POLICIES FOR CONTROLLING A VARYING SPEED INDUCTION MOTOR [0001] The present invention relates to the field of method for controlling electric machines such as motors or generators. [0002] Method for controlling machines such as motors and generators, are known in the art, in particular method for controlling induction motor or induction generator, more recently and more particularly air-cored machines are proposed. [0003] Induction machines, especially air-cored, may have the drawback of requiring high supply voltages. [0004] That is, there is a need for limiting the requirement of high supply voltages for motors and generators, in particular for induction motors and induction generators. [0005] An aim of the present invention is to overcome the disadvantages of the prior art mentioned above, and in particular to provide a method for controlling machines, able to at least reduce the requirement of high supply voltages and to improve the performances of the machine. [0006] A first aspect of the present invention relates to a method for controlling an induction-based machine (IM) or an air-cored resonant- induction machine (ACRIM), having a rotor and a stator, comprising the steps of: - retrieving an actual rotor speed ωr and a setpoint of rotor speed ωrSP, - setting a stator speed ω _s based on the actual rotor speed ωr, so as to stand the machine in an electrical resonance, - using a modified indirect field-oriented control algorithm (irFOC), with the set stator speed ω _s and the actual rotor speed ω _r as input, so that the actual rotor speed ω _r tends to the setpoint of rotor speed ωrSP. [0007] In other words, it concerns a method for controlling an induction-based machine (IM) or an air-cored resonant-induction machine (ACRIM), having a rotor and a stator, and equipped with capacitors, comprising the steps of: - retrieving an actual current rotor speed ^^ _^^ , and a setpoint of rotor speed, - selecting and setting a stator speed ^^ _^^ based on the actual current rotor speed ^^ _^^ , so as to realize a policy of machine control, - using a modified indirect field-oriented control algorithm, with the stator speed ^^ _^^ as input. [0008] This allows a reduction of the required voltage, in particular with the resonant policy. This eases the formulation of various policies, especially with the resonant policy described below, for which it also reduces the input voltage as the power factor can now be always 1. That is, this allows to design and manufacture a machine (motor or generator) with reduced voltage in use, being due to air-cored, lighter, and cheaper. In particular, for high voltage applications, this allows to decrease the requirement of voltage from e.g.800 V to 600 V (of course depending on the initial and selected power factor). That is, the policy is a way or manner to control the machine with a given aim. [0009] The inventors investigate the closed-loop control of motor, in particular an Air-Cored Resonant Induction Motor. The idea is to operate at resonance, which might allow for a good efficiency while keeping the supply voltage low. This is achieved by interpreting the resonance condition as a relation giving the stator velocity (or stator speed) as a function of the rotor velocity (or rotor speed), and feeding it to a suitably modified version of (indirect) Field-Oriented Control algorithm. [0010] Induction Motors (IM) have many advantages: they are dependable, rugged, low-maintenance, do not rely on rare-earth material, and also relatively cheap to manufacture. Nevertheless, for applications where power-to-weight matters, e.g. electric vehicles, they cannot compete with Permanent Magnet Synchronous Motors (PMSM). As a significant part of the IM weight is due to the iron core, it is appealing to get rid of it and consider air- cored motors. This has the extra benefit of eliminating iron losses. But this of course comes at a price: as the inductance values are much smaller due to the absence of ferromagnetic material, the supply voltage needs to be much higher. To circumvent the problem, the stator can be endowed with capacitors, so as to take advantage of resonance inductive coupling: this is the Air-Cored Resonant Induction Motor (ACRIM). [0011] The present invention is concerned not with seeking interesting static operating points for the ACRIM as known in the prior art (see Z. Jin, M. Iacchetti, A. Smith, R. Deodhar, Y. Komi, A. Abduallah C. Umemura, and K. Mishima, “Air-cored resonant induction machines: Comparison of capacitor tuning criteria and experimental validation,” IEEE Transactions on Industry Applications, vol. 57, no. 4, pp. 3595–3606, 2021), but with the dynamic control around such points. To this end, it is interpreted that the resonance condition as a relation giving the stator velocity ^^ ^^ in function of the rotor velocity ^^ _^^ , and feed it to a suitably modified version of (indirect) Field- Oriented Control (FOC) algorithm, also called irFOC. It is considered only the case where the rotor velocity, besides the currents, are measured, but this is also applicable to “sensorless”, providing some adaptations. [0012] The modified algorithm exchanges the role of stator speed and the rotor flux, with respect to the standard irFOC. [0013] The model and peculiarities of the ACRIM will be discussed hereafter, the resonance equation established, the resonance equation studied, and interpreted as the basis for a control policy. [0014] Advantageously, the setpoint of rotor speed, and a setpoint of rotor speed ^^ _{^^ ^^ ^^} is desired or imposed by the user (e.g. when the machine is used as a generator). [0015] Advantageously, the method comprises: - receiving an actual (mechanical) rotor speed ^^ _^^ from a speed sensor. The sensor may be an electrical sensor or a mechanical sensor. The rotor speed may also be estimated or deduced. [0016] Advantageously, the method comprises: - deducting an actual rotor speed ( ^^ _^^ ) from an actual mechanical rotor speed. [0017] Advantageously, the modified indirect field-oriented control algorithm is an indirect field-oriented control algorithm modified for receiving at least as input the stator speed ^^ _^^ . [0018] Advantageously, the modified indirect field-oriented control algorithm further accepts at least as input the setpoint of rotor speed (mechanical or electrical), and/or controller gains, and/or actual currents, and/or the actual rotor speed (mechanical or electrical). [0019] Advantageously, the modified indirect field-oriented control algorithm outputs at least a torque setpoint (as usual), a rotor flux setpoint, and a stator currents setpoint (on each dq components). [0020] Advantageously, the method further comprises the steps of setting a (current) stator speed at the setpoint or desired stator speed. [0021] Advantageously, in the electrical resonance, the reactance of the machine appears as suppressed. In other words, the policy of machine control is the electrical resonance for which the reactance of the machine appears as suppressed, in particular from a user point of view. The reactance is for an equivalent electrical circuit representing the electrical circuit of the machine (i.e. the rotor and the stator). [0022] This allows to reduce the requirement of voltage as mentioned above. [0023] Advantageously, the policy of machine control is realized for the actual rotor speed. [0024] It is to be noted that the link between the mechanical rotor speed ^^ _^^ and the actual (current) rotor speed ^^ _^^ is well known: with ^^ being the number of pairs of poles. Advantageously, the policy of machine control is realized for the actual current rotor speed between zero and a maximum current rotor speed. In other words, the actual rotor speed is imposed between zero and a maximum rotor speed. For example, for the policy with maximum efficiency, there is no specific limit. For the resonance policy, some preferred range of use may be present, depending on the type of machine and its parameters. [0025] In the present invention, the role of the magnetic rotor flux are exchanged with respect to the standard irFOC algorithm. [0026] Advantageously, the magnetic rotor flux ^^ _{^^ ^^} has a variable magnitude (as setpoint) dependent on a requested torque ^^ _{^^ ^^ ^^ ^^} , the actual rotor speed ^^ _^^ and the actual stator speed ^^ _^^ , and/or some known (i.e. predetermined, such as e.g. a rotor resistance ^^ _^^ , and/or the number of pairs of poles n) motor characteristics (that is to say, actually , which means that the magnetic rotor flux ^^ _{^^ ^^} is an internal output. In the initial (conventional) irFOC, the setpoint ^^ _^^ is kept constant or equivalently is adjusted using ^^ _^^ = Note that in the preceding notation ^^ ^^ [0027] In our invention, ^^ _^^ is imposed by the resonance; so we turn ^^ _{^^ ^^} or ^^ _^^ variable, still using the above condition. [0028] This allows to create the resonance 'everywhere', and to stand at the resonance. That is, this allows to place the machine at the resonance and to stay at the resonance, providing adjustment or regulation to stay nearby.

[0029] Advantageously, the stator speed OJ _S is defined from an arbitrary policy.

[0030] This allows the writing of various policies while keeping the same algorithm helps reducing the size of microprocessor or equivalent. The inversion of the role of (p _sp and a) _s , allows to write policies because a) _s , is an essential component of the motor regime and is not anymore limited by the choice of cf> _sp (via the setpoint of current I _d ).

[0031] Advantageously, the rotor can also be equipped with one or more capacitors. Advantageously, the stator can be equipped with one or more capacitors.

[0032] Advantageously, only the stator is equipped with one or more capacitors.

[0033] That is, when the stator is equipped with one or more capacitor, the modified indirect field-oriented control algorithm is modified accordingly. That is to say, the rotor and/or the stator can be equipped with one or more capacitors.

[0034] Advantageously, the machine is a motor or a generator.

[0035] A second aspect of the present invention concerns a method for controlling an induction-based machine (IM) or an air-cored resonant-induction machine (ACRIM), having a rotor and a stator, comprising the steps of: retrieving an actual rotor speed (a> _r ) and a setpoint of rotor speed (w _r5P ), - setting a stator speed (<z> _s ) based on the actual rotor speed (X) and possibly the torque setpoint (T _emSP ), so as to stand the machine in a policy of machine control,

- using a modified indirect field-oriented control algorithm (irFOC), with the set stator speed (<z> _s ) and the actual rotor speed (<i> _r ) as input, so that the actual rotor speed (a> _r ) tends to the setpoint of rotor speed (,^ _rSP ).

[0036] This allows to provide a method for controlling the machine with the above-mentioned advantages. There is the possibility to use the torque setpoint as mentioned above.

[0037] Advantageously, the policy of machine control is selected in the group consisting of a maximum efficiency policy, a maximum torque policy, or as constraints as a maximum voltage policy, or a maximum current policy.

[0038] This allows to provide different policies of machine control, so as to take the benefit of each. This also allows to improve the performances of the machine by mixing policies with respect to the context, e.g. maximum torque or resonant during acceleration, and maximum efficiency at steady points.

[0039] Advantageously, the stator speed is defined from an arbitrary policy.

[0040] Advantageously, the policy of machine control is an aggressivity policy wherein an aggressivity factor (a) is selected between 0 and 1 , 0 and 1 being included, and the aggressivity policy is a linear weighted interpolation of the maximum efficiency policy and the maximum torque policy. Both initial policies being constant slip policies, the resulting policy remains a constant slip policy (a)g = (1 — a) * o)™ ^{ax r]} + a * o)™ ^{aXl Te} ). [0041 ] This allows selecting a driving mode, or changing the behavior of the machine between accelerations and steady speed modes in order to maximize applications satisfactions.

[0042] Advantageously, the policy of machine control is a voltage barrier policy, wherein the voltage barrier is a voltage limit wherein a voltage of the induction-based machine or of the air-cored resonant-induction machine is limited by a used power supply of the induction-based machine or of the aircored resonant-induction machine.

[0043] Advantageously, the policy of machine control is a low torque setpoint policy. This policy allows to deal properly with low torque conditions, e.g. when close to stopping, or when the load is low in general and the speed setpoint is close. Such conditions require proper adjustments of the flux to avoid losing the control.

[0044] Advantageously, the actual rotor speed setpoint (d> _rSP ) is an input.

[0045] The main advantages of these policies are that they all fit nicely in our new modified version of the irFOC algorithm (the a) _s FOC algorithm), consisting overall in a new paradigm.

[0046] Other features and advantages of the present invention will appear more clearly from the following detailed description of particular non-limitative examples of the invention, illustrated by the appended drawings where: figure 1 represents schematically an ACRIM stator, figure 2 represents the resonance in the f _s - fr plane, figure 3 represents a close-up view of the resonance in motor mode, figure 4 represents a resonance policy in motor mode, figure 5 represents a locus of eigenvalues in motor mode and efficient useful zone, figure 6 represents a step response of a Pl-controlled fast subsystem, figure 7 represents disk-based stability margins of compensated transfer, figure 8 represents test scenario with velocities expressed in Hertz, figure 9 represents a block diagram schematically illustrating the method of control, figure 10 represents a speed scenario, figure 1 1 represents a selection of slip, figure 12 represents voltage limits in the <w _s (<w _r ) graph without capacitor, figure 13 represents voltage limits in the <w _s (<w _r ) graph with capacitor, figure 14 represents <w _s (<w _r ) graph for a voltage barrier policy, figure 15 represents <w _s (<w _r ) graph for the voltage barrier policy with intensity constraints, figure 16 represents a summary of adjustment. figure 17 represents torques.

[0047] Figure 1 represents schematically an ACRIM stator.

[0048] I. Generalities

[0049] The ACRIM stator is represented in figure 1 . It shows the added capacitors, here to each phase of the stator.

[0050] II. Model and peculiarities of the ACRIM

[0051 ] A. Model of the ACRIM

[0052] An ACRIM is in principle an induction motor, though with very different parameters, fit with a series capacitor C _s in each stator phase (represented 10a, 10b, 10c in figure 1 ), as illustrated in figure 1. The motor is also equipped with resistances 1 1 a, 1 1 b, 11 c and with inductances 12a, 12b, 12c. It therefore obeys nearly the same equations, the only difference being the presence of the capacitor voltage U ^dq in the stator voltage law (equation 1 a), and of its evolution equation (1 b). Therefore, the ACRIM model reads in the classical dq (stator synchronous) frame the following equations (1 a-1 g), where <a, b > = a ^T b is the scalar product.

[0053] The equations (1 a, 1 b, 1 c, 1d, 1 e, 1f and 1 g) are:

[0054] The variables U ^dq (capacitor voltage), (stator flux), ijj ^dq (rotor flux referred to stator), i ^dq (stator current), i ^dq (rotor current referred to stator) and u ^dq (stator voltage) are 2 x 1 vectors; aj _r (rotor velocity also called actual rotor speed), OJ _S (stator velocity or stator speed also called actual current stator speed), T _em (electromagnetic torque) and Ti (load torque) are scalars; C _s , R _s , R _r , L _s , L _r , L _m ,J, n are constant parameters (possibly slowly-varying for R _s ,

R _r ); finally, J-. = ; notice that < a, J * b > is then the 2D cross-product of a and b. It is to be noted that SP means setpoint in the indices.

[0055] For simpler expressions in the sequel, we also introduce the identity matrix and the slip velocity (jdg : — (jd _s — (jd _r . The physical control input is the impressed potential vf ^p , and is related to u ^dq by vf ^p = it^ = R{d _s )u ^dq (we assume the motor is star-connected); the frame angle 0 _S is defined the rotation matrix with angle 0 _S . Being chosen at will, the frame velocity can also be seen as a control input. On the other hand, the load torque Ti is an unknown disturbance. Equations (1 ) are the fundamental physical relations obtained from an energy approach or a microscopic approach. They are not in state form, which is not needed at this point. A suitable state-space representation can then be derived for the purpose at hand (control, state estimation ...), as will be done hereafter. At steady state, the twelve variables U ^dq , $ ^dq , is ^q , ~i ^dq , a) _r , T _em and the four inputs u ^dq , a) _s , T _t are linked by the twelve relations (bar notation means ‘at steady state’).

[0056] The equations (2a, 2b, 2c, 2d, 2e, 2f and 2g) are reproduced below. [0057] Four primary quantities must therefore be chosen, the remaining twelve quantities being then determined from the steady-state relations. A convenient way to do this is as follows. First, we obviously have T _em = T ₍ by (2f); second, multiplying both sides of (2c) by J and using J ² = -I yields third, using (2e) and (2g) yields

[0058] We then choose the operating point aj _r , Ti, and either the magnitude and direction of ijj ^dq (which is the usual practice), or OJ _S and the direction of ip ^dq (which will be more convenient in the ACRIM case); notice that as the equations are invariant by a rotation of the vector variables, the direction of ijj ^dq is immaterial and can be chosen to zero without loss of generality. Finally, the remaining variables are obtained by using successively (2c), (2e), (2d), (2b) and (2a).

[0059] B. The resonance equation

[0060] The problem with an air-cored motor is that it requires in general a much higher voltage than a conventional motor. Thanks to the stator capacitors, the problem can be alleviated by operating at a resonance point, where the current vector i ^dq is parallel to the voltage vector u ^dq , i.e. where the power factor is ±1 . Equivalently, this means the total impedance seen from the stator terminals is purely resistive. In the following, we address the case where only the stator is equipped with a capacitor (so-called tuning option (c)), but a similar strategy is possible for the other cases (capacitor at rotor only, and capacitor on both).

[0061 ] We now show that the resonance condition i ^dq parallel to corresponds in fact to a relation between OJ _S and a) _r . Eliminating U ^dq from (2a)-(2b), and using <a, b > gives the last line stemming from (2d). On the other hand, eliminating ip^ ^q from (2c) and (2e) yields

[0062] Multiplying both sides by (R _r * I - a) _g L _r * /), and using 2 = -I, yields hence

[0063] Finally, injecting this in (4) produces

[0064] Therefore, the resonance condition amounts to that is, where a the so-called leakage factor.

[0065] Table 1 shows the experimental data (rated values & parameters). Table I

IS, i-i KIV. I K i A i. .M; •:< >K EAT: I - X AI.VI S AM P IA E AS-.I.I I.H 'S.

[0066] C. Peculiarities of the ACRIM

[0067] We illustrate the peculiarities of the ACRIM with the example of an experimental motor, whose characteristics are listed in Table I. Because there is no ferromagnetic core, the inductance and rated flux values are much smaller than for a conventional IM with the same rated power; moreover, the leakage factor a is much larger (typically about 0.05 for a conventional IM). On the other hand, the rated frequencies are also much higher, so as to ensure a reasonable efficiency. The moment of inertia J _m of the motor alone is very small, because of the rotor lightweight construction; of course, as soon as the motor is coupled to a mechanical load, the total moment inertia J is usually much higher. The role of the stator capacitors is to decrease the rated voltage without changing the rated current: without them, this voltage would jump to 198 V, with a power factor of 0.45. Notice the capacitors do not change the efficiency; indeed by (2b), meaning the power in the capacitors is zero.

[0068] It should be emphasized that the ACRIM rated point strongly depends on the capacitance value C _s , and is the result of a delicate trade-off.

[0069] III. A control policy based on the resonance condition

[0070] We now study in detail the resonance condition (5). To keep things simple, we first rewrite it as

[0071] This equation can be seen as a relation giving (l _s in function of (17 as the roots of the fourth-order polynomial equation

[0073] As we want to establish a policy giving (1J in function of (17, we must understand how to select the “good” root among the four possibly existing real roots. To this end, we first rewrite the condition as

—2 „

_ 2 Q(lc — 1 (1 = - - -

1 — (7Cl(l _s

1 > 2 1

Which can be justified if and only if - < (l _s < — . Clearly,

[0074] Figure 2 represents the resonance in the f _s - f _r plane wherein

[0075] That is to say, figure 2 represents the locus of the solutions of the resonance constraint (eq. 5 below); it also illustrated with respect to the motor vs generator mode that we may have between 1 and 4 solutions,

[0076] Remembering that by (3) £l _g and T, must have the same sign, and that (17 * T, is positive in motor mode and negative in generator mode, this yields the graph displayed in Fig. 2. We thus see that, for a given (17, there is one resonant point in generator mode and one in motor mode when |(17 | is below a minimum value (in our example about 2n x 793 rad s’ ¹ ); two resonant points in generator mode and two in motor mode when |(17| is above this minimum value but below a maximum value (about 2n x 972 rad s’ ¹ ); and only two resonant points in generator mode when |(17| is above this maximum value.

[0077] That is, all these policies are related to the fact that OJ _S is calculated as a function of a) _r . For example for resonance, we use equation (5) which is a 4 ^th order polynomial equation in a) _s . The policies of the prior art revert to another calculation/choice of OJ _S but the rest of the algorithm, i.e. the modified indirect FOC (irFOC) remains valid.

[0078] In other words,

1 - the speed controller gives the T _emSP (as in irFOC),

2 - then from <w _s chosen, we find 6Og — 60s 60 r, from which we deduce 0 _sp

(as

3 - from which the (current setpoints) are classically derived (see eq (8) below).

[0079] That is, above mentioned 1 & 3 are classical (from irFOC), and the present invention is about the resolution of OJ _S for resonance, and step 2. [0080] Figure 3 represents a close-up view of the resonance in motor mode. Interestingly it shows that the resonance may occur close to the optimum efficiency line resulting practically in a favorable situation.

[0081 ] It is also interesting to note that the central symmetry with respect to the origin, hence we need to study only the case where nJ is positive. If we further concentrate on the motor mode, we get the close-up view in Fig. 3. In the intermediate zone where there are two resonant points for a given a) _r , the “good” point is the more efficient one (i.e. the closer to the optimal efficiency line); where there is only one point, this is the only possibility to operate at resonance, though the efficiency is not very good.

[0082] Figure 4 represents the resonance policy in motor mode. It particularly illustrates the possibility to have discontinuous solution (w _s ) when the rotor e.g. accelerates.

[0083] The resonance policy giving a) _s in function of a) _r is then obtained by taking the symmetric of the graph of Fig. 3 with respect to the diagonal (i.e. the f _s = fr line), and excluding the “useless” zone, which gives Fig. 4. Notice the discontinuity when switching between the two branches of the “useful motor zone” of Fig. 3.

[0084] IV. The control scheme

[0085] We introduce here the notion of a» _s -driven FOC, which is a modification of standard FOC suited for the ACRIM.

[0086] A. A state form adapted to FOC

[0087] Choosing \p^ ^q and besides <w _r and U^ ^q as state variables, eliminating ip^ ^q and i^ ^q , and using < if ^q Jif ^q >=0, the ACRIM model (1 ) reads in state form

[0088] This state form is well-adapted for designing a control law along the two-time-scale approach used in FOC. Indeed, assuming that a> _r and ip ^dq are “slow” variables, the term

[0089] [0090] in (6c) is seen as a “slow” (vector) disturbance d ^dq that can be rejected by a controller with integral effect. The “fast” variables i ^dq and U ^dq then evolve according to

[0091 ] The control problem is then split into two simpler subproblems: on the one hand controlling (7a) with a “fast” current loop; on the other hand controlling (6a)-(6b) with a “slow” velocity loop, as if i ^dq were the control input.

[0092] Figure 5 represents a locus of eigenvalues in motor mode and efficient useful zone, as some eigen values are in the positive real halfplane, the open loop control is unstable, and the close loop mandatory; in the negative half-plane, the other eigen values are standard to deal with. [0093] B. Dynamic peculiarities of the ACRIM

[0094] Besides the static peculiarities mentioned in section ll-C, the dynamic behavior of the ACRIM is also very different from the conventional IM. Indeed, it is strongly unstable in much of its useful operating region, and in particular around its rated point. This can be seen by studying the seven eigenvalues of the tangent linearization of (6).

[0095] To this end, Fig. 5 displays the locus of these eigenvalues, for a> _r stepping through the efficient useful zone of Fig. 3, and T, varying from 0 to twice the rated torque. Notice that the thickness of the eigenvalue traces is due to the variation of T, for each single a) _r .

[0096] Very roughly speaking, the four traces with large negative real parts correspond to the “fast” subsystem (6c)-(6d); the resonance phenomenon is the cause of the very large imaginary parts. The three traces with mostly positive real parts correspond to the “slow” subsystem (6a)-(6b); notice there is a small hardly visible real trace near the origin, also mainly in the right half-plane. In fact the two subsystems are quite coupled, and the resonant coupling is responsible for the unstable behavior; indeed, the aircored motor by itself, without the capacitors, is stable.

[0097] A practical consequence of the ACRIM dynamic instability, is that the motor cannot be operated at the resonance with e.g. a simple openloop V/f control law; closed-loop control is thus imperative.

[0098] C. Classical FOC and a> _s -driven FOC

[0099] We first review the classical (indirect) FOC method in a setting adapted to our needs. The goal is to run the motor at the rotor velocity setpoint a> _r sp and rotor flux set point with o) _r sp and </> _S p possibly time-varying, despite the (usually unknown) load torque Ti, </> _S p being the magnetic rotor flux setpoint. [0100] The method comprises three steps:

1 ) Velocity loop; from a) _rSP and the measurement of a) _r , the electromagnetic torque setpoint T _emSP is produced; this loop correctly controls <w _r provided the actual torque T _em is close to the desired T _emSP . 2) Flux orientation; from T _emSP the control OJ _S and the current setpoint are produced.

3) Current loop; from i ^qq _p and the measurement of the control u ^qq is produced.

[0101] The rationale is the following. Assume the current loop is fast enough so that i ^qq rapidly converges to i ^qq _p . If the current setpoint i ^qq _p is taken as

With A yet to determine, the flux error ■= $ ^qq - ijj ^qq _p satisfies from (6b)

[0102] If we choose a) _g = A, i.e. OJ _S = A + oj _r , the flux error obviously exponentially converges to zero with the time constant L _r I R _r . In other words, with this choice of the control input OJ _S , the flux orients itself to the desired ip ^qq _p , hence the name “Field-Oriented Control”. On the other hand,

R

[0103] Therefore, setting A := — ^- T _emSP yields T _em -+ T _emSP TL'lpsp

[0104] The mechanical equation (6a) then tends to

J da) _r

— T _emSP T ₍ n dt hence can be controlled by the velocity loop.

[0105] We emphasize the method relies on a two-time-scale assumption: the current loop must be much faster than the rotor flux time constant L _r I Rr and the velocity loop. For the conventional IM, this is usually easily doable, as the open-loop stator current dynamics is already rather fast. For the ACRIM, this is somewhat more delicate, as the natural time scales are not so-well separated (see previous section).

[0106] The principle of a» _s -driven FOC is simply to exchange the role of C/)SP and OJ _S : the goal is now to run the motor at a desired a> _r sp as before, but with a given OJ _S (implicitly aj _g ); and </> _S p is now determined in step 2) using the same but reversed relation = R _r T _emSP /na) _g . Steps 1 ) and 3) are otherwise unchanged. There are nevertheless two issues to address: on the one hand, an approximate derivative of </> _S p must be generated for use in (8).

[0107] On the other hand, the choice (0sp)2 = R _r Temsp I na) _g is of course possible only when the right hand-side is positive. In some cases, for instance during a large deceleration, Temsp may become negative, and the only way to stay at resonance is to switch from the ’’motor” branch (dashed line) in Fig. 3 (where > a» _r , i.e. a) _g > 0) to the ’’generator” branch (plain line) (where o) _g < 0); as this happens in rather exceptional conditions, we rule out this case to keep things simple.

[0108] D. The velocity loop

[0109] Exactly as in standard FOC, the velocity loop in a» _s -driven FOC is just a PI controller as in standard FOC

[01 10] For the test motor, the tuning k := 0.74 and ki := 8.22 gives a good result.

[01 1 1 ] Figure 6 represents a step response of a Pl-controlled fast subsystem. It is the classical output of the MATLAB function step, it illustrates the response of a MIMO system (4 inputs a.k.a. ‘from’, 4 outputs a.k.a. ‘to’) to a step function; one can subjectively judge the stability of each response.

[01 12] E. The fast current loop

[01 13] The current loop for the ACRIM is more complicated because of the capacitors. Indeed, the “fast” subsystem (7a) is truly Multiple- Input Multiple-Output, with dimension 4, and moreover depending on a» _s . In the standard case without capacitors, it simply consists of two uncoupled onedimensional subsystems (but for the harmless term oL _s (Jd _s di ^d< i _s , which can be compensated if desired), easily controlled by the simple PI controller

[01 14] Nevertheless, it turns out that this simple PI controller still does the job, at the expense of a specific tuning depending on &L _S and C _s . On the test motor, the tuning K := 3.8954 and Ki := 5784.2 ensures both a good dynamic behavior and a comfortable robustness: the 2 %-settling time in tracking and disturbance rejection is about 1 .9 ms, as illustrated in figure 6.

[01 15] Figure 7 represents disk-based stability margins of compensated transfer. It is the classical Bode plot; the black dot is the most critical point clarifying the awaited margins.

[01 16] The gain margin is 11.2 dB, the phase margin is 59° as illustrated in figure 7 (we use disk-based margins, which are true robustness indicators for MIMO systems, as explained in P. Seiler, A. Packard and P. Gahiner, “An introduction to disk margins [Lecture Notes], IEEE Control systems Magazine, vol. 40 no 5, pp 78-95, 2020). These figures are for a>s near its rated value, but remain satisfying in a large range of stator velocities; the tuning could moreover be scheduled with ais if deemed necessary.

[01 17] Figure 8 represents test scenario with velocities expressed in Hertz. The main information is on the last graph revealing that the power factor (PF) is indeed one, and that, all over the experiment; one can also observed among other things, how the input stator voltage is reduced and transferred to the capacitor (it _s vs. U _s ).

[01 18] V. Simulation results

[01 19] We illustrate the good behavior of the proposed control scheme in simulation on the following scenario, as illustrated in figure 8: at t = 0 s, the motor starts in the rated steady state; it must stay there until t = 2 s, and is then ramped down to 700 Hz in about 23 s, where it must stay until the end of the simulation. In addition, the load torque is suddenly changed to half the rated torque at t = 6 s, then back to the rated torque at t = 8 s.

[0120] The velocity and currents feeding the controller are corrupted by band-limited white noise, to assess its performance in the face of measurement noise.

[0121 ] The controller performs very well, the actual rotor velocity o) _r very closely following its setpoint a> _rS p, as shown in figure 8 (top view). [0122] The power factor PF is always equal to one (see figure 8 bottom view), hence the motor always operates as desired at the resonant stator frequency; as a consequence, the stator voltage is kept small, whereas the capacitor voltage is much larger.

[0123] Also notice that at about t = 15.5 s, the velocity crosses the discontinuity visible in figure 4, resulting as anticipated in a sudden drop of the efficiency jy.

[0124] That is, we have presented a strategy for the closed-loop control of an Air-Cored Resonant Induction Motor. It relies on a policy selecting the “good” stator velocity a) _s as a function of the rotor velocity a) _r , which is then fed to a modified version of (indirect) Field-Oriented Control. The specific policy used in the specific example, namely always being at resonance, it not the only conceivable one, but is just an effective means to limit the supply voltage. The control scheme would still work with a different policy.

[0125] Figure 9 represents a block diagram schematically illustrating the method of control.

[0126] That is, the method according to the present invention uses a standard irFOC algorithm modified to take as inputs at least the actual rotor speed and the setpoint of rotor speed. The actual rotor speed (<w _r ) may be retrieved from a sensor, and the rotor speed setpoint is the speed of use desired by the user or imposed in case of functioning as a generator. The further inputs may be the actual stator currents in dq frame (or adding classically as input an extra rotor speed yielding an arbitrary rotor position by integration, hence allowing the Park’s transformation). The modified irFOC algorithm may deliver as output the stator u _s voltage to be realized (either dq or per phase, usually feed to the PWM module i.e. Pulse width modulation module). Then, the irFOC algorithm internally classically transforms the rotor speed error into an electrical torque setpoint; and specifically, first estimates and imposes depending on the selected policy, a stator electrical speed co _s , from which a variable rotor flux setpoint is inferred (as (pjp = R _r T _emS p/na) _g ) [0127] Figure 9 shows the irFOC, the sensor for retrieving a> _r , the computation of OJ _S , the computation of V^sp, the integrator, the inverter, the PWM module, the torque command, the flux command, the motor and the voltages and currents.

[0128] The invention also relates to the following.

[0129] In the context of the control of an asynchronous induction motor (possibly air-cored and equipped with capacitors or not as proposed mentioned above), it is possible to use the modified irFOC (co _s FOC) algorithm for which we propose various policies. In particular, they may have advantages with respect to the above mentioned (full) resonance policy, which is perfect for providing a unity power factor (PF=1 ), but may imposes a poor or lower efficiency. The various polices may also cope with some new difficulties imposed by new designs of motor.

[0130] As in practice, efficiency is favored compared to power factor, and that as well efficiency might not yield enough torque, it is proposed here some trade-off policies.

[0131 ] It is understood that as efficiency, the best torque policy (also called maximum efficiency policy) is also made by a constant slip (aj _g = a) _s - a) _r ). Therefore, it is proposed to follow a constant slip arbitrarily defined upon an ‘aggressivity’ parameter “a” within [0,1 ]. This parameter can be changed anytime. This is possible in general and completely compatible with the modified IRFOR also called a) _s FOC.

[0132] However, it is also understood that due to the potential presence of a capacitor, the constant slip strategy is mostly impossible to sustain at low speed and some deviations must be proposed. At low speed the requested torque might be too high and violate the voltage constraint or limit, therefore something must be done to maintain the control.

[0133] There are two cases. First, if we want to keep on the induced line then we need to reduce the requested torque. This torque is simply given from a close-form. But it may get to 0, turning the acceleration/deceleration impossible or too slow. Second, if we have a minimum torque to respect, then we need to relax the a) _g constant constraint (or leave the line) and augment |<w ₅ |. The corresponding OJ _S is given from solving a polynomial equation of degree 6. The solution can be stored in a LUT (look-up table) if the minimum torque is fixed. If the minimum torque is changing, e.g. if we want a slower deceleration, then either we need a 2- dimensional LUT, either to compute the slip on the fly, or use an approximation with a margin experimentally set.

[0134] Overall, a list of policies is proposed, that should not be seen independent. Instead, they can be seen mostly as options and often be merged or combined, wherever necessary.

[0135] It is possible to use modified irFOC (a> _s F0C) adapted algorithm as detailed above to work at given aj _g (here often constant), and the algorithm remains the same.

[0136] 1 - Analysis

[0137] All the following analysis are made in the ‘motor’ half plane. The generator case is not yet considered but the reasoning might be equivalent. All the graphs are in dq.

[0138] We call <Jl) _g — <Jl) _s — (l) _r the slip; despite the slip is also largely referred <i> _s but we prefer calling the normalized slip this entity.

Classically, we also define respectively, the leakage factor, and the rotor time constant, j2

[0139] a = l — - ²² -

L _s L _r

[0140] z _r = f ^L

[0141 ] Eventually, we note the efficiency, q. The electromagnetic-torque is T _em . The setpoint is noted sp. [0142] The classical irFOC algorithm uses a constant 1/ ^s to drive the selected solution. Therefore, due to the classical equations,

[0144] It is immediate to see that aj _g may wander around as directly proportional to the requested torque. Consequently a) _g may not be always optimum with respect to other aspects (efficiency, voltage & current limits, energy ... ).

[0145] Figure 10 represents a speed scenario.

[0146] Figure 11 represents a selection of slip made by the irFOC algorithm for the speed scenario of figure 10.

[0147] In the contrary, the co _s FOC algorithm, does not see l^ ^p constant anymore and allows an explicit & wanted selection of a) _g ; it is possible to call this generic selection a policy.

[0148] From the a) _g selection, it is possible to derive the flux setpoint as well as the current ones; then the ‘field-oriented control’ (FOC) approach processes as usual. As stated above, first an estimation <|) of the expected flux is followed using a filter of order 1 ,

[0149] time constant.

[0150] Second the stator current setpoints become

[0152] 2 - Various policies

[0153] 2.1 - founder policies

[0154] There are at least two policies of interest, the best efficiency q & best torque (according to the current limit). It is well known that the best efficiency can be written in terms of a constant a) _g . It turns out, and maybe this is less known, that the best maximum torque also (‘best’ is with respect to the produce current). Explicitly,

[0158] In the a» _s (o» _r ) graph, those appear as parallel lines, accordingly, the best torque line is always above & different. From top to bottom in the Figure 1 (right picture), the best torque line, the best efficiency line, the main diagonal (<w ₅ =0), then its symmetric in the generator mode halfplane.

[0159] In terms of application these 2 policies are quite useful to prevent efficiency loss, hence energy, or to get the maximum torque at steady state. It is also simple to make a linear weighted combination of both. By using a parameter ‘a’ within [0,1 ]; for aggressivity; semantically ‘a=0’ favors best efficiency, while ‘a=1 ’ favors best torque, the slip becomes:

[0161 ] A choice of ‘a’ selects a <o ₅ =cst line between the 2 optimum lines. It is also equivalent to choosing an iso torque line and a maximum torque T _e “ _m value; so, it sets dynamically the maximum torque. Note that the parameter ‘a’ can be changed anytime according to the user request.

[0162] 2.2 - Need for other policies and summary

[0163] However it also may happen, that during a particular scenario, difficulties are encountered, such as the voltage or the current limits. The current limit is not that rigid. Indeed, it is possible to see its risk as the probability to melt the wires (which depend also of the temperature and the time spend in high current demand); but as a simplification, it is possible to write it in terms of a limit of current not to overpass. For the voltage, it is easier as we are indeed strictly limited by the power supply (the v _DC ).

[0164]

[0165] Figure 12 represents voltage limits in the a> _s graph without capacitor.

[0166] Figure 13 represents voltage limits in the a> _s graph with capacitor.

[0167] Voltage limit in the m _s (m _r ) graph, without & with capacitor. Here for delivering a nominal torque of 5N.m (at steady state) with V _dc = 400 V (1 OkW air-core machine). The iso-lines represent the scalar field of the voltage norm (in dq). The pulsation a) _s must stay within the hatched lines. See also how the hatched lines crosses our best slip policies. Therefore, we cannot always strictly stick on those preferred lines

[0168] 2.3 - Avoid too low a> _s

[0169] This is important when having a capacitor (see Figure 12). This is further studied in § 2.1 1 below.

[0170] Figure 14 represents <w _s (<w _r ) graph for the voltage barrier policy.

[0171 ] 2.4 - Voltage barrier policy

[0172] The voltage is naturally limited by the used power supply V _DC . The voltage barrier is the location of this voltage limit in the <w _s (<w _r ) graph, as introduced in Figures 12-13. It comes from different terms. According to the voltage expression,

[0174] at low speed, a) _s 0 and the term — C _s <l> _s gets too high; at high speed a) _s -> oo, and the term L _s a> _s gets too high.

[0175] Both lead to too high voltage, and turn the control impossible (voltage saturation) in concerned areas.

[0176] To overcome this issue, we must avoid those areas. There are at least two possible strategies to respect the voltage constraint, for example on a deceleration (with a capacitor - see Figure 13):

[0177] 1 : Reduce the requested torque (T _em ), and keep on the dashed line.

[0178] 2: Once the minimum acceptable torque is reached, modify the slip (or <w _s ) by quitting the chosen slip line and following the hatched line (with a margin) from above until reaching the best slip line back (as illustrated in Figure 14 showing the strategy (policy) summary to pass the voltage barrier).

[0179] If we want to follow properly the voltage constraint, we must solve in a) _s the following polynomial equation = ⁰ > issued from eq. (eq.41), with:

(eq. A2)

[0181] To select the solution among the possible 6, select a real one, either negative or positive depending on the sign of ?/£, take the one that minimizes |o» _s |.

[0182] For the current constraint, it is simpler, a) _s must stay within some boundaries to produce an arbitrary given torque (obviously bounded). These boundaries appear to be straight lines parallel to the diagonal. These lines gather or shrink onto the maximum torque line (which happen to be also the minimal current line) - see Figure 15.

[0183] Figure 15 represents <w _s (<w _r ) graph for the voltage barrier policy with intensity constraints (hatched lines).

[0184] Figure 15 shows intensity constraints (hatched lines). They depend on the maximum tolerated current, here |/ _af , _c | = 644, and the requested torque, here T _em = 1.2 Nm. The minimum current for a given torque is always achieved at the same a) _g , here on the dotted line. [0185] It may happen that the intensity & the voltage constraints are incompatible (no solution). In this case, it is mandatory to reduce the requested torque.

[0186] 2.5 - Torque adjustments

[0187] To satisfy the constraints at low speed, OJ _S can stay on the induced iso torque line while reducing the maximum torque to satisfy the V- constraint. Indeed, at constant slip, the intensity constraint, once satisfied, remains so, only the voltage constraint matters.

[0188] Explicitly it is from eq. A1 :

[0190] Therefore with a predefined OJ _S , our torque setpoint must satisfy |T/£| < T™ ^Xv . If not, it may be clipped easily.

[0191 ] For the current constraint, the maximum torque to respect is:

[0193] Then the maximum torque to avoid overpass as used for the speed controller, is simply the minimum of these later (during acceleration), or the maximum when all are negative (for deceleration), and of course being limited by its own initial value. This can be summarized as,

[0195] 2.6 - Favor efficiency, favor heat

[0196] The best efficiency line has its related maximum torque, therefore depending on application and motor characteristic, we can favor <w°, whenever the requested torque allows it, i.e. is less than its associated maximum torque. Note that there is also another possible treatment when is low - see below § 2.10. [0197] Meanwhile it will induce more current. Therefore, if e.g. your machine tends to heat, it might be more interesting to keep on the maximum torque production for fewer needed current. Overall it depends on applications and machines, but several policies are explicitly possible.

[0198] 2.7 - Speed controller anti-windup adjustment

[0199] 2.7.1 - Adjusting the maximum torque

[0200] Manipulating the torque setpoint as proposed in §2.5 imposes some adaptation of the speed controller anti-windup. Indeed, initially the maximum torque was set to a constant value. With this strategy, the torque setpoint is now limited (at low speed). As it is usually lowered (cropped), the former classical anti-windup (with higher constant maximum torque) cannot be triggered anymore.

[0201 ] It is thereby necessary to change the maximum (and minimum torque) according to eq. A3.

[0202] 2.7.2 - Adjusting the integral itself

[0203] Similarly the anti-windup behavior must be adjusted to avoid being unnecessarily triggered. The countermeasure depends on the antiwindup model, possibly on the controller model (e.g. PI vs. IP), it suffices to modify the integral value itself, and forcing a non-triggered status.

[0204] 2.8 - Current controllers and anti-windup adjustment

[0205] When entering the voltage barrier, the current controllers are naturally under anti-windup (due to voltage saturation). Therefore, especially if one is using IP controllers, they may not adapt anymore to a change of the current setpoint. Consequently, we must do it explicitly as we did for the speed controller.

[0206] 2.9 - the resonant policy

[0207] With a motor equipped with a capacitor per phase, as proposed above in a generalization of the resonance condition. It is possible to resonate on a large range of speed (<w _s ). Meanwhile, it is not always interesting, because sometime the efficiency may become low; sometimes it may however help reducing the voltage. This could be interesting depending on applications.

[0208] 2.10 - the blind jumping policy

[0209] To jump the voltage barrier without solving a polynomial of degree 6, it is possible to propose similarly a trial an error approach based on some simple criterions. Here we considered the saturation imposed by a) _s 0, not the saturation imposed by too large |<w _s |, i.e. the one we can jump.

[0210] It is possible to remark that the voltage barrier is almost vertically symmetrical. Therefore when encountering it, i.e. when the voltage almost saturates, just symmetrize the current OJ _S (reversing its sign - if going toward 0), and continue augmenting |<w _s | as long as the saturation persists. Interestingly one can keep on the maximum torque line (as above) by reducing |<w _s | without saturation, or augmenting it with saturation, and so until the requested I _d = I _q is reached.

[021 1 ] As a better approximation, one can also solve the polynomial and make a table for the minimum admissible torque. Jump the barrier using this table and adjust OJ _S afterward as explained here. Of course an option can also be to learn this table online using this approach, i.e. for each a> _r , learn the a) _g making the voltage saturation.

[0212] 2.1 1 the a> _s adjustment control policy

[0213] Figure 16 represents a summary of adjustment.

[0214] This strategy primary role is to avoid the voltage saturation without the knowledge of the motor parameter nor computing solutions of large polynomials. The idea follows the preceding ones, we simply decide to augment or reduce OJ _S (hence aj _g ) on simple qualitative criterions. We will explain it for the positive case (it is easy to generalize to the negative one). Simply speaking we want to reduce OJ _S when approaching the upper voltage barrier; and jump the barrier (or reduce the torque) when decelerating (see Figure 14). Inspiring from the idea of the PID controller we can propose an adjustment as:

[0215] a) _s « — a) _s + Aa> _s

With &a) _s = f 6a) _s dt the adjustment. The idea is the following, at time t, we look at the current value of the voltage. If we are far from the voltage saturation, we leak Aa> _s toward 0. If we are almost at saturation, we decrease Aa> _s gently. At least if we are close to saturation, we decrease Aa> _s stronger. Therefore we need 2 thresholds for defining ‘almost’ and ‘close’, namely V^ _ost and V^ _se with and 3 gains (K _leak , K _almost , K _close ) with ^almost < Ketose - Possible values are for instance for V _dc = 400 and a control at 20kHz, V°% _ost = 10 V, V^ _se = 5 V, K _leak = 0.02, K _almost = 0.01, K _close = 0.05.

Now, we have 3 main cases: In these equations smooth is a function turning the adjustment differentiable at V = V^ _ost (otherwise it acts like the absQ function). It could be the x ■-> x ^a with a > 1 (if a = 1 there is no smoothing and it remains equivalent to using the abs() function - see Figure 16), or any of the classical differentiable approximation of the absolute function x »-> |x| . The Figure 16 summarizes the situation (for the positive cases).

[0216] 2.12 - the torque setpoint adjustment control policy

This strategy is like the a> _s -control adjustment but does not impact OJ _S . Its primary role is as well to avoid the voltage saturation. The idea follows the preceding ones, we simply decide to augment or reduce the torque setpoint T/£ on simple criterions. We will explain it for the positive case (it is easy to generalize to the negative one). Simply speaking we want to reduce |T/£| when approaching the voltage barrier. As previously:

With AT _e ^s £ = f ST^dt the adjustment. The idea is the following, at time t, we look at the current value of the voltage. If we are far from the voltage saturation, we leak AT _e ^s £ toward 0. If we are close to saturation, we decrease AT _e ^s £. Therefore we need 1 threshold for defining ‘close’, we can reused V^ _se previously introduced - see § 2.1 1 ; and 2 gains (K^ _k ,K^ _e ). Possible values are for instance for V _dc = 400 and a control at 20kHz, V^ _se = =

In these equations smooth is a function turning the adjustment differentiable at V = V^ _se (otherwise it is like the absQ function). It could be the x ■-> x ^a with a > 1, or any of the classical differentiable approximation of the absolute function x ■-> |x|.

Eventually we do not want that this strategy reduces the torque too much. If it is so, we prefer jumping (if possible). With T™ ⁿ this minimum torque value, we can write it as:

Note that this strategy is complementary to the OJ _S control one, but the balance between the two, depends on application. Its role is to avoid jumping the voltage barrier whenever possible; because this jump will impose the torque to go to 0 during the jump while creating some instabilities.

[0217] 2.13 - maximization of |T^ ⁿ |

[0218] Figure 17 represents torques.

[0219] It is interesting to maximize |T _e ^ ⁿ | to be able to follow as many scenarios (speed setpoints) as possible. However it is completely dependent on applications. The above §2.5 exposes the starting torque using ay _r = 0 when one wants to keep strictly on the policy. If this torque is enough, it is fine, otherwise the torque must be adjusted.

[0220] If this reduced torque is too low, as said, either one has a minimum torque, and using it imposes aj _g (if possible) to start on the voltage barrier. If one does not have this torque, a) _g must be augmented until leaving the voltage barrier. The difficulty is that the voltage barrier is also defined from the requested torque. Therefore, it is a space of 2-dimension in (T _em , o> _s ), a trade-off needs to be defined (larger OJ _S are safer, but less efficient). Unfortunately at this stage, we cannot resolve this trade-off as application dependent. Figure 17 shows the possible (T _em , a> _s ) when starting. Note that it will change as soon as the motor will start (a) _r 0). Therefore a margin is better to use. This graph is obtained using a> _r = 0 in the polynomial we have - eq. A2.

[0221 ] That is, figure 17 shows the possible (T _em , <w _s ) when starting, one needs to be in the higher subspace (blank space).

[0222] 3. Conclusion

[0223] It is possible to define a meta policy as a parameter ‘a’ (somehow for aggressivity) within [0,1 ]. This parameter can be changed anytime.

[0224] This parameter induces in the graph <w _s (<w _r ), a line parallel and inside the two optimum lines respectively for optimum efficiency & optimum torque. This policy simply consists in following this induced line, i.e. to work at o) _g = cst(a).

[0225] At low speed the requested torque might be too high and violate the voltage constraint, therefore something must be done.

[0226] We have 2 cases:

[0227] First, if we want to keep a) _g constant (or on the induced line) then we need to reduce the requested torque. This torque is simply given from a close-form. But it may get to 0, turning the acceleration/deceleration impossible or too slow.

[0228] Second, if we have a minimum torque to respect, then we need to relax the aj _g constant constraint (or leave the line) and augment |<w ₅ |. The corresponding OJ _S is given from solving a polynomial equation of degree 6. The solution can be stored in a LUT if the minimum torque is fixed. If the minimum torque is changing, e.g. if we want a slower deceleration, then either we need a 2-dimensional LUT, either to compute a) _s on the fly, or use an approximation with a margin experimentally set. If one wants to avoid computing the solution of this polynomial equations, one can follow a OJ _S and|or torque adjustment control approaches, consisting in a close loop manner to adjust them slightly at each loop, based on simple criterion (proximity to voltage saturation). [0229] It is possible to use the modified irFOC (to _s FOC) adapted algorithm as mentioned above to work at given ro ₅ (here constant).

[0230] It is of course understood that obvious improvements and/or modifications for one skilled in the art may be implemented, still being under the scope of the invention as it is defined by the appended claims.