I-f Control

The first type of vector control addressed is the I-f control, where a reference current is imposed in the (d,q) reference frame with a constant amplitude value, rotating at a reference speed ω^*.

Figure 2.37: Current vector rotating at speed ω^* (Source: [22]).

The amplitude is guaranteed by the current closed-loop control, realised by

means of PI regulators (one for the current component along the d-axis and one for the current component along q-axis). The speed is open-loop imposed (as for V/f control).

The reference values are used to impose a rotating vector with amplitude Is [A]

rms:

⎧

⎪⎨

⎪⎩

i^∗_d=√ 2 · I_s i^∗_q = 0

(2.144)

The control scheme is shown below:

Figure 2.38: I-f control scheme

The performances of the I-f vector control are good if working in no-load con-ditions but it is subjected to high instability when under load. For this reason this type of control strategy is not used in practice but it is perfect to test (always in no-load conditions) the flux estimators and observers and in particular for tuning the PI current regulators (in (d,q) reference frame).

When working in no-load conditions, the current vector is in phase with the stator and rotor flux vectors (considering an IM), this allows to verify that the observed flux phase is correct. While, if considering SPM, the current is aligned

to the magnet direction or to the maximum inductance direction if considering SynchRel machines.

An analogy between the I-f control and the Field-Oriented Control can be high-lighted: in no-load conditions, the reference (d,q) directions are the same. The situation changes when under load.

In fact, under load, a phase shift is present between current and rotor flux, for IM. This lets the machine suddenly reach the limit torque; if the limit is overcome, the control is lost.

Without loss of generality, for synchronous machines, in an IPM machine, the presence of load torque causes a phase shift between current and rotor d-axis. Even in this case, once the limiting condition is reached (corresponding to the MTPA angle) and overcome, the control is lost.

Losing the control means that the imposed reference current is no more followed by the system.

PI Current Controller Tuning

In section 2.6.1 PID controllers have been introduced, here a real application will be presented.

As mentioned, the I-f control is also used to tune PI current regulators; the procedure will be described in the following, without loss of generality, considering an IPM machine (since it is the selected machine for this thesis work). It can be easily applied to all the other AC drives described in section 2.5, by simply changing the considered motor electrical and magnetic model.

From voltage equation (2.77) and the simplified IPM magnetic model from equa-tion (2.108) (considering the SPMSM convenequa-tion), it follows:

⎧

⎪⎨

⎪⎩

v_d= R_si_d+ L_d^di_dt^d − ωL_qi_q vq= Rsiq+ Lqdiq

dt + ωλm+ ωLdid

(2.145)

applying Laplace transform and collecting the current terms (the s-dependence has been neglected for simplicity)

⎧

⎪⎨

⎪⎩

V_d= I_d(R_s+ sL_d) − ωL_qI_q

Vq = Iq(Rs+ sLd) + ωλm+ ωLdId

(2.146)

From the flux-based model described in section 2.5.3 and neglecting the satura-tion and cross-saturasatura-tion effects, the IPM model can be represented as:

Figure 2.39: IPMSM model with SPMSM convention

where the other equations terms as to be intended as additive disturbances that shall be feedforward compensated in the PI current regulator.

For the PI tuning, the complete current loops shall be considered, including the PI transfer function and the converter transfer function. For the latter, assuming that the sampling strategy (that will be explained in chapter 5) 1 Sample, 1 Refresh or simply 1S1R will be used, it can be demonstrated ([23]) that the inverter can be

modelled as follows:

V (s)

V^∗(s) = 1

1 + sτinverter,delay

(2.147) where the delay includes transport and sampling delays, and for the 1S1R strat-egy it is computed as follows:

τinverter,delay = T_sw+ T_sw

2 = 1.5 · T_sw = 1.5

f_sw (2.148)

This can be justified knowing that the inverter output voltages are a pulse train with different amplitudes, but only after half-period the average value is the desired one.

Therefore, from the PI transfer function of equation (2.140), the current loops are:

Figure 2.40: Current loops with feedforward

For sake of simplicity, but without loss of generality, the procedure is described for both current loops, since they have same open-loop transfer function, considering

a generic inductance average value L and neglecting feedforwards:

Gcurrent,OL(s) = GP I(s) · GIN V(s) · GIP M(s) =

= K_p s ·

(︃

s + K_i Kp

)︃

· 1

1 + sτinverter,delay

· 1 L · 1

s + ^R_L^s

(2.149)

The open-loop transfer function presents:

• A zero from the PI current regulator in: zPI = -_K^K_pⁱ

• A pole from the PI current regulator integral action in the origin

• An electrical pole from the IPM dynamic model in pIPM = -^R_L^s

• A pole from the inverter dynamic model in pINV = -_τ ¹

inverter,delay

It is known from the control system theory, that the system frequency response depends on the pole with the smallest module, the dominant pole, limiting the system bandwidth. To increase it the PI current regulator parameters shall be tuned to compensate the dominant pole, choosing the PI zero as follows:

K_i K_p = R_s

L (2.150)

When evaluating the frequency response of a transfer function, it shall be con-sidered the cut-off frequency, corresponding to the frequency where the transfer function has unitary module:

|G_current,OL(jω)|

⃓

⃓

⃓

⃓ω=ωc

= 1 (2.151)

The cut-off frequency can be approximated to the closed-loop bandwidth, thus the objective becomes to maximize the system bandwidth to obtain a faster re-sponse. In any case the phase margin, guaranteeing the system stability, shall be respected: this is evaluated from the following formula [18]:

γ= 180^◦+^̸ G_current,OL(jω_c) (2.152)

The typical value of the phase margin that allows a stable and not oscillating system response is γ ≥ 60^◦.

Since the only remaining pole, apart from the one in the origin, is the inverter pole, the PI regulator shall be tuned to respect the previous condition, thus ωb,current

≤ |pinverter|.

Indeed, typically ωb is chosen equal to 2π· ^f₂₀^sw and eventually increased basing on experimental results.

From the open-loop transfer function module:

|G_current,OL(jω)|

⃓

⃓

⃓

⃓ω=ωc

= K_p

|jω_c|L · 1

|1 + jω_c· τinverter,delay| = 1 (2.153) and since:

• |jωc| = ωc

• |1+jωc· τinverter,delay| = |1+_pinverter^jωc | ≃ 1 because ωc ≤ |pinverter| Therefore, the open-loop transfer function module becomes:

G_current,OL(jω)

⃓

⃓

⃓

⃓ω=ωc

= K_p

ωc· L = 1 (2.154)

From the previous assumption, for which ωc ≃ ω_b,current, the proportional con-stant can be determined as:

K_p = ω_b,current· L (2.155)

and from condition in equation 2.150, the integral constant is determined as:

Ki = R_s

L · Kp (2.156)

As stated before, this procedure is used for the PI current regulator of both id and iq current loops. It is worth mentioning that the inductances values used

for tuning the controllers are the average constant values of the d-axis and q-axis inductances.