Method for controlling a plurality of binary selectable components that contribute additively to an output quantity, in particular a voltage in a modulating electric battery

Abstract

A control method for controlling a system comprising a plurality of N binary selectable components, each contributing additively to an output quantity that the system must make coincide as closely as possible with a setpoint quantity. Each selectable component (Ck) is activated by a binary command uk. The method comprises a repetition of the steps: - acquiring a setpoint quantity, - acquiring at least one state (SOCk) of each selectable component, - determining a set of constraints to be respected, - using a Pontryagin Maximum Principle algorithm to determine adjoint states with a Hamiltonian quantity to be minimized, - using a linear integer mixed solver module to determine a minimum of the Hamiltonian quantity while respecting the set of constraints, in order to deduce an optimal control vector u* to be applied. Figure from the abstract: Fig. 1

Description

Title of the invention: Method for controlling a plurality of binary selectable components that contribute additively to an output quantity, in particular a voltage in a modulating electric battery

[0001] The present invention relates to a control method for controlling a plurality of selectable organs in a binary manner and contributing additively to an output quantity, in particular for example a voltage in a modulating electric battery.

[0002] The output quantity must follow an evolving setpoint over time.

[0003] More specifically, the present invention relates to a control method to control a plurality of electrochemical cells of an electric battery, selectable in a binary manner and contributing additively (series arrangement) to an output voltage of such a modulating electric battery.

[0004] We are particularly interested here in electric batteries with switchable cells, sometimes referred to in the trade by the acronym SWIBA (for Switched Battery).

[0005] It is important to note here that binary selection is not a continuous control. Indeed, in the possible control solutions, over time there will be jumps between 0 and 1 with a discontinuous effect. It generally follows that optimal computational solutions based on analytical calculations involving continuous or differentiable functions will not be directly usable in the present context.

[0006] When the number of selectively available organs (respectively, battery cells / modules) becomes large, the selection combinations quickly become very extensive, and the selection decision is no longer trivial. In practice, if the number of organs is N, the combinations evolve as 2N. If N > 20, the combinations already exceed one million, and if N equals 32, the combinations exceed one billion. At these scales, it is no longer reasonable to perform calculations without involving at least one powerful computing unit, especially for real-time decision-making.

[0007] A SWIBA switchable battery, i.e. modular or configurable, is a battery that can vary the output voltage depending on whether certain cells of the set of cells available in the SWIBA battery are used or not, the cells being arranged in a series configuration.

[0008] In known embodiments, when a modular battery has a certain number of cells, each of which can be activated or not, with a number of For cells of 30 or more, it is common to use permutation-based decision rules based on the previous usage time of each cell or on the measured voltages or respective charge states of these cells.

[0009] However, these rules do not take into account the actual state observed of each of the cells, such as for example the case of early degradation of a cell.

[0010] Furthermore, attempts have also been made to apply optimal analytical control methods, but these methods use real numbers and do not give good results if one tries to round the results of the real numbers to integers as is necessary for the on / off control of cell activation.

[0011] The inventors sought to propose a computational solution that could be implemented on a microcontroller of an embedded computer in a time of a few milliseconds or a few tens of milliseconds, in order to be able to run in real time the decision logic for selecting the battery cells.

[0012] To this end, a control method is proposed for controlling a system comprising at least a plurality of binary selectable components that are activated to contribute together additively to an output quantity, which the system must make coincide as closely as possible with a setpoint quantity, each selectable component being put into service (i.e., activated) by a binary command uk, the method comprising a step of defining a time horizon, and the method being characterized in that it comprises, according to a first periodicity, a repetition of the steps: a) acquire a setpoint value as the reference value for the output value, b) acquire at least one state of each selectable component, c) determine a set of constraints to be respected, including in particular a constraint of respecting the setpoint value for the output quantity, d) use a Pontryagin Maximum Principle type algorithm over the time horizon to determine adjoint states with a Hamiltonian quantity formulation H,

[0013] where the function ¢, expressed as a vector -, -, ÿv) is a function a generalized cost function representing one or more cost functions to be minimized and compliance with all constraints, e) Use a linear integer mixed solver module to determine a minimum of the Hamiltonian quantity while respecting the set of constraints, in order to deduce an optimal control vector, f) apply the optimal control vector, g) calculate gradients of the adjoint states of the Pontryagin Maximum Principle type algorithm for the next iteration.

[0014] For the record, 'Pontryagin' is the name of a Russian scientist also written in technical literature as Pontryagin or IIoHTpsrHH.

[0015] Thanks to these provisions, the software embodying said method can be implemented on an embedded computer microcontroller and can perform an iteration of steps a) to g) in a reasonable time for real-time execution, namely typically less than 50 ms, preferably less than 10 ms.

[0016] It should be noted that the time horizon in question here is much longer than the first periodicity. The time horizon extends from a time t0 (or '0') to a time tf (or T). The time horizon is denoted by the interval in brackets [0,T]. For a vehicle, this could be a driving cycle lasting several tens of minutes or even a few hundred minutes, for example, the driving cycle known as WLTC, which lasts 30 minutes.

[0017] We will also see that the set of constraints to be respected can be written, in line with the literature relating to optimal control, by a matrix relation (AX < b), A being a matrix, X and b being vectors, X representing the generalized state variables.

[0018] It is noted that each selectable component can be in a state that is in operation (i.e., 'sought' or 'activated') or, conversely, in a state that is 'not sought' or 'not activated'. There is no possible intermediate state between the two states activated and not activated. In other words, we have uk=0 or uk=l, and intermediate values ​​are not possible.

[0019] The Pontryagin Maximum Principle type algorithm is known by the acronym PMP used later in this document, and in technical field reference documents.

[0020] The mixed-integer linear resolver module is known by the acronym MILP (from the English Mixed-Integer Linear Programming) used later in this document and in technical field reference documents.

[0021] It should be noted that the number N of selectable organs can be at least equal to 10, or at least equal to 20, or even at least equal to 30, and even greater than 30. We will see later that the interest in using the present process increases with the number N.

[0022] The phrase "contribute together in an additive manner" must be understood broadly, which also covers "contribute together in a subtractive manner".

[0023] According to one aspect, the selectable components are electrochemical cells of a switchable-cell electric battery, and the output quantity is a voltage across the terminals of the electric battery, the selected electrochemical cells being arranged in series in the electric battery, the electric battery being intended to provide a setpoint voltage varying over time.

[0024] Each cell is switchable in the sense that it is activated, in series with the others, when it is activated, and it is isolated and bypassed when it is not activated or deactivated. This binary selection corresponds to a command denoted uk for cell k, k ranging from 1 to N.

[0025] A double electrical switch, or two logically coupled electrical switches, is provided to switch from the activated state to the inactivated state and vice versa.

[0026] The application of the optimal control vector u* (u*i to u*N) means that for each cell, the control uk is applied to the corresponding electrical switch Tk.

[0027] According to one aspect, each selectable component is affected by static and non-static, or even dynamic, characteristics. Some characteristics can be known in advance and do not need to be measured or estimated in real time as a function of time. On the other hand, the non-static characteristics must be constantly re-evaluated, i.e., measured by sensors on the actual system or alternatively extracted from the output of a dynamic behavioral model.

[0028] According to one aspect, the static characteristics include the nominal cell capacity (QQk) and the non-static characteristics include the voltage (vk), the state of charge (SoCk), the capacitance loss (Qlossk), the internal resistance (rk).

[0029] According to one aspect, concerning the Hamiltonian quantity H, the generalized cost function includes a term that aims to minimize a sum of the resistances of the activated cells, a term that aims to minimize a drop in the state of charge of each cell and / or the complete battery, a term that aims to minimize an aging rate of each cell and / or the complete battery.

[0030] The weighting between the three criteria stated above can be done by means of Lagrange multipliers as is known in the technical field of optimal control calculation.

[0031] It is noted that it is not excluded to use other criteria to be minimized, on the one hand for the case of the battery and on the other hand for the general case of a system with multiple selectable components.

[0032] According to one aspect, the Hamiltonian quantity H can be written as follows: v^N VN ' i \ v^N H = L, ( rtuk ) + Lj Mi ( qqJ-Uk + Lj Mk-f2 ( h Qlossk).uk

[0033] with rk: electrical resistance of the cell of rank k, Xk: adjoint state associated with the charge state of the cell denoted q or SoC, Qk • time derivative of the cell's charge state k = . state assistant associated with the loss of capacity of the k-rank cell QIOSS] : time derivative of the k-cell capacity loss = f2 (i, Qlossk).ukso = rk + xk.ft( + (i, Qlossk)

[0034] QQk the nominal capacity of the k-rank battery.

[0035] It must be understood that the functions fi and f2 can be arbitrary, the example The information provided in the description is in no way exhaustive.

[0036] According to one aspect, the setpoint quantity Vreq and the output quantity Vout conform to a constraint formalism written as follows: voutVmin and vout < Vmax with yr — v'out Aq

[0037] Vmin and Vmax being acceptable limits, i.e. Vmin = Vreq — AV accuracy and Vmax ~ Vreq AVaccuracy, AVaccuracy being a configurable tolerated deviation.

[0038] This expression of the constraint on the output quantity allows us to take into account the discretization involved by the binary control and the on / off mode activation of the selected cells.

[0039] The parameterization of AVaccuracy allows adjusting the desired precision and consequently the computation time that will be used (the more tolerant one is, the faster the algorithm can be carried out).

[0040] In other words: V N ^min = Vreq - AVaccuracy — Vout = JL] uk'Vk — Vmax = Vreq + AVaccuracy

[0041] According to one aspect, the first periodicity is between 0.5 Hz and 4 Hz. According to one example, the first periodicity may be close to 1 Hz. For example, the first periodicity may correspond to a recurrence of steps a) to g) every second.

[0042] Thus, the decision is repeated every second, to determine which cells should be involved in supplying the output voltage.

[0043] The calculation time is on the order of a few milliseconds or a few tens of milliseconds; on a standard embedded serial computer, used in automotive applications, it is generally less than 50 ms.

[0044] The present invention also relates to a control system comprising at least one switchable cell electric battery and an electronic control unit in which the method as described above is implemented.

[0045] The present invention also relates to a vehicle comprising at least one switchable cell electric battery and an electronic control unit in which the method as described above is implemented.

[0046] The invention will be further detailed by describing non-limiting embodiments, and based on the accompanying figures illustrating variants of the invention, in which: - [Fig.1] schematically illustrates an example of the structure of a series-switchable cell battery, for which the method according to the present invention can be advantageously applied; - [Fig.2] schematically illustrates an example of a generalized system with a plurality of binary selectable organs, to which the method according to the present invention can be applied; - [Fig.3] illustrates an example of a schematic logic diagram in the context of the application of the proposed process to a battery with switchable cells; - [Fig.4] illustrates an example of a schematic logic diagram in the context of applying the proposed process to a generalized system with a plurality of selectable components; - [Fig.5] illustrates an example of a functional block diagram of the promoted process; - [Fig.6] shows an example of chronograms illustrating a result obtained with the proposed method.

[0047] In the various figures, the same reference numerals designate identical or similar elements. For the sake of clarity, some elements are not necessarily shown to scale.

[0048] As seen in [Fig.1], a switchable cell battery, identified as 10, has been shown. The switchable cell battery is composed of a plurality of N cells denoted Ck, namely Cb C2, C3, up to CN.

[0049] The Ck cells, k varying from 1 to N, are generally arranged in series mode.

[0050] Each cell Ck is switchable in the sense that it can be activated, in series with The others, when it is activated. Conversely, each cell Ck can be isolated and bypassed when it should not be involved and must be excluded from the circuit.

[0051] For each cell, there is an electrical switch, labeled Tk, which allows the cell to be activated or deactivated. For each electrical switch, labeled Tk, there is a double contact, namely a first contact that bypasses the cell, shown vertically in [Fig. 1], and another contact, shown horizontally in [Fig. 1], which connects the main branch to the positive terminal of the cell.

[0052] Another solution is to have two independent electrical switches controlled according to coupled logic. Only one switch (or one contact) is closed at a time, and there is always one of the switches (or contacts) that is closed at any given time.

[0053] Low-resistance power transistors (RDSon) can be used. The use of electromechanical relays is also possible.

[0054] Binary selection by activation of switch Tk corresponds to a logical command denoted uk for cell k.

[0055] The control vector denoted u (ui to uN) or even simply u corresponds to the state of the electrical switch Tk corresponding to each cell, it evolves over time, from the instant to to the instant tf, a time interval which is called the horizon denoted 'hor' in the present presentation.

[0056] The output quantity is a voltage across the terminals of the electric battery, denoted Vout. We can write: _ V' Yes-

[0057] Only activated cells contribute to the establishment of the output voltage.

[0058] The number of cells involved at a given time is me = yN. UK

[0059] According to a particular example, the number of cells N is 36. In general, N can be considered to be greater than 20, which makes the combinatorial possibilities particularly broad and the optimal solution difficult to identify.

[0060] Each cell Ck is characterized by its nominal capacity QQk, which is a static characteristic.

[0061] Moreover, as is generally known, each cell Ck is characterized by its current voltage vk and by its state of charge SoCk.

[0062] Among the non-static characteristics, we will also find on the one hand the loss of capacity noted Qlossk, (the actual capacity observed is therefore QQk - Qlossk) and on the other hand the internal resistance noted rk, representative of the state of health SoH and / or in practice of the aging of the cell.

[0063] Among the non-static characteristics, the cell temperature can also be used.

[0064] For the purposes of this disclosure, a cell may be a module of several unit cells arranged in parallel and / or in series.

[0065] In the illustrated example, these are lithium-ion electrochemistry-based cells, but the process can be applied to any type of battery cell electrochemistry.

[0066] Figure 2 illustrates a configuration similar in structure and problem to Figure 1, but which differs from a modular battery. According to a first example, each selectable element is an on / off controlled valve that can allow a fluid flow FLk to flow into a main fluid circuit where it is added to the other fluid flows from the other valves to give the total outgoing flow FLtot. The logic control denoted uk represents the open or closed state of the valve. rank k. Generally, the evolution of the command over time is noted uk(t) for t from 0 to T (time horizon 'hor').

[0067] According to another configuration, still based on [Fig.2], each selectable element is an electronic switch. The current that the switch allows to pass when it is conducting is added to the currents (FL1 FL2 FL3 etc.) that come from the other electrical switches, because the connection is made in parallel with regard to the current.

[0068] In all cases we have selectable organs which contribute additively to providing an output quantity which can be an electrical voltage or of another nature in the generalized case.

[0069] In general, it is noted that each selectable organ is affected by static characteristics and non-static characteristics.

[0070] The setpoint quantity Vreq and the output quantity Vout conform to a constraint formalism written as follows: Vmin > Vout and Vout < Vmin and Vout < Vmin. with Vreq = Vaccuracy and Vmax = Vreq + Vaccuracy,

[0071] kVaccuracy is a tolerated deviation, a consequence of the obligatory discretization involved by the binary.AVaccuracy command is configurable and / or calibrable.

[0072] Optimization criteria.

[0073] To optimize the overall use of the cells, we seek to minimize the decrease in the state of charge during consumption phases, or to maximize the increase in the state of charge during the charging phase, cell by cell and also globally for the complete battery. We set dSoCk _ 4¾ _ . _ / » j dt = dt = 4k = J}(QQj-uk with: state of charge of cell k, : time derivative of the state of charge / = current flowing in the main branch of the battery, (the same for all cells involved), QQk = nominal capacity of the k-rank cell, binary control as defined previously / j = specific function of the cell's load dynamics

[0074] An example of the fi function is given at the end of the description.

[0075] Furthermore, we seek to minimize the sum of the internal resistances of the assembly cells involved.

[0076] This is written using the following formulas. [0077> R=

[0078] with rk being the internal resistance of the cell k

[0079]

[0080]

[0081]

[0082]

[0083]

[0084]

[0085]

[0086]

[0087]

[0088]

[0089]

[0090]

[0091]

[0092]

[0093]

[0094]

[0095]

[0096]

[0097]

[0098] R is the total resistance seen across the battery terminals. It is also possible to try to maximize the use of cells with the highest nominal capacities. QQ = mean(QQk)=-^ Furthermore, we aim to minimize the aging effects on each cell. This is expressed using the following formulas. dt ~ ' QQ, with = Qlossk = / 2 ( i, Qlossk ) uk f2 = specific function of the cell's aging dynamics. uk as defined previously. An example of the function f2 is given at the end of the description. In view of the above, the Hamiltonian quantity H that we will seek to minimize therefore uses a term which aims to minimize the sum of the resistances of the activated cells, a term which aims to minimize the decrease in the state of charge of each cell and / or the complete battery, and a term which aims to minimize the aging rate of each cell and / or the complete battery. The 3 criteria above are weighted using Lagrange multipliers. The Hamiltonian quantity H can be written as follows: H=E* ( r^k ) + E'^4 / i ( ) uk+E^ / 2 ( i, Qiossk) uk H = Ej ( r k u k ) + L] 4¾ + E] ^Qlossk Advantageously, we note that the expression of the Hamiltonian is linear in uk and therefore we can factor uk. We then define a generalized cost function denoted which is representative of one or more cost functions to be minimized. The generalized cost function is expressed as a vector L / L 4 -, 4 -, k = rk + 4 / j (ÿ) + (i' Qlossk) The Hamiltonian quantity H can therefore be written in vector form H = s , , with <$T being the transpose of the vector and u designating the command vector. Logic for determining the optimal order Advantageously, according to the present invention, a Pontryagin Maximum Principle (PMP) logic is used on the time horizon [0,T] to determine adjoint states (Xk,qk) with a Hamiltonian quantity H defined above and a modulus mixed integer linear (MILP) solver to determine the minimum of the Hamiltonian quantity.

[0099] Note that Pontryagin's principle, as its title states, concerns the search for a maximum, but it is of course also valid for finding a minimum, provided that the mathematical formulas have been adapted to the specific case. Here, we are looking for a minimum of the Hamiltonian quantity.

[0100] A class of optimization problems concerns the determination of extrema of functionals, that is, of functions of functions. Here, we seek to identify the extrema with respect to the control function uk(t) on the horizon [0,T]. uk is the system input, also called the control. Here, the peculiarity is that the control takes binary values.

[0101] The generalized state variables are Qloss^ with = SoC = state cell load. In a generic way, state variables are denoted by x.

[0102] Pontryagin's extrema conditions correspond to conditions on partial differential equations as is known in the technical field, therefore not described in detail here.

[0103] The reader may consult the reference work by Michael Ross "A Primer on Pontryagin's Principle in Optimal Control" ISBN 978-0984357116.

[0104] The solution proposed by Pontryagin uses adjunct states called “co-states” in the literature.

[0105] In the present formulas, the adjoint states in question are the vectors X and q, also written Xket qk with k varying from 1 to N.

[0106] The adjoint states in question are initialized in step 12 of the logic diagram in [Fig.3]. After each new computation cycle, the adjoint states are updated as symbolized by frame 19 in [Fig.3].

[0107] At step 11 of the logic diagram in [Fig.3], the control unit acquires the voltage setpoint Vreq over the entire prediction horizon to allow the initialization of costates (adjoint states).

[0108] In step 13 of the logic diagram of [Fig.3], the control unit acquires the voltage and current setpoints at each sampling instant t. The control unit also acquires for each cell its terminal voltage v and its state of charge q.

[0109] At step 14 of the logic diagram of [Fig.3], the control unit acquires, at each sampling instant t, for each cell the internal resistance r and the capacitance loss Qloss.

[0110] At step 15 of the logic diagram in [Fig.3], control unit 1 performs the calculation of the generalized cost function ¢.

[0111] At step 16 of the logic diagram in [Fig.3], the control unit calculates the lower and upper bounds of the target range for the output voltage Vout.

[0112] At step 17 of the logic diagram of [Fig.3], the control unit calculates all the constraints in the form of a matrix equation, of the type Ax < b.

[0113] At step 18 of the logic diagram in [Fig.3], the control unit runs the linear integer mixed resolver module.

[0114] Regarding the mixed integer linear resolver module, the intlinprog() function in Matlab can be chosen. Alternatively, the open source solutions known as "Branch&Bond" or "Simplex" can also be chosen, without excluding other equivalent solutions.

[0115] It is noted that the acquisition of system states can be carried out by means of sensors, i.e., by taking physical measurements on the battery or on the system under consideration. Alternatively, the acquisition of certain system states can be carried out by reading the outputs of a behavioral model that mathematically reproduces the behavior of the system based on other state information, and / or using behavioral equations of the system.

[0116] Turning to [Fig.4], we find a generalization of the principle set out in [Fig.3], the setpoint quantity Vreq acquired in step 11, can be a voltage, a flux, a current without limit of nature.

[0117] All comments made about [Fig.3] can be applied, mutatis mutandis, to [Fig.4]. The generalized states are identified by the letter x. The function f characterizes the time derivatives of each state.

[0118] The system states are measured or retrieved from behavioral models. Note that the adjoint state X can be a multidimensional entity, in particular a vector of dimension n, equal to the number of state variables to be considered.

[0119] The function g represents a cost function to be minimized. The control u is a vector of binary values, which forms the specific characteristic of the method presented here.

[0120] On [Fig.5], the control unit 1 receives inputs of voltage and current respectively Vreg and Ireg.

[0121] The control unit 1 calculates the optimal command u*k and transmits it to the battery 10.

[0122] The u*k commands are applied to the Tk switches of the battery cells 10. An illustrated example, the battery 10 delivers an output voltage Vout and a current lout to an electrical machine 2, optionally via an inverter.

[0123] The control unit 1 receives the battery states vk, qk, rk.

[0124] The control unit 1 includes a parameter database or calibration tables. This includes in particular the nominal capacity of each cell QQk, the definition of the behavioral modeling functions fl and f2.

[0125] Calibration tables may also include the ^Vaccuracy parameter discussed above.

[0126] Note the looping concerning the adjoint states Xket qk., which are updated at each iteration.

[0127] On the timing diagrams of [Fig.6], the upper portion marked 6A shows the evolution over time of the setpoint voltage as well as the output voltage Vout and the evolution over time of the current lout delivered by the battery.

[0128] The portion located just below, marked 6B, represents the evolution of the total apparent resistance R of the battery.

[0129] The portion located below, marked 6C, represents the evolution of the state of charge as well as the number of cells involved.

[0130] The lower portion of [Fig.6] labeled 6D illustrates the evolution over time of the activation state for each of the N=36 cells.

[0131] The dashed line at mid-height of the current graph (portion 6A) represents zero current (i=0). A positive current corresponds to a current leaving the battery, and a negative current corresponds to a regenerative phase with a current entering the battery. At time t1, the current switches from the positive to the negative domain, and the state of charge curve SoC shows that it passes through a minimum at this time. A little later, at time t2, the current switches back from the negative to the positive domain, and the state of charge curve SoC shows that it passes through a peak.

[0132] According to a non-limiting example, the voltage across each cell can be between 5 volts and 25 volts, for example preferably between 10 volts and 16 volts.

[0133] Benefits and applications

[0134] Advantageously, the proposed method leads to an equalization of the charge states over time, even if starting from an initial situation where the Sock charge states are very different between the different cells. This equalization is inherent to the method, and there is no need to resort to any particular permutation or prioritization logic.

[0135] The principle is applicable regardless of the number N, when considering increasingly larger numbers. The computational difficulty increases proportionally to the number N², which is far below the general combinatorics which increases as 2N.

[0136] Applications can thus be envisioned where N is greater than 50, for example N equal to 100, or even more.

[0137] The input quantity can be an electrical power to be delivered and not just an electrical voltage, in which case the optimization mechanism will also determine the pair u and i corresponding to the setpoint power to be delivered.

[0138]

[0139] Example of Function f ± We can choose for fi: f — _ —J—

[0140]

[0141] with i = current expressed in [ Amps ] Q, = cell capacity k expressed in [ Ah] K

[0142]

[0143] SoCk expressed between [ 0 ... 1 ] without units giving f; in [sec1]

[0144]

[0145]

[0146] Example of Function f 2 We can choose for f2: f _K WSqQ , M lo IVa.y, A / N! ATL «aUS il 2 H (1+A^.Qlos^) /

[0147]

[0148]

[0149] with i = current expressed in [ Amp ] Jtab = table function of the SoC, typically between [0.001 and 0.05] Kq, A and mq = constants, equal for example: Kq = 4.e26, Aq = 200, and mq = 4

[0150]

[0151]

[0152] QlosS}.: loss of cell capacity k expressed in [Asec] SoHk expressed between [ 0... 1] without units giving f2 in [Asec / sec].

Claims

Demands

1. A control method for controlling a system comprising at least a plurality of binary selectable components that are activated to contribute together additively to an output quantity, which the system must make coincide as closely as possible with a setpoint quantity, each selectable component (Ck) being activated by a binary control uk, the method comprising a step of defining a time horizon, and the method being characterized in that it comprises, according to a first periodicity, a repetition of the steps: a) acquiring a setpoint quantity (vreq) as a setpoint for the output quantity (v), b) acquiring at least one state (qk, SOCk) of each selectable component, c) determining a set of constraints to be respected (AX < b), including in particular a constraint of respecting the setpoint quantity for the output quantity,d) use a Pontryagin Maximum Principle (PMP) algorithm over the time horizon to determine adjoint states (Xk,qk) with a Hamiltonian quantity formulation H, where the function ¢, expressed as a vector (¢^ ¢-,, -, ¢^ -, ¢^ is a generalized cost function representing one or more cost functions to be minimized and the constraint set, e) use a mixed integer linear solver module (MILP) to determine a minimum of the Hamiltonian quantity while respecting the constraint set, to deduce an optimal control vector u* (u*i to u*N), f) apply the optimal control vector u*, g) calculate gradients of the adjoint states of the Pontryagin Maximum Principle algorithm for the next iteration.

2. A method according to claim 1, characterized in that the selectable elements are electrochemical cells of a switchable-cell electric battery, and the output quantity is a voltage across the terminals of the electric battery, the selected electrochemical cells being arranged in series in the electric battery, the electric battery being intended to provide a setpoint voltage varying over time.

3. A method according to any one of claims 1 to 2, characterized in that each selectable element is affected by static and non-static characteristics.

4. A method according to claim 3 when it depends on claim 2, wherein the static characteristics include the nominal cell capacity (Qk) and the non-static characteristics include the voltage (vk), the state of charge (SOCk), the capacitance loss (Qlossk), the internal resistance (rk).

5. A method according to claim 4, wherein, with respect to the Hamiltonian quantity H, the generalized cost function <e>includes a term that aims to minimize a sum of the resistances of the activated cells, a term that aims to minimize a drop in the state of charge of each cell and / or the complete battery, and a term that aims to minimize an aging rate of each cell and / or the complete battery.

6. A method according to claim 5, wherein the Hamiltonian quantity H is written as follows: H = Lj ( rkuk ) + L1 4 / ) ( 0¾ J -¾+A ^- / 2 (K Qlossk ) .uk rk ; electrical resistance of the k-th cell : adjoint state associated with the cell's state of charge, denoted q or SoC $k : time derivative of the cell's state of charge k _ y ; adjoint state associated with the cell's capacity loss of rank k Qloss : time derivative of the cell's capacity loss kk = f2 (i,Qlossk)uk QQk being the nominal capacity of the k-th battery, therefore 4 = + W, ( 4 ) + V' <4 Oss* ' ' n. 1 y ! KL

7. A method according to any one of claims 1 to 6, wherein the setpoint Vreq and the output Vout conform to a constraint formalism written as follows: Vout — min and Vont — Vmax with , Vmin and Vmax being acceptable bounds, i.e. Vmin = V req — lux tiracy C{ Vmux — Vreq+ AVaccwacy, AVaccuracy being a parameterizable tolerated deviation.

8. A method according to any one of claims 1 to 7, wherein the first periodicity is between 0.5 Hz and 4 Hz.

9. Control system comprising at least one switchable cell electric battery and an electronic control unit (1) in which the method according to any one of claims 1 to 8 is implemented.

10. Vehicle comprising at least one switchable cell electric battery and an electronic control unit (1) in which the method according to any one of claims 1 to 8 is implemented.< / e>

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