Δ V 3 .

[0140] The second conversion function is denoted G v 3rd.

[0141] The second conversion function G v 3r is here a first-order function.

[0142] More specifically, the second conversion function G v3r exhibits a gain dependent on the measured current value of the first desired initial value, the second desired initial value, the number of turns of each winding 42 to 46 and the inductance of the insulation interfaces 48 to 52.

[0143] According to the example described, the second transfer function G v3r is such that: G v 3 r s = V 3 s φ 3 s φ ^ 2 = 0 D ^ 1 = 0 = e s + d

[0144] Or : d = 1 C 3 R L 3 , And e = 4 πC 3 ⋅ n 1 n 3 ⋅ I 13 , R , eq . sin D 1 , eq . π + φ 3 , eq + I 13 , I , eq . cos D 1 , eq . π + φ 3 , eq + I 23 , R , eq . sin D 1 , eq . π + φ 3 , eq + I 23 , I , eq . cos D 1 , eq . π + φ 3 , eq R The subscript denotes the actual value of the first harmonic (of current here). I The subscript denotes the imaginary value. I 13, R , eq is the current flowing in the inductance L 13 of the figure 6 (see details with reference to this figure), and I 23, R , eq is the current flowing in the inductance L 23 of the figure 6 (see details with reference to this figure).

[0145] The third correction subunit 86 thus forms a proportional integral corrector.

[0146] The third addition subunit is capable of obtaining a third candidate value φ 3 ,c by adding the third desired initial value φ 3 ,eq with the third correction value Δ φ 3 .

[0147] It comes like this: φ 3 , c = φ 3 , eq + Δ φ 3

[0148] The third adjustment subunit 90 is specific to obtaining a third value to be applied φ 3 ,a .

[0149] To do this, the third adjustment subunit 90 adjusts the third candidate value φ 3, c to obtain the third value to apply φ 3, a .

[0150] This adjustment aims to ensure that the third value to be applied φ 3, a is between two extreme third values.

[0151] The two extreme third values ​​are noted φ 3 ,min And φ 3, max .

[0152] Each of the third extreme values φ 3 ,min And φ 3, max is the result of applying a respective affine function to the first desired initial value D 1 ,eq .

[0153] More specifically, the third extreme value φ 3, min is the result of the second calculation function FC2 applied to the first desired initial value D 1 ,eq , so that in this case: φ 3 , min = π D 1 , eq − 1 2

[0154] For the third extreme value φ 3, min is the result of a third calculation function FC3 applied to the first desired initial value D 1 ,eq . The third calculation function FC3 is such that: φ 3 , max = π 3 2 − D 1 , eq

[0155] The third adjustment subunit 90 is then appropriate to give as the value to be applied the unchanged candidate value when the candidate value is between the two extreme values ​​and otherwise the extreme value closest to the candidate value.

[0156] Put another way, it comes from: φ 3 , app = φ 3 , c si φ 3 , c ∈ φ 3 , min φ 3 , max φ 3 , app = φ 3 , min si φ 3 , c ≤ φ 3 , min φ 3 , app = φ 3 , max si φ 3 , c ≥ φ 3 , max

[0157] This results in a third control law to be applied, this third control law having as its duty cycle the third predefined value and as its phase shift the third value to be applied φ 3 ,a .

[0158] The third application subunit 92 is specific to applying the third control law to be applied.

[0159] As for the second control unit 58, compared to the other two control units 54 and 60, it operates in open loop.

[0160] This implies that the second control unit 58 comprises only a second calculation subunit 94, a second adjustment subunit 96 and an application subunit 98.

[0161] The functions of these second subunits 94, 96 and 98 are similar to those described for the corresponding third subunits (subunits 82, 90 and 92).

[0162] Because of its open-loop operation, the second control unit 58 does not have second subunits having the role of third subunits.

[0163] As explained in the demonstration section, the regulation system 28 thus ensures stable operation of the converter 10 for any operating point.

[0164] In addition, the regulation system 28 can be applied to other configurations of the converter 10.

[0165] Examples of such configurations are given in Part 2 of the appendix. APPENDIX 1 - GENERAL CASE 1.1 - Converter topology

[0166] The 10 converter is a hybrid DC-DC multiport converter featuring a DC-AC current switch on the current port side and two voltage inverters on the voltage port side.

[0167] The abbreviation "DC" refers to the corresponding English term "Direct Current," while the abbreviation "AC" refers to the corresponding English term "Alternating Current."

[0168] There figure 2This shows the presence of a temporary current source 26 at current port 20 (port 1) and H-bridges 38 and 40 on the first voltage port 22 (port 2) and the second voltage port 24 (port 3). Current port 20 represents a power source (e.g., a photovoltaic module), the first voltage port 22 (port 2) is bidirectional (e.g., battery system and electrical grid), and the second voltage port 2 (port 3) is a DC load.

[0169] Ports 20, 22, and 24 of this converter 10 are coupled due to the presence of an inductor on each isolation interface 48 to 52. This inductor can be either a leakage inductor from the transformer 26 alone or in series with an external inductor. Consequently, this converter 10 behaves as a multivariable system with multiple inputs and outputs (MIMO). In other words, changing a control parameter affects ports 20, 22, and 24 of the converter 10.

[0170] Controlling the power flow of this 10-volt converter can be achieved by regulating the input current. i 1 of the current port 20 and the output voltage V 3 of the second voltage port 24.

[0171] Because the system is a coupled 10 multiport converter, the algebraic sum of all input and output powers of the system should approximately equal 0 (or equal to the system losses).

[0172] Thus, the power flowing from and into the first voltage port 22 is imposed according to the following formula (neglecting the power stored in the magnetic core 47 of the transformer 26): P 2 = − P 1 − P 3

[0173] Or : P 1 is the power flow from and into current port 20, and P 3 is the power flow from and into the second voltage port 24.

[0174] The waveform of the AC signals from converter 10 circulating in windings 42, 44, 46 of transformer 26 are visible on the figure 5 as well as the control signals of the reference switches T1, T21, and T31.

[0175] The trapezoidal shape of the current's evolution i L 1 shown on the figure 5 corresponds to the current flowing in the first winding. The parameters v 2 and v 3 are respectively the AC voltages on the transformer side 26 of the voltage ports 22 and 24. D 1 is the duty cycle of the control of the switches of the first current port 20 (0.5 < D 1 < 1). φ 2 and φ3 are respectively the phase shifts (in radians) of the control signals of the reference switches T21 and T31 with respect to the reference switch T1. The duty cycles of the control signals of the voltage ports 22 and 24 are fixed at D 2 = D 3 = 50% on the operating mode, so the voltages v 2 and v 3 are the visible tensions on the figure 5 . T s designates the switching period ( fs is the switching frequency).

[0176] The shapes of the waveforms of the figure 5 are an approximation of the actual shape of the signals. In reality, the current i L 1 is not perfectly trapezoidal, as it is slightly affected by the switching of voltage ports 22 and 24 at instants t 1 and t 5 for the first voltage port 22 on one side and t 2 and t6 for the second voltage port, 24 on the other side.

[0177] There figure 6 shows the star-delta equivalence of the transformer winding circuits 26 at port 1, with v 2 = S 2. V 2 and v 3 = S 3 . V 3 . S 2 and S 3 respectively the switching functions of ports 2 and 3. n 1 , n 2 and n 3 respectively denote the number of turns of each winding 42, 44 and 46 of transformer 26.

[0178] In these equivalent circuits, the current port 20 is replaced by an equivalent voltage source. The expression for the voltage v 1 on this port at transformer 26 is detailed later. The voltage at point vx can be obtained using the following relation R1: v x = L a . v 1 + L b . v 2 ′ + L c . v 3 ′

[0179] Or: L a = 1 1 L 2 ′ + 1 L 3 ′ L 1 + 1 1 L 2 ′ + 1 L 3 ′ with L k ′ = n 1 n k 2 L k L b = 1 1 L 1 + 1 L 3 ′ L 2 ′ + 1 1 L 1 + 1 L 3 ′ , L c = 1 1 L 1 + 1 L 2 ′ L 3 ′ + 1 1 L 1 + 1 L 2 ′ , And the inductances of the equivalent delta circuit are calculated as follows: L ij = NA , ∀ i = j L i ′ + L j ′ + L i ′ L j ′ ∑ k ≠ i , j n 1 L k ′ , ∀ i ≠ j 1.2 - Operating Principle

[0180] For a given operating point, the operating cycle of this converter 10 is divided into four successive time intervals.

[0181] The first time interval groups together the instants t such that 0 ≤ t ≤ t 0 .

[0182] Right now t = 0, switches T1 and T4 are in the conducting state. Switches T2 and T3 are already in the conducting state (generally before t = 0 due to the previous implementation of the cycle). During this time interval, all switches on current port 20 are in the conducting state, which implies that v 1 = 0.

[0183] Thus, using relation R1, we get: v x = L b . v 2 ′ + L c . v 3 ′

[0184] At the voltage ports, switches T22, T24, T32, and T34 should be in the conducting state, so that: v 2 ′ = − V 2 . n 1 n 2 , v 3 ′ = − V 3 . n 1 n 3 et v x < 0

[0185] This allows the current i L 1 increases by - I 1 to I 1 in the interval. Neglecting resistance R 1 in series with the leakage inductance at port 1, the current i L 1 is expressed according to the following relation R2: i L 1 t = − v x L 1 t − I 1 = L b . V 2 . n 1 n 2 + L c . V 3 . n 1 n 3 L 1 t − I 1

[0186] Or: I 1 is the value of the input current i 1 from the current port 20 to the selected operating point.

[0187] This current is considered constant at each operating point since it is limited by the inductance L f , which is a value much greater than the value of the inductance L 1 .

[0188] When i L 1 is approaching the value I1, the current begins to flow more through diodes D1 and D4 and less through diodes D2 and D3.

[0189] HAS t = t 0 , diodes D2 and D3 switch to the blocked state.

[0190] The second time interval groups together the instants t such as t 0 ≤ t ≤ T s 2 .

[0191] During this time interval, the input current I 1 passes through T1, T4, D1 and D4 and i L 1 = I 1. Switches T2 and T3 transition to the blocked state between t 0 and t 1. Zero current switching (also referred to by the acronym ZCS, which stands for "Zero Current Switching"). The switching of voltage bridges 22 and 24 is then implemented to ensure that the current i L 1 of transformer 26 switches into the next time interval.

[0192] HAS t = t 1 , the H-bridge 38 of the tension port 2 tilt and v 2 ′ = V 2 . n 1 n 2 .

[0193] At t = t 2, the H-bridge 40 of the tension port 3 tilts and v 3 ′ = V 3 . n 1 n 3 .

[0194] Ideally, this does not impact the current value i L 1 since it is imposed by the current source 26 ( L f » L 1) .

[0195] In this model, the ideal case is considered. Consequently, the approximation is made that the current i L 1 is perfectly trapezoidal and the resistance R1 is negligible, which leads to the following expression for the AC voltage at port 1: v 1 = v x = L b . v 2 ′ + L c . v 3 ′ 1 − L a > 0

[0196] The third time interval groups together the instants t such as T s 2 ≤ t ≤ t 3 .

[0197] Right now t = T s 2 Switches T2 and T3 are put in the conducting state with zero current. As a result, all switches on current port 20 are again in the conducting state and the voltage v 1 = 0. The voltage at the star point becomes v x = L b . v 2 ' + L c . v 3 ' > 0, with v 2 ′ = V 2 . n 1 n 2 And v 3 ′ = V 3 . n 1 n 3 The current i L 1 has been decreasing since I 1 verse - I 1 and verify the following relation R3: i L 1 t = − v x L 1 t − T s 2 + I 1 = − L b . V 2 . n 1 n 2 + L c . V 3 . n 1 n 3 L 1 t − T s 2 + I 1

[0198] Right now t = t 3, diodes D1 and D4 are in the blocked state.

[0199] The fourth time interval brings together the instants t such as t 3 ≤ t ≤ T s .

[0200] The input current passes entirely through T2, T3, D2 and D3 in this time interval and i L 1 = -I 1 .

[0201] Right now t = t 4, Switches T1 and T4 are put in the conducting state with zero current.

[0202] At those moments t = t 5 and t = t 6 , Voltage ports 2 and 3 are switched respectively and v 2 ′ = − V 2 . n 1 n 2 And v 3 ′ = − V 3 . n 1 n 3 .

[0203] The AC voltage at port 1 is written as: v 1 = v x = L b . v 2 ′ + L c . v 3 ′ 1 − L a < 0

[0204] This cycle is then repeated for each switching period. T s . 1.3 - Operating conditions

[0205] To ensure that power transfer between ports occurs with smooth switching across the entire operating range for all bridges, while preventing overvoltages at port 1, three main conditions must be verified.

[0206] To avoid voltage surges and ensure smooth switching at port 1, the switches should not be in the blocked state until the transformer current has completely reversed. i L1 and the blocking of the diodes in series. Otherwise, these switches would block a significant portion of the current in the inductive section, causing a sudden overvoltage at port 1.

[0207] Such a condition can therefore be written as follows: t 4 ≥ t 3 → D 1 T s ≥ T s 2 + t 0

[0208] Or t 0 = 2 L 1 I 1 L b . V 2 . n 1 n 2 + L c . V 3 . n 1 n 3 with reference to relation R2.

[0209] This results in relation R4: D 1 ≥ 2 L 1 I 1 L b . V 2 . n 1 n 2 + L c . V 3 . n 1 n 3 . T s + 1 2

[0210] AC voltages at voltage ports v 2' and v 3' should be reversed after the current has completely reversed i L 1 and the blocking of the diodes and associated switches for port 1.

[0211] If v 2 and / or v 3 are overturned before, the current i L Port 1 may not be able to reverse, so no power can be exchanged between the ports. Furthermore, if the voltages v 2 and / or v3 are reversed between the blocking of the diodes and the associated switch; the diodes can then return to the conducting state, thus causing overvoltages at port 1. This condition also ensures smooth switching on the voltage ports, and the associated conditions can be written as: t 5 ≥ t 4 → D 1 T s 2 + φ 2 T s 2 π + T s 4 ≥ D 1 T s And t 6 ≥ t 4 → D 1 T s 2 + φ 3 T s 2 π + T s 4 ≥ D 1 T s

[0212] This results in the two conditions corresponding to the following relations R5 and R6: φ 2 ≥ π . D 1 − 1 2 And φ 3 ≥ π . D 1 − 1 2

[0213] It can be deduced from relations R2 and R3 that the slope of the current i L 1 during its repayment is proportional to -vx = - ( L b . v 2 ' + L c . v 3 ' ) .

[0214] Thus, the reversal of this current is guaranteed if both the AC voltages of ports 2 and 3 have the same sign (both negative for a positive slope and positive for a negative slope).

[0215] Furthermore, the slope of the current iL 1 will be maximized in this way. Thus, conditions R5 and R6 will always be met simultaneously.

[0216] It thus appears that these conditions depend on the operating point of converter 10, based on the fact that I 1, V 3, D 1, φ 2 and φ 3 vary depending on the desired power, making the control of this topology complex. 1.4 - Generalized average model of the 10-converter

[0217] The state equations of the system under study are the system corresponding to the following relation R7: L f di 1 dt = V 1 − V dc , 1 − R f . i 1 C 3 dV 3 dt = − V 3 R L 3 + S 3 ⋅ i 13 ⋅ n 1 n 3 + S 3 ⋅ i 23 ⋅ n 1 n 3 L 12 di 12 dt = v 1 − n 1 n 2 S 2 ⋅ V 2 − R 12 ⋅ i 12 L 13 di 13 dt = v 1 − n 1 n 3 S 3 ⋅ V 3 − R 13 ⋅ i 13 L 23 di 23 dt = n 1 n 2 S 2 ⋅ V 2 − n 1 n 3 S 3 ⋅ V 3 − R 23 ⋅ i 23

[0218] Or: V dc , 1 = v 1 pour 0 ≤ t ≤ T s 2 − v 1 pour T s 2 ≤ t ≤ T s , S 2 t = 1 pour t 1 ≤ t < t 5 − 1 pour 0 ≤ t < t 1 et t 5 ≤ t < T s , And S 3 t = 1 pour t 2 ≤ t < t 6 − 1 pour 0 ≤ t < t 2 et t 6 ≤ t < T s .

[0219] The average model commonly used for power electronic circuits takes into account the average values ​​of state variables to transform this discrete model into a continuous model. This averaging model cannot be implemented for a converter 10 as it results in a zero transformer current (since it is an AC-type variable).

[0220] Therefore, the generalized average model of this system is developed to study it as a continuous model while representing AC signals with better accuracy than the classical average model.

[0221] Thus, DC signals are represented by average values ​​(zero-order coefficient of Fourier series) and AC signals are represented by their fundamentals (first-order coefficient of Fourier series).

[0222] The k-th coefficient of the Fourier series of a variable x is denoted 〈 x 〉 k and it is complex and verifies the following relation R8: x k = x kR + j x kI

[0223] The large-signal model of the system can thus be derived from relation 7 to obtain the system corresponding to the following relation R9: d i 1 0 dt = V 1 L f − V dc , 1 0 L f − R f L f . i 1 0 d V 3 0 dt = − V 3 0 C 3 R L 3 + 2 C 3 . n 1 n 3 . S 3 1 R . i 13 1 R + 2 C 3 . n 1 n 3 . S 3 1 I . i 13 1 I + 2 C 3 . n 1 n 3 . S 3 1 R . i 23 1 R + 2 C 3 . n 1 n 3 . S 3 1 I . i 23 1 I d i 12 1 R dt = − R 12 L 12 . i 12 1 R + ω s . i 12 1 I + v 1 1 R L 12 − n 1 n 2 S 2 1 R L 12 . V 2 d i 12 1 I dt = − ω s . i 12 1 R − R 12 L 12 . i 12 1 I + v 1 1 I L 12 − n 1 n 2 S 2 1 I L 12 . V 2 d i 13 1 R dt = − R 13 L 13 . i 13 1 R + ω s . i 13 1 I + v 1 1 R L 13 − n 1 n 3 S 3 1 R L 13 . V 3 0 d i 13 1 I dt = − ω s . i 13 1 R − R 13 L 13 . i 13 1 I + v 1 1 I L 13 − n 1 n 3 S 3 1 I L 13 . V 3 0 d i 23 1 R dt = − R 23 L 23 . i 23 1 R + ω s . i 23 1 I + n 1 n 2 S 2 1 R L 23 . V 2 − n 1 n 3 S 3 1 R L 23 . V 3 0 d i 23 1 I dt = − ω s . i 13 1 R − R 23 L 23 . i 23 1 I + n 1 n 2 S 2 1 I L 23 . V 2 − n 1 n 3 S 3 1 I L 23 . V 3 0

[0224] Or: The index R denotes the real part of a complex number, the index I denotes the imaginary part of a complex number. S 2 0 = S 3 0 = 0 , S 2 1 R = 2 π cos D 1 π + φ 2 , S 2 1 I = − 2 π sin D 1 π + φ 2 , S 3 1 R = 2 π cos D 1 π + φ 3 , S 3 1 I = − 2 π sin D 1 π + φ 3 , v 1 1 R = 1 π . 2 L b 1 − L a . n 1 n 2 . V 2 . cos D 1 π + φ 2 + 2 L c 1 − L a . n 1 n 3 . V 3 0 . cos D 1 π + φ 3 + L b . V 2 . n 1 n 2 + L c . V 3 0 . n 1 n 3 . sin ω s t 0 1 − L a , v 1 1 I = 1 π . − 2 L b 1 − L a . n 1 n 2 . V 2 . sin D 1 π + φ 2 − 2 L c 1 − L a . n 1 n 3 . V 3 0 . sin D 1 π + φ 3 + L b . V 2 . n 1 n 2 + L c . V 3 0 . n 1 n 3 . cos ω s t 0 1 − L a − L b . V 2 . n 1 n 2 + L c . V 3 0 . n 1 n 3 . 1 1 − L a , And V dc , 1 0 = − 2 . L b . V 2 . n 1 n 2 + L c . V 3 0 . n 1 n 3 . D 1 1 − L a + 2 . L b . V 2 . n 1 n 2 + L c . V 3 0 . n 1 n 3 . 1 1 − L a − 2 L b 1 − L a . n 1 n 2 . V 2 . φ 2 π − 2 L c 1 − L a . n 1 n 3 . V 3 0 . φ 3 π + 4 L 1 1 − L a . T s . i 1 0 .

[0225] The large signal model can be represented in matrix form with: X ˙ = A . X + B . U Or : X = [〈 i 1 〉 0 〈 V 3 〉 0 〈 i 12 〉 1 R 〈 i 12 〉 1 I 〈 i 13 〉 1 R 〈 i 13 〉 1 I 〈 i 23 〉 1 R 〈 i 23 〉 1 I ] T< , And U = [ V 1 V2] T< .

[0226] The system control input parameters are the duty cycle D 1 and the phase shifts φ 2 and φ 3 .

[0227] The system control output parameters are 〈 i 1 〉 0 and 〈 V 3 〉 0 .

[0228] Therefore, the average equations obtained are non-linear.

[0229] Linearization must be performed at the operating point in order to use conventional linear controllers.

[0230] The small-signal model of this system is obtained by introducing small perturbations in the system variables at the operating point and using a Taylor series expansion, such that: x = x eq + x ^

[0231] Or: Variables marked with a "^" symbol represent small associated signals (perturbations around the operating point), and x eq denotes the value of 〈 x 〉 at the operating point which is sometimes also called the equilibrium point.

[0232] The disturbances in the voltage sources around the average values ​​can be neglected in this study ( V 1 ^ = V 2 ^ = 0 ).

[0233] This is primarily due to their slow variation compared to the rapid dynamics of the control system (e.g., the voltage of the photovoltaic panel, the battery system, or the electrical grid). The resulting linearized mathematical model has an order of 8. 1.5 - Reduced-order model of the converter 10

[0234] A reduced-order model of a system is a simplified model that can be more easily used in simulations. Furthermore, it makes the design of the system's controllers much simpler.

[0235] Consequently, it will be possible to recalculate the system controllers in real time when a change in the operating point occurs. However, the main drawback of order reduction is lower accuracy for the mathematical model.

[0236] Reduced-order mean modeling relies on separating the system dynamics in the frequency domain into two parts: low-frequency dynamics (slow variables) and high-frequency dynamics (fast variables).

[0237] After this separation, only the dominant dynamic of the system is taken into account for the study of the system's behavior.

[0238] For converter 10, the DC variables can be considered as slow variables and the AC variables as fast variables. In this case, the slow variables represent the input and output parameters of the system, while the fast variables represent the internal operation of converter 10.

[0239] Since the purpose of converter control 10 is to regulate input and output parameters, dominant low-frequency dynamics are retained and fast dynamics are ignored in the reduced-order model.

[0240] The two subsystems can thus be represented as follows, by separating the state vector X of the unreduced system into two parts: X = X s X f T Or : The subscript "s" denotes the slow-dynamic subsystem, the subscript "f" denotes the fast-dynamic subsystem. X s = [〈 i 1 〉 0 〈 V 3 〉 0 ] T< , And X f = [〈i 12 〉 1 R 〈 i 12 〉 1 I 〈 i 13 〉 1 R 〈 i 13 〉 1 I 〈 i 23 〉 1 R 〈 i 23 〉 1 I ] T<

[0241] Thus, the system of relation R9 becomes: X ˙ = A . X + B . U → X ˙ s = A ss . X s + A sf . X f + B s . U X ˙ f = A fs . X s + A ff . X f + B f . U Or: The matrices A ss , A sf , A fs and A ff are parts of the matrix A designated above, A ss being the gain matrix between the variables Ẋ s and X s , A sf the gain matrix between the variables Ẋ s and X f , A fs the gain matrix between the variables Ẋ f and X s and A ff the gain matrix between the variables Ẋ f and X f , and B s and B f are parts of the matrix B linking Ẋ s and Ẋ f to the input matrix U.

[0242] Matrices A ss , A sf , A fs , A ff , B s And B f are obtained using a rearrangement of the matrices A And B.

[0243] To obtain the reduced-order model, the fast-dynamic subsystem is first solved at a chosen operating point (equilibrium point) ( Ẋ f,eq = 0), assuming that the slow variables are constant and equal to their average values, that is: X s = X s , eq , so that 〈 i 1 〉 0 = I 1 ,eq And < V 3 〉 0 = V 3 ,eq

[0244] The average response X f,eq The fast-dynamic subsystem is thus obtained.

[0245] Next, for the slow-dynamic subsystem, the fast variables are replaced by the previously calculated average response. Thus X f = X f,eq .

[0246] The slow-dynamic subsystem is then linearized around the chosen operating point while ignoring the dynamics of the fast variables. This yields a reduced-order linearized model of the converter 10, whose expression is given by a relation R10 corresponding to the following two-equation system: d ι 1 ^ 0 dt = − R f L f − 4 L 1 L f . 1 − L a . T s . ι 1 ^ 0 + 2 L c L f . 1 − L a . n 1 n 3 D 1 , eq − 1 + φ 3 , eq π . V ^ 3 0 + 2 L b L f . 1 − L a . n 1 n 2 . V 2 + 2 L c L f . 1 − L a . n 1 n 3 . V 3 , eq . D ^ 1 + 2 L b L f . 1 − L a ⋅ n 1 n 2 . V 2 . φ ^ 2 π + 2 L c L f . 1 − L a . n 1 n 3 . V 3 , eq . φ ^ 3 π d V ^ 3 0 dt = − V ^ 3 0 C 3 R L 3 − 4 C 3 ⋅ n 1 n 3 . I 13 , R , eq . sin D 1 , eq . π + φ 3 , eq + I 13 , I , eq . cos D 1 , eq . π + φ 3 , eq + I 23 , R , eq . sin D 1 , eq . π + φ 3 , eq + I 23 , I , eq . cos D 1 , eq . π + φ 3 , eq . D ^ 1 − 4 πC 3 . n 1 n 3 . I 13 , R , eq . sin D 1 , eq . π + φ 3 , eq + I 13 , I , eq . cos D 1 , eq . π + φ 3 , eq + I 23 , R , eq . sin D 1 , eq . π + φ 3 , eq + + I 23 , I , eq . cos D 1 , eq . π + φ 3 , eq . φ ^ 3

[0247] It thus appears that the order of the mathematical model has been reduced from 8 to 2 using the reduced-order modeling technique. This simplifies the analysis of the dynamic behavior and the design of the converter 10 control.

[0248] Applying the Laplace transform to the R10 relation model leads to a reduced transfer function linking, on the one hand, the input current I 1 of port 1 to the cyclic ratio D 1 by a first transfer function G i 1r and on the other hand the DC output voltage V 3 of port 3 at the phase shift φ3 by a second transfer function G v 3rd ( s ) .

[0249] The first transfer function G i 1r is expressed according to the following relation R11: G i 1 r s = I 1 s D 1 s φ ^ 2 = 0 φ ^ 3 = 0 = a . s + b s + c s + d

[0250] Or : a = 2 L b L f . 1 − L a . n 1 n 2 . V 2 + 2 L c L f . 1 − L a . n 1 n 3 . V 3 , eq , b = a C 3 R L 3 − 8 C 3 . L c L f . 1 − L a . n 1 n 3 2 . D 1 , eq − 1 + φ 3 , eq π . I 13 , R , eq . sin D 1 , eq . π + φ 3 , eq + I 13 , I , eq . cos D 1 , eq . π + φ 3 , eq + I 23 , R , eq . sin D 1 , eq . π + φ 3 , eq + I 23 , I , eq . cos D 1 , eq . π + φ 3 , eq , And c = R f L f + 4 L 1 L f . 1 − L a . T s .

[0251] This expression of the first transfer function G i 1r can be further simplified by ignoring the dynamics of the tension V 3, that is to say, considering that this value V 3 does not vary much around its nominal value. This yields a first reduced transfer function of order 1 corresponding to the following relation R12: G i 1 r 2 s = I 1 s D 1 s φ ^ 2 = 0 φ ^ 3 = 0 V ^ 3 = 0 = a s + c

[0252] The second transfer function G v 3r is expressed according to the following relation R13: G v 3 r s = V 3 s φ 3 s φ ^ 2 = 0 D ^ 1 = 0 = e s + d

[0253] Or : d = 1 C 3 R L 3 , And e = 4 πC 3 . n 1 n 3 . I 13 , R , eq . sin D 1 , eq . π + φ 3 , eq + I 13 , I , eq . cos D 1 , eq . π + φ 3 , eq + I 23 , R , eq . sin D 1 , eq . π + φ 3 , eq + I 23 , I , eq . cos D 1 , eq . π + φ 3 , eq

[0254] The order of the first and second reduced transfer functions G v 3rd and G i 1r2 is independent of the number of ports of the converter 10.

[0255] Thus, the reduced-order model just described for a 10 hybrid-supplied converter can be generalized to an n-port MAB 10 converter, where 1 port is a current-supplied port while the other (n-1) are voltage-supplied ports.

[0256] These first and second reduced transfer functions are then still of the first order regardless of the number of ports of the 10 MAB converter, so that the voltages of the voltage supply ports do not vary much around the average values ​​at a certain operating point.

[0257] These reduced-order models can be used for converter analysis 10 and for designing closed-loop controllers. 1.6 - Control Strategy

[0258] Two parameters need to be controlled in this system (the output control parameters), namely the DC input current of port 1 and the DC output voltage. V 3 of port 3.

[0259] However, there are three input control parameters, and these are the duty cycle D 1 and the phase shifts φ 2 and φ 3 .

[0260] Thus, at a chosen operating point, there is an infinite combination of values ​​for the input control parameters that can yield the desired values ​​while respecting the relationships R4 to R6. The elements of such a combination will be called D 1 ,eq , φ 2 ,eq And φ 3 ,eq for input control parameters and I1 ,eq And V 3 ,eq for output control parameters.

[0261] Furthermore, from the R9 system, it can be noted that the output control parameters I 1 ,eq And V 3 ,eq depend on all input control parameters D 1 ,eq , φ 2 ,eq And φ 3 ,eq This shows that the studied converter structure 10 is coupled and that a change in one of the control variables has an effect on all ports.

[0262] One way to control a converter 10 is to make D 1,eq equal to a minimum allowed value D 1, min in relation R4 with a safety margin ε at each point of operation. It thus comes down to: D 1 , eq = D 1 , min + ϵ

[0263] The value of ε can be chosen arbitrarily, for example set to 0.01.

[0264] The corresponding values ​​of φ 2 ,eq And φ 3 ,eq are then calculated from the system of relation R9 at the chosen operating point.

[0265] Next, using relations R5 and R6, a minimum allowed value is obtained for φ 2 and φ 3.

[0266] The calculated values ​​of the three input control parameters are then sent to the system.

[0267] Adding a feedback loop to the converter control 10 allows the ports to be decoupled.

[0268] The approximations made to develop the mathematical model will lead to an error in the steady state if only the open control loop is applied.

[0269] A PI controller is therefore used for each of the control loops to suppress this error (see figures 3 And 4These controllers are calculated based on the reduced-order model presented earlier. Adjustment subunits 70 and 90 are added to ensure that conditions R4 to R6 are always satisfied.

[0270] In the case of figures 3 And 4 , the maximum allowed for the value of D 1 is noted D 1, max and is equal to 1.

[0271] Similarly, the maximum allowed for the values ​​of φ 2 and of φ 3 is noted φ max and is calculated as follows: t 5 ≤ T s et t 6 ≤ T s → φ max = π . 3 2 − D 1

[0272] The transfer function of the conversion subunit 66 is expressed as follows: C 1 s = K p 1 . T i 1 . s + 1 T i 1 . s

[0273] Using the first transfer function G i 1r2 of relation R12, a first-order transfer function is obtained.

[0274] He is chosen T i 1 = 1 c .

[0275] The closed-loop transfer function CLTF 1 thus becomes: CLTF 1 s = C 1 s G i 1 s 1 + C 1 s G i 1 s = K p 1 . a s + K p 1 . a = 1 τ 1 . s + 1

[0276] Or : τ 1 = 1 K p 1 . a is the time constant of the closed-loop system.

[0277] However, the response time of this closed-loop transfer function to obtain 95% of the reference current is such that t r 1.95% = 3 . τ 1.

[0278] The winning K p 1 is then chosen for the desired value of t r 1.95%, so this gain verifies: K p 1 = 3 a . t r 1 , 95 %

[0279] As for the transfer function of the conversion subunit 86, it is expressed as follows: C 3 s = K p 3 . T i 3 . s + 1 T i 3 . s

[0280] Using the second transfer function G v 3r of relation R13, a first-order transfer function is obtained.

[0281] He is chosen T i 3 = 1 d .

[0282] The closed-loop transfer function CLTF3 thus becomes: CLTF 3 s = C 3 s G v 3 s 1 + C 3 s G v 3 s = K p 3 . e s + K p 3 . e = 1 τ 3 . s + 1

[0283] Or : τ 3 = 1 K p 3 . e is the time constant of the closed-loop system.

[0284] However, the response time of this closed-loop transfer function to obtain 95% of the reference voltage is such that t r 3.95% = 3. τ 3 .

[0285] The winning K p 3 is then chosen for the desired value of t r 3.95%, so this gain verifies: K p 3 = 3 e . t r 3 , 95 % 1.6 - Experimental Results

[0286] There figure 7 shows the results of simulations of closed-loop current control I 1.

[0287] The values ​​of the parameters of the simulated converter 10 are given in Table 1 below. [Table 1] Settings Value in the simulation V 1 200 V V 2 400 V fs 20 kHz L f 0,016 H R f 10 mΩ L 1 83 µH R 1 10 mΩ L 2 83 µH R 2 10 mΩ L 3 230 µH R 3 10 mΩ C 3 100 µH R c 3 1 mΩ L m 8.3 mH n 1 100 turns n 2 83 turns n 3 124 turns P 3, nominal 3 kW (received) P 1, nominal 3.5 kW (delivered) P max (between two ports) 4 kW R L 3 120 Ω

[0288] The response time was chosen so that the values ​​of the PI1 controller parameters for relations R14 are such that t r 1.95% = 3 ms.

[0289] The observation of the figure 7 shows that the open loop response time is indeed equal to the chosen value of 3 ms, which validates the development of the mathematical model of the controller.

[0290] There figure 8 shows the results of the same simulation for closed-loop control of the output voltage V 3.

[0291] The response time was chosen so that the values ​​of the PI3 controller parameters for relations R15 are such that t r 3.95% = 10 ms.

[0292] The observation of the figure 8 shows that the open loop response time is indeed equal to the chosen value of 10 ms, which validates the development of the mathematical model of the controller.

[0293] THE figures 9 And 10 present the evolution of each of the voltage values v 1 , v 2 , v 3 and I L1 respectively obtained by simulation and theoretical.

[0294] Comparing the two figures clearly shows that the signals obtained by simulation conform to the theoretical forms.

[0295] The minimal difference between the characteristics chosen for the calculation of the controller parameters and the characteristics of the simulated responses is due to simplification assumptions that were made, including the first harmonic approximation of the AC signals and the order reduction of the model.

[0296] To improve the model's accuracy, higher-order harmonics could be considered. However, this leads to increased model complexity.

[0297] The model described provides the best compromise between accuracy and complexity. 2 - OTHER CASES

[0298] In the structure of converter 10, the coupling between ports 20, 22 and 24 is generated by the presence of the inductances of the isolation interfaces 48, 50 and 52.

[0299] However, the absence of an inductance on one of the ports is a special case that can be regulated with the same regulation strategy.

[0300] In such a case, the conversion functions without order reduction would be written: G i 1 r s = 2 n 1 n 2 V 2 L f s + R f L f + 4 L 1 L f T S And : G v 3 s = 4 b πC 3 s + R 3 L 3 − a b w s s 3 + 2 R 3 L 3 + 1 R L 3 C 3 s 2 + 2 R 3 L 3 2 + w s 2 + 2 R 3 R L 3 C 3 L 3 + 8 π 2 C 3 L 3 s + 8 R 3 π 2 C 3 L 3 + R 3 2 R L 3 C 3 L 3 2 + w s 2 R L 3 C 3 a = 2 L 3 n 3 n 2 V 2 cos φ 23 , b = − 2 L 3 n 3 n 2 V 2 sin φ 23 , And φ 23 being the phase shift between the voltages V 2 and V 3.

[0301] It should be noted here that in this case, only the voltage port without inductance (called the master port) is subject to the control constraints imposed by the current port. Therefore, the external phase shift of this port has minimum and maximum permissible values. The external phase shifts of the other voltage ports can have any calculated value to obtain a desired value for their DC output voltage.

[0302] Furthermore, the sources / loads connected to the ports can be changed. For example, load 39 can be replaced by a voltage source and port 3 can be made bidirectional.

[0303] Similarly, a load 39 can replace the voltage source of port 2.

[0304] In each of these cases, at least one voltage port is connected to a stable voltage source, the average value of which does not vary too much over time.

[0305] An electrical grid or a battery are examples of such a stable voltage source.

[0306] Finally, the mathematical model and control strategy proposed in this patent can be generalized to an MAB converter consisting of a total of n ports, n being any integer and one port being powered by current.

[0307] In such a case, the 10-port active multi-bridge converter includes the 20-port input current and at least one output voltage port.

[0308] Each output voltage port is associated with a control unit conforming to the control unit described with reference to the figure 4 , so that the control system 28 comprises (n-1) control unit(s).

[0309] In each case, the proposed converter can be used in domestic applications or in electric vehicles.

Claims

1. A regulation system (28) for a multi-active bridge converter (10) with hybrid supply, the multiple active bridge converter (10) comprising: - an input current port (20) comprising an H-shape switch bridge (30) for which a reference switch (T1) is defined, the reference switch (T1) being called the first reference switch (T1) and being controlled by a first control law (LC1), the first control law (LC1) having a first duty cycle (D1), - at least one output voltage port (22, 24) comprising an H-shape switch bridge (38, 40) for which a reference switch (T21, T31) is defined, the reference switch being called the second reference switch (T21, T31) and being controlled by a respective second control law (LC2, LC3), each second control law (LC2, LC3) having a second predefined duty cycle (D2, D3) and a phase shift (φ2, φ3) with respect to the first control law (LC1), - a transformer (26) having windings (42, 44, 46), each winding (42, 44, 46) being connected to a port (20, 22, 24) by a respective isolation interface (48, 50, 52), the regulation system (28) being apt to regulate the active multi-bridge active (10) to a setpoint, the setpoint comprising a setpoint current value (I1,ref) for the current of the current port (20) and at least one setpoint voltage value (V3,ref) for the at least one voltage port (22, 24), the regulation system (28) comprising: - a measuring unit for measured values, the measured values comprising the current (I1) of the current port (20) and the voltage (V2, V3) of each voltage port (22, 24), - a first control unit (56) of the first reference switch (T1), the first control unit (56) comprising: - a first subunit (62) for calculating a first desired initial value (D1,eq), the first desired initial value (D1,eq) being a desired value for the first duty cycle (D1) equal to the sum of the result of a first calculation function (FC1) applied to the set values (I1,ref, V3,ref) and a safety margin (ε), - a first subunit (64) for determining the difference between the measured current value (I1) and the setpoint current value (I1,ref), in order to obtain a determined difference of current (ΔI1), - a first correction subunit (66) apt to convert the determined difference of current (ΔI1) into a first corrective value (ΔD1) for the first duty cycle (D1) by applying a first conversion function (Gi1r), - a first addition subunit (68) for adding the first desired initial value (D1,eq) and the first corrective value (ΔD1), so as to obtain a first candidate value (D1,c), - a first adjustment subunit (70) suitable for adjusting the first candidate value (D1,c) so as to obtain a first value to be applied (D1,app), comprised between two first extreme values (D1,min, D1,max), the first lowest extreme value (D1,min) being the result of the first calculation function (FC1) applied to the measured values (I1, V2, V3), - a first application subunit (72) for applying the first control law having as duty cycle the first value to be applied (D1,app), and - for each voltage port (22, 24) having a desired voltage value (V3,ref), a second control unit (58, 60) of the second reference switch (T21, T31) of the voltage port (22, 24) considered, each second control unit (58, 60) comprising: - a second calculation subunit (82) of a second desired initial value (φ3,eq), the second desired initial value (φ3,eq) being a desired value for the phase shift (φ2, φ3) equal to the result of a second calculation function (FC2) applied to the first desired initial value (D1,eq), - a second determination subunit (84) for determining the difference between the measured voltage value (V2, V3) for the voltage port (22, 24) considered and the setpoint voltage value (V3,ref) for the voltage port (22, 24) considered, in order to obtain a determined difference of voltage (ΔV3), - a second correction subunit (86) apt to convert the determined difference of voltage (ΔV3) into a second corrective value (Δφ3) for the phase shift (φ2, φ3) by applying a second conversion function (Gv3r), - a second addition subunit (88) for adding the second desired initial value (φ3,eq) and the second corrective value (Δφ3), so as to obtain a second candidate value (φ3,c), - a second adjustment subunit (90) suitable for adjusting the second candidate value (φ3,c) so as to obtain a second value to be applied (φ3,app) between two second extreme values (φ3,min, φ3,max), the second lowest extreme value (φ3,min) being the result of the second calculation function (FC2) applied to the first desired initial value (D1,eq), and - a second application subunit (92) for applying the second control law having as phase shift (φ2, φ3) the second value to be applied (φ3,app).

2. The regulation system according to claim 1, wherein each adjustment subunit (80, 90) is apt to give as the value to be applied (D1,app, φ3,app), the candidate value (D1,c, φ3,c) unchanged when the candidate value (D1,c, φ3c)is comprised between the two extreme values (D1,min, D1,max, φ3,min, φ3,max) and otherwise the extreme value (D1,min, D1,max, φ3,min, φ3,max) closest to the candidate value (D1,c, φ3,c).

3. The regulation system according to claim 1 or 2, wherein the second highest extreme value (φ3,max) is the result of a third calculation function (FC3) applied to the desired first initial value (D1,eq).

4. The regulation system according to claim 3, wherein at least one of the second calculation function (FC2) and the third calculation function (FC3) is an affine function.

5. The regulation system according to any one of claims 1 to 4, wherein each correction subunit (66, 86) is a proportional integral corrector.

6. The regulation system according to any one of claims 1 to 5, wherein each conversion function (Gi1r, Gv3r) is a first order function.

7. The regulation system according to any one of claims 1 to 6, wherein each H-bridge (32, 38, 40) has two midpoints (32M1, 32M2, 38M1, 38M2, 40M1, 40M2) each isolation interface (48, 50, 52) including two lines connecting a respective midpoint (32M1, 32M2, 38M1, 38M2, 40M1, 40M2) with one end (42E1, 42E2, 44E1, 44E2, 46E1, 46E2) of the associated winding (42, 44, 46), one of the two lines including a resistor (R1, R2, R3) in series with an inductance (L1, L2, L3).

8. The regulation system according to claim 7, wherein each winding (42, 44, 46) has turns, the first conversion function (Gi1r) having a gain dependent on the measured values for the voltages (V2, V3), on the second desired initial value (φ3,eq), on the number of turns of each winding (42, 44, 46) and on the inductances (L1, L2, L3) of the isolation interfaces (48, 50, 52).

9. The regulation system according to claim 7 or 8, wherein each winding (42, 44, 46) has turns, the second conversion function (Gv3r) having a gain dependent on the measured current value (I1), on the desired first initial value (D1,eq), on the second desired initial value (φ3,eq), on the number of turns of each winding ( 42, 44, 46 ) and on the inductances ( L1, L2, L3) of the isolation interfaces ( 48, 50, 52 ).

10. The regulation system according to any one of claims 7 to 9, wherein each winding (42, 44, 46) includes turns, the first calculation function (FC1) further depending on the number of turns of each winding (42, 44, 46) and on the inductances (L1, L2, L3) of the isolation interfaces (48, 50, 52).

11. The regulation system according to any one of claims 1 to 10, wherein at least an input voltage port (22, 24) is a bi-directional port.

12. The regulation system according to any one of claims 1 to 11, wherein each determination subunit (64; 84) is a subtractor.

13. The regulation system according to any one of claims 1 to 12, wherein the safety margin (ε) is a predefined value.

14. A multi-active bridge converter (10) with hybrid supply, the multi-active bridge converter (10) comprising: - an input current port (20) comprising an H-shape switch bridge (30) for which a reference switch (T1) is defined and operates at a current, the reference switch (T1) being called the first reference switch (T1) and being controlled by a first control law (LC1), the first control law (LC1) having a first duty cycle (D1), - at least one output voltage port (22, 24), each voltage port (22, 24) comprising an H-shape switch bridge (38, 40) for which a reference switch (T21, T31) is defined, each reference switch being called the second reference switch (T21, T31) and being controlled by a respective second control law (LC2, LC3), each second control law (LC2, LC3) having a predefined second duty cycle (D2, D3) and a phase shift (φ2, φ3) with respect to the first control law (LC1), - a transformer (26) having windings (48, 50, 52), each winding (48, 50, 52) being connected to a port (20, 22, 24) by a respective isolation interface (48, 50, 52), and - a regulation system (28) according to any of claims 1 to 13.

15. A regulation method for a multi-active bridge converter (10) with hybrid supply, the multi-active bridge converter (10) comprising: - an input current port (20) comprising an H-shape switch bridge (30) for which a reference switch (T1) is defined, the reference switch (T1) being called the first reference switch (T1) and being controlled by a first control law (LC1), the first control law (LC1) having a first duty cycle (D1), - at least one output voltage port (22, 24) comprising an H-shape switch bridge (38, 40) for which a reference switch (T21, T31) is defined, the reference switch being called the second reference switch (T21, T31) and being controlled by a respective second control law (LC2, LC3), each second control law (LC2, LC3) having a second predefined duty ratio (D2, D3) and a phase shift (φ2, φ3) with respect to the first control law (LC1), - a transformer (26) having windings (42, 44, 46), each winding (42, 44, 46) being connected to a port (20, 22, 24) by a respective isolation interface (48, 50, 52), the method being apt to regulate the multiple active bridge converter (10) to a setpoint, the setpoint comprising a setpoint current value (I1,ref) for the current of the current port (20) and at least one setpoint voltage value (V3,ref) for the at least one voltage port (22, 24), the method comprising the steps of: - measurement of measured values, the measured values comprising the current (I1) of the current port (20) and the voltage (V2, V3) of each voltage port (22, 24), - control of the first reference switch (T1), the control step comprising: - a first calculation of a first desired initial value (D1,eq), the first desired initial value (D1,eq) being a desired value for the first duty cycle (D1) equal to the sum of the result of a first calculation function (FC1) applied to the setpoint values (I1,ref, V3,ref) and of a safety margin (ε), - a first determination of the difference between the measured current value (I1) and the setpoint current value (I1,ref), so as to obtain a determined difference of current (ΔI1), - a first correction converting the determined current deviation (ΔI1) into a first corrective value (ΔD1) for the first duty cycle (D1) by applying a first conversion function (Gi1r), - a first addition of the first desired initial value (D1,eq) and of the first corrective value (ΔD1), so as to obtain a first candidate value (D1,c), - a first adjustment suitable for adjusting the first candidate value (D1,c) so as to obtain a first value to be applied (D1,app) between two first extreme values (D1,min, D1,max), the first lowest extreme value (D1,min) being the result of the first calculation function (FC1) applied to the measured values (I1, V2, V3), - a first application ( 72 ) of the first control law having as duty cycle the first value to be applied (D1,app), and - For each voltage port (22, 24) having a set voltage value (V3,ref), controlling the second reference switch (T21, T31) of the voltage port (22, 24) considered, each step of controlling the second switch (T21, T31) comprising: - a second calculation of a second desired initial value (φ3,eq), the second desired initial value (φ3,eq) being a desired value for the phase shift (φ2, φ3) equal to the result of a second calculation function (FC2) applied to the first desired initial value (D1,eq), - a second determination of the difference between the measured voltage value (V2, V3) for the voltage port (22, 24) considered and the setpoint voltage value (V3,ref) for the voltage port (22, 24) considered, in order to obtain a determined difference of voltage (ΔV3), - a second correction converting the determined difference of voltage (ΔV3) into a second corrective value (Δφ3) for the phase shift (φ2, φ3) by applying a second conversion function (Gv3r), - a second addition of the second desired initial value (φ3,eq) and of the second corrective value (Δφ3), so as to obtain a second candidate value (φ3,c), - a second adjustment suitable for adjusting the second candidate value (φ3,c) so as to obtain a second value to be applied (φ3,app) between two second extreme values (φ3,min, φ3,max), the second lowest extreme value (φ3,min) being the result of the second calculation function (FC2) applied to the first desired initial value (D1,eq), and - a second application of the second control law having as phase shift (φ2, φ3) the second value to be applied (φ3,app).