Brushless motors are rated by attaching thermocouples to the stator windings and measuring maximum temperature while under load. The motor is attached to a dynamometer and the torque is increased until the steady-state temperature rise is equal to the maximum allowable temperature rise of the insulation system that is used in the motor. This is generally performed at rated voltage and rated speed. The intermittent torque zone just means that you can't run the motor in that area continuously without the motor getting too hot for the insulation system.
There are a number of reasons that the speed-torque curves in Figs. 1 and 2 look different. Figure 1 probably tested the motor at rated voltage and full speed and then let the torque pull the speed down. The intersection between rated speed and rated torque is the point where their motor's steady state temperature increase equaled the rated temperature rise of the insulation system. Then they probably just assumed that the rated torque holds for all speeds. This is a conservative assumption because in reality your rated torque should slightly increase as your speed decreases.
Figure 2 probably did maximum temperature tests at different speeds and that is why the line separating the continuous and intermittent torque zones isn't horizontal. The reason Fig. 2 has a rectangle shape is because the are limiting the torque in some way. In general if you continue to increase the torque on a brushless motor it will decrease the speed like in Fig. 1, forming a triangle shape. However, if you are current limited (say you have a current limit on your control) then that will prevent you from getting a speed-torque curve that takes the triangle shape and you will get the rectangle shape as in Fig. 2.
So my best guess is that Fig. 1 is showing you the speed-torque curve of a brushless motor with no current limit and Fig. 2 is showing you the speed-torque curve of a brushless motor with a current limit.
You said yourself that depending on what part of the charging process you are in, you keep the current constant or try to maintain that voltage. That's going to require some kind of controller, though not necessarily PID or some subset thereof.
The characteristics of a charging battery change very slowly relative to what even a slow microcontroller can measure and react to. Batteries also don't exhibit second order effects like inertia, like motor speed as a function of current does. Both these together allow very simple control schemes to work well.
Probably about the simplest control scheme for a switching power supply is pulse on demand. It is always stable and robust, although results in more ripple than a more finely tuned control scheme can accomplish.
When the output is below the regulation threshold, you do a pulse, else you don't. To avoid inductor saturation, you may always not do a pulse at the next slot immediately after a previos one, but that's a detail.
I've done pulse on demand switching power supplies with the PIC 10F204 a bunch of times. The code spins in a loop checking the comparator output as long as it is indicating the output is above the regulation threshold. When the output falls below the threshold, the code following the loop is executed, which produces a pulse. The instruction cycles to jump back to the top of the loop and do the next comparator check usually take enough time so that it's OK to do the next pulse righ away if the comparator indicates the output is still below the threshold.
Sometimes this can go meta-stable by producing two pulses in a row before the feedback catches up to the output having gone higher, but in all cases it remains stable as long as the maximum load isn't exceeded.
This sort of system is fine for battery charging, except that you have two thresholds, one for voltage and one for current. You only do a pulse if the output is below both. The higher level logic can adjust the limits as the battery progresses thru the charging procedure.
Best Answer
Step back and think what Id does for you and equally the relationship between Vd and Vq
Injection into the quadrature axis will facilitate torque production while injection into the direct access with facilitate field weakening.
To maximise the torque production you ideally want zero Id as a direct current component will result in a reduction in torque production. How? via a vector rotation. Id and Iq are always in quadrature and thus if there is some Id for a given Iq, the overall frame of reference must rotate.
The two PI(D) controllers within the FOC current loop take in a current error and produce a voltage demand. This Vd and Vq, via an inverse Clarke & Park produce the three phase voltage demands that shall be applied to the stator to facilitate current flow. As you increase in rotor velocity the displacement power factor increases as the \$j\omega_e L \$ vector length increase with regards to the \$V_{bemf} \$ and the IR vector. This natural rotation as to where the current shall be injected occurs as the Id component is kept as close to zero as possible (resulting in an increasing Vd term).
Lets look at this rotation.
The inverse Clarke & Park equations are such.
\$ V_a = V_d \cdot Cos\Theta - V_q\cdot Sin\Theta \$
\$ V_b = V_d \cdot Cos(\Theta - \frac{2\pi}{3}) - V_q\cdot Sin(\Theta - \frac{2\pi}{3}) \$
\$ V_c = V_d \cdot Cos(\Theta + \frac{2\pi}{3}) - V_q\cdot Sin(\Theta + \frac{2\pi}{3}) \$
for a "perfectly aligned" rotor and zero Vd component, the anticipated waveforms can be seen below
With a 20% Vd component added the waveforms can be seen below:
there is a clear phase-shift. If I overlay the two sets of Vabc it becomes very clear.
The waveforms now lead where they should be due to how a direct component rotates the vector. There is also a slight magnitude increase since the envelope of the 3phase sinewave follows \$\sqrt{V_d^2 + V_q^2}\$ and under current control the Vq component would have been reduced.
So it has been established that:
What would happen if aspects of Iq was used as the Id demand. For starters it would need to be aspects of Vq to influence Vd since the output of the PI(D) is voltage not current. What would happen if we had one PI controller that takes in an Iq demand and generates a Vq based upon the present Iq. From this produce Vd based upon say ... \$Vd = \frac{Vq}{Vd+Vq}\$ akin to the original query.
We would have a response which varies the phase relationship of the applied voltage based upon the quadrature current demand. Whenever there was a demand change or a load change the controllers response would result in a phase shift from the desired angle and thus compromise the ability of the system to generate torque effectively.
By implementing independent controllers for Id and Iq, their specific contributors to the overall system response can be controlled. Iq is free to change based upon a demand from an outer velocity loop while Id is controlled to rotate the vector due to any possible rotation due to acceleration or velocity.
Since Id (and thus Vd) has an impact of the torque angle, an Id demand can be used to produce field weakening to enable the system to reach higher velocities at the expense of torque production.